11institutetext: 1 Department Of Mathematics, Penn State, State College, PA
16802, USA 22institutetext: 2 Department Of Mathematics, Texas State, San
Marcos, TX 78666, USA

# Companion-Based Multi-Level Finite Element Method for Computing Multiple
Solutions of Nonlinear Differential Equations

Wenrui Hao1 Sun Lee1 Young Ju Lee2

###### Abstract

The use of nonlinear PDEs has led to significant advancements in various
fields, such as physics, biology, ecology, and quantum mechanics. However,
finding multiple solutions for nonlinear PDEs can be a challenging task,
especially when suitable initial guesses are difficult to obtain. In this
paper, we introduce a novel approach called the Companion-Based Multilevel
finite element method (CBMFEM), which can efficiently and accurately generate
multiple initial guesses for solving nonlinear elliptic semi-linear equations
with polynomial nonlinear terms using finite element methods with conforming
elements. We provide a theoretical analysis of the error estimate of finite
element methods using an appropriate notion of isolated solutions, for the
nonlinear elliptic equation with multiple solutions and present numerical
results obtained using CBMFEM which are consistent with the theoretical
analysis.

###### Keywords:

Elliptic Semilinear PDEs Finite Element Method Multiple Solutions

###### MSC:

49M37 65N30 90C99

##  1 Introduction

Nonlinear partial differential equations (PDEs) are widely used in various
fields, and there are many versions of PDEs available. One such example is
Reaction-Diffusion Equations, which find applications in physics, population
dynamics, ecology, and biology. In physics, Simple kinetics,
Belousov–Zhabotinskii reactions, and Low-temperature wave models are examples
of applications. In population dynamics and ecology, the Prey-predator model
and Pollution of the environment are relevant. In biology, Reaction-diffusion
equations are used to study Cell dynamics and Tumor growth volpert2014elliptic
. Schrodinger equations kevrekidis2015defocusing ; wang2015new  and
Hamiltonian systems kapitula2004counting ; simon1995concentration  are also
important topics in the field of Quantum mechanics. Another important area in
the realm of nonlinear PDEs is pattern formation, which has numerous
applications, such as the Schnakenberg model deutsch2005mathematical , the
Swift-Hohenberg equation lega1994swift , the Gray-Scott model
wei2013mathematical , the FitzHugh-Nagumo equation jones2009differential , and
the Monge–Ampère figalli2017monge ; gutierrez2001monge  equation, which finds
various applications.

In this paper, we focus on computing multiple solutions of elliptic semi-
linear equations with nonlinear terms expressed as polynomials. Although we
limit our nonlinear term to a polynomial, it is still an interesting case of
the above applications that has yet to be fully explored.

To solve these nonlinear PDEs, various numerical methods have been developed,
such as Newton’s method and its variants, Min Max method, bifurcation methods
zhao2022bifurcation , multi-grid method henson2003multigrid ; xu1996two  or
subspace correction method chen2020convergence  or a class of special two-grid
methods cai2009numerical ; huang2016newton ; xu1994novel ; xu1996two ;
xu2022new , deflation method farrell2015deflation , mountain pass method
breuer2003multiple ; choi1993mountain , homotopy methods chen2008homotopy ;
hao2014bootstrapping ; hao2020homotopy ; hao2020spatial ; wang2018two , and
Spectral methods grandclement2009spectral . However, finding multiple
solutions can be a challenging task, primarily due to the difficulty of
obtaining suitable initial guesses for multiple solutions. It is often
uncertain whether good numerical initial guesses for each solution exist that
can converge to the solutions. Even if such initial guesses exist, finding
them can be a challenging task.

To address this challenge, we introduce a novel approach called the Companion-
Based Multilevel finite element method (CBMFEM) for solving nonlinear PDEs
using finite element methods with conforming elements. Our method is based on
the structure of the full multigrid scheme brandt2011multigrid  designed for
the general nonlinear elliptic system. Given a coarse level, we compute a
solution using a structured companion matrix, which is then transferred to the
fine level to serve as an initial condition for the fine level. We use the
Newton method to obtain the fine-level solution for each of these initial
guesses and repeat this process until we obtain a set of solutions that
converge to the stationary solutions of the PDE. Our approach is different in
literature, such as those presented in breuer2003multiple  or li2017new ,
which attempt to find additional solutions based on the previously found
solutions.

The main advantage of our method is that it can generate multiple initial
guesses efficiently and accurately, which is crucial for finding multiple
solutions for nonlinear PDEs. Furthermore, our method is robust and can be
easily applied to a wide range of elliptic semi-linear equations with
polynomial nonlinear terms.

In this paper, we also present a mathematical definition of isolated solutions
for elliptic semilinear PDEs with multiple solutions, which leads to well-
posedness of the discrete solution and provides a priori error estimates of
the finite element solution using the framework introduced in xu1994novel  and
xu1996two .

We organize the paper as follows: In §2, we introduce the governing equations
and basic assumptions. In §3, we present the error estimate of the nonlinear
elliptic equation using the FEM method. In §4, we introduce the CBMFEM,
including the construction of the companion matrix and filtering conditions.
Finally, in §5, we present numerical results obtained using CBMFEM, which are
consistent with the theoretical analysis. Throughout the paper, we use
standard notation for Sobolev spaces $W^{k,p}(\Omega)$ and the norm
$\|\cdot\|_{k,p}$. If $k=0$, then $\|\cdot\|_{0,p}$ denotes the $L^{p}$ norm.
The symbol $C^{k}(\Omega)$ denotes the space of functions, whose first $k\geq
0$ derivatives are continuous on $\Omega$. Additionally, we denote
$\stackunder[1pt]{v}{\scriptscriptstyle\sim}$ as the vector while
$\stackunder[0pt]{\stackunder[1pt]{v}{\scriptscriptstyle\sim}}{\scriptscriptstyle\sim}$
is the matrix.

##  2 Governing Equations

In this section, we introduce the governing equations that we will be solving.
Specifically, we are interested in solving the quasi-linear equations, where
we assume that polygonal (polyhedral) domain $\Omega$ is a bounded domain in
${\rm I\\!R}^{d}$ with $d=1,2,$ or $3$.

$-\Delta u+f(x,u)=0,\quad\mbox{ in }\Omega,$ (1)

subject to the following general mixed boundary condition:

$\alpha u-\beta\nabla u\cdot\textbf{n}=\gamma g,\quad\mbox{ on
}\partial\Omega,$ (2)

where n is the unit outward normal vector to $\partial\Omega$, $\alpha$,
$\beta$ and $\gamma$ are functions that can impose condition of $u$, on
$\partial\Omega$, such as the Dirichlet boundary $\Gamma_{D}$, pure Neumann
boundary $\Gamma_{N}$, mixed Dirichlet and Neumann boundary or the Robin
boundary $\Gamma_{R}$, i.e.,

$\partial\Omega=\overline{\Gamma}_{D}\cup\overline{\Gamma}_{N}\quad\mbox{ or
}\quad\partial\Omega=\overline{\Gamma}_{R},$ (3)

with $\overline{\Gamma}_{D}$, $\overline{\Gamma}_{N}$, and
$\overline{\Gamma}_{R}$ being the closure of $\Gamma_{D}$, $\Gamma_{N}$, and
$\Gamma_{R}$, respectively. Specifically,

$\beta|_{\Gamma_{D}}=0,\quad\alpha|_{\Gamma_{N}}=0,\quad\mbox{ and
}\quad(\alpha\beta)|_{\Gamma_{R}}>0.$ (4)

We shall assume that $g$ is smooth, and in particular when
$\overline{\Gamma}_{N}=\partial\Omega$, compatibility conditions will be
assumed for the functions $f$ and $g$ if necessary, leykekhman2017maximum .
For the sake of convenience, we denote $\Gamma_{1}=\Gamma_{R}$ or $\Gamma_{N}$
throughout this paper and assume that for some $c>0$, a generic constant, the
followings hold:

$\beta=\gamma\quad\mbox{ and }\quad 0\leq\frac{\alpha}{\beta}\leq c\quad\mbox{
on }\quad\Gamma_{1}.$ (5)

We also provide some conditions for the function $f$, which is generally a
nonlinear polynomial function in both $x$ and $u$. We shall assume that
$f(x,u):\overline{\Omega}\times{\rm I\\!R}\mapsto{\rm I\\!R}$ is smooth in the
second variable. We shall denote the $k$-th derivative of $f$ with respect to
$u$ by $f^{(k)}$, i.e. $f^{(k)}=\frac{\partial^{k}f}{\partial u^{k}}$. and
that there are positive constants $C_{1}$ and $C_{2}$ such that

$|f(x,u)|\leq C_{1}+C_{2}|u|^{q},$ (6)

where $q$ is some real value, such that $1\leq q\leq\infty$ for $d=1$, $1\leq
q<\infty$ for $d=2$ and $1\leq q<5$ for $d=3$. This is sufficient for defining
the weak formulation (see (8)). To apply the finite element method, we
consider the weak formulation of Eq. (1) which satisfies the fully elliptic
regularity (see schatz1996some  and references cited therein), i.e., solution
is sufficiently smooth. We introduce a space $V$ defined by:

$V=\\{v\in H^{1}(\Omega):v|_{\Gamma_{D}}=0\\}.$ (7)

The main problem can then be formulated as follows: Find $u\in V$ such that

$\mathcal{F}(u,v)=a(u,v)+b(u,v)=0,\quad\forall v\in V,$ (8)

where $a(\cdot,\cdot),b(\cdot,\cdot):V\times V\mapsto{\rm I\\!R}$ are the
mappings defined as follow, respectively:

$a(u,v)&=\int_{\Omega}\nabla u\cdot\nabla
v\,dx+\int_{\Gamma_{1}}\frac{\alpha}{\beta}uv\,ds,\quad\forall u,v\in V\\\
b(u,v)=\int_{\Omega}f(x,u)v\,dx-\int_{\Gamma_{1}}gv\,ds,\quad\forall u,v\in
V.$ (9)

##  3 Finite element formulation and a priori error analysis

We will utilize a finite element method to solve (8), specifically a
conforming finite element of degree $r\geq 1$. The triangulation of the domain
$\Omega$ will be denoted by $\mathcal{T}_{h}=\\{T\\}_{i=1,\cdots,N_{e}}$. As
usual, we define

$h={\rm{max}}_{T\in\mathcal{T}_{h}}{\rm diam}(T).$ (10)

Let $V_{h}$ be the subspace of $V$ that is composed of piecewise globally
continuous polynomials of degree $r\geq 1$. We shall denote the set of
vertices of $\mathcal{T}_{h}$ by $\mathcal{V}_{h}$. Then, the dimension of
$V_{h}$ is denoted by $N_{h}$, the total number of interior vertices and the
space $V_{h}$ can be expressed as follows:

$V_{h}={\rm span}\\{\phi_{h}^{i}:i=1,\cdots,N_{h}\\},$ (11)

where $\phi_{h}^{i}$ is the nodal basis on the triangulation
$\mathcal{T}_{h}$, i.e.,

$\phi_{h}^{i}(x_{j})=\delta_{ij},\quad\forall x_{j}\in\mathcal{V}_{h}.$ (12)

The discrete weak formulation for (8) is given as: Find $u_{h}\in V_{h}$ such
that

$\mathcal{F}(u_{h},v_{h})=a(u_{h},v_{h})+b(u_{h},v_{h})=0,\quad\forall
v_{h}\in V_{h}.$ (13)

We note that for any $u_{h}\in V_{h}$, there exists a unique
$\stackunder[1pt]{u}{\scriptscriptstyle\sim}_{h}=(u_{h}^{1},\cdots,u_{h}^{N_{h}})^{T}\in{\rm
I\\!R}^{N_{h}}$ such that

$u_{h}=\sum_{i=1}^{N_{h}}u_{h}^{i}\phi_{h}^{i}.$ (14)

To obtain a solution $u_{h}$ to (13), we need to solve the following system of
nonlinear equations:

$\stackunder[1pt]{F}{\scriptscriptstyle\sim}_{h}(\stackunder[1pt]{u}{\scriptscriptstyle\sim}_{h})=\left(\begin{array}[]{c}F_{h}^{1}(\stackunder[1pt]{u}{\scriptscriptstyle\sim}_{h})\\\
\vdots\\\
F_{h}^{N_{h}}(\stackunder[1pt]{u}{\scriptscriptstyle\sim}_{h})\end{array}\right)=\stackunder[1pt]{0}{\scriptscriptstyle\sim},$
(15)

where

$F_{h}^{i}(\stackunder[1pt]{u}{\scriptscriptstyle\sim}_{h}):=a(u_{h},\phi_{h}^{i})+b(u_{h},\phi_{h}^{i})=0,\quad\forall
i=1,\cdots,N_{h}.$ (16)

###  3.1 A priori error analysis

In this section, we will discuss the convergence order of the finite element
solutions for solving (8). Throughout this section, we introduce a notation
for a fixed $\delta>0$:

$N_{\delta}=\\{v\in V:\|u-v\|_{1,2}<\delta\\},$ (17)

where $u$ is the solution to the equation (8). We will make the following
assumption:

###### Assumption 3.1

There exists a solution $u\in W^{1,2}(\Omega)\cap C^{1}(\Omega)$ of the
problem (8) and there is a constant $\Gamma$ such that
$\|u\|_{0,\infty}\leq\Gamma$ and sufficiently smooth. Furthermore, in
particular, $u$ is isolated in the following sense: there exists
$\delta=\delta_{u}>0$ such that for all $w\in V$ such that
$0\neq\|w\|_{1,2}<\delta$, there exist $\eta_{u}>0$ and $v_{w}\in V$, such
that

$\left|\mathcal{F}(u+w,v_{w})\right|\geq\eta_{u}\|w\|_{1,2}\|v_{w}\|_{1,2}>0.$
(18)

###### Remark 1

We note that, to the best of our knowledge, this is the first time the notion
of isolation has been introduced in the literature. In chen2008analysis , a
similar definition is presented, but it allows $v$ to be any function in $V$,
not necessarily dependent on $w$ in (18). This can lead to several issues. To
illustrate one of the issues, we consider the problem of solving the following
equation:

$-u^{\prime\prime}-u^{p}=0\quad\mbox{ in
}\quad\left(0,\frac{2\pi}{\sqrt{p}}\right)\quad\mbox{ and }\quad
u(0)=u\left(\frac{2\pi}{\sqrt{p}}\right)=0.$ (19)

The weak form is given by $\mathcal{F}(u,v)=\int_{\Omega}\nabla u\nabla
vdx-\int_{\Omega}u^{p}vdx=0$ and $u=0$ is an isolated solution in the sense of
(18), namely, for any $w\in V$, we choose $v_{w}=w$, so that we have

$\left|\mathcal{F}(u+\varepsilon
w,w)\right|=\left|\int_{\Omega}\nabla(\varepsilon w)\nabla
w-\varepsilon^{p}\int_{\Omega}w^{p+1}\right|\geq\eta_{u}\|\varepsilon
w\|_{1,2}\|w\|_{1,2},$ (20)

for some $\eta_{u}$, no matter how $\varepsilon$ is small, due to Poincaré’s
inequality and Sobolev embedding, i.e., ($\|w\|_{0,p+1}\lesssim\|w\|_{1,2}$ in
1D). On the other hand, if we choose $w=\cos(\sqrt{p}x)-1$, $0<\varepsilon\ll
1$ and $v=\sin(\sqrt{p}x)$, then we have that

$\mathcal{F}(u+\varepsilon w,v)=\varepsilon\int_{\Omega}\nabla w\nabla
vdx-\varepsilon^{p}\int_{\Omega}w^{p}vdx=-\varepsilon^{p}\int_{\Omega}w^{p}vdx=0,$
(21)

which implies that $0=\left|\mathcal{F}(u+\varepsilon
w,v)\right|\geq\eta_{u}\varepsilon\|w\|_{1,2}\|v\|_{1,2}=O(\varepsilon)$. This
will not make sense.

We begin with the following lemma as a consequence of our assumption:

###### Lemma 1

Under the assumption that $\|u_{h}\|_{0,\infty},\|u\|_{0,\infty}\leq\Gamma$,
we have

$\|f^{(1)}\|_{0,\infty},\|f^{(2)}\|_{0,\infty}<C(\Gamma),$ (22)

where $C$ is a constant that depends on $\Gamma$.

We shall now consider the linearized problem for a given isolated solution $u$
to the equation (8): For $q\in V^{*}$, find $w\in V$ such that

$A(u;w,v):=a(w,v)+\int_{\Omega}f^{(1)}(u)wv\,dx=(q,v),\quad\forall v\in V.$
(23)

This corresponds to the following partial differential equation: find $w$ such
that

$-\Delta w+f^{(1)}(u)w=q\quad\mbox{ in }\Omega,$ (24)

subject to the same type of boundary condition to the equation (8), but with
$g$ replaced by zero function in (2). We shall assume that the solution to the
equation (23) satisfies the full elliptic regularity schatz1974observation ,
i.e.,

$\|w\|_{2,2}\lesssim\|q\|_{0,2}.$ (25)

We shall now establish the well-posedness of the linearized problem as
follows.

###### Lemma 2

$A(u;\cdot,\cdot)$, defined in Eq. (23), satisfies the inf-sup condition,
i.e.,

$\inf_{w\in V}\sup_{v\in V}\frac{A(u;v,w)}{\|v\|_{1,2}\|w\|_{1,2}}=\inf_{v\in
V}\sup_{w\in V}\frac{A(u;v,w)}{\|v\|_{1,2}\|w\|_{1,2}}\gtrsim 1.$ (26)

###### Proof

Based on babuska1972survey , we need to prove that

  * (i)

there exists a unique zero solution, $w=0$, to $A(u;w,v)=0$ for all $v\in V$;

  * (ii)

$A(u;\cdot,\cdot)$ satisfies the Garding-type inequality, i.e., there exist
$\gamma_{0},\gamma_{1}>0$ such that

$|A(u;v,v)|\geq\gamma_{0}\|v\|_{1,2}^{2}-\gamma_{1}\|v\|_{0,2}^{2},\quad\forall
v\in V,$

For the second condition, the Garding-type inequality holds due to the
Poincare inequality evans1990weak . Secondly, we will prove the first
condition using the proof by contradiction. Let us assume that there exists a
non-zero solution $w\in V$ such that

$A(u;w,v)=0,\quad\forall v\in V.$ (27)

Then, for $\epsilon$ sufficiently small, we define $u_{\epsilon}=u+\epsilon
w\in N_{\delta}(u)$. Then, by Assumption 3.1, we can choose $v_{w}\in V$, for
which the inequality (18) holds and observe that

$\displaystyle\mathcal{F}(u+\epsilon w,v_{w})$ $\displaystyle=$ $\displaystyle
a(u+\epsilon w,v_{w})+b(u+\epsilon w,v_{w})=a(u+\epsilon w,v_{w})$
$\displaystyle\quad+\int_{\Omega}\left[f(u)+f^{(1)}(u)(\epsilon
w)+\frac{1}{2}f^{(2)}(\xi)(\epsilon
w)^{2}\right]v_{w}\,dx-\int_{\Gamma_{D_{{}_{c}}}}gv_{w}\,ds$ $\displaystyle=$
$\displaystyle a(u,v_{w})+b(u,v_{w})+A(u,\epsilon
w,v_{w})+\frac{1}{2}\epsilon^{2}\int_{\Omega}f^{(2)}(\xi)w^{2}v_{w}\,dx$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon^{2}}{2}\|f^{(2)}\|_{0,p_{1}}\|w\|_{0,2p_{2}}^{2}\|v_{w}\|_{0,p_{3}},$

where the last inequality used the generalized Hölder inequality with $p_{i}$
for $i=1,2,3$, satisfying the identity
$\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=1.$ Since we can choose
$\epsilon$ to be arbitrarily small, this contradicts Assumption 3.1. Thus, the
proof is complete.

After establishing the inf-sup condition for the linearized equation at the
continuous level, we can establish the discrete inf-sup condition using the
standard techniques such as the $W^{1,2}$ stability and $L^{2}$ norm error
estimate of Ritz-projection for sufficiently small $h$ (see
brenner2008mathematical ; leykekhman2017maximum ).

###### Lemma 3

Under the Assumption 3.1, the following discrete inf-sup condition holds if
$h<h_{0}$ for sufficiently small $h_{0}$. Specifically, there exists
$\alpha_{0}$, which is independent of $h$, such that

$\inf_{w_{h}\in V_{h}}\sup_{v_{h}\in
V_{h}}\frac{A(u;v_{h},w_{h})}{\|v_{h}\|_{1,2}\|w_{h}\|_{1,2}}=\inf_{v_{h}\in
V_{h}}\sup_{w_{h}\in
V_{h}}\frac{A(u;v_{h},w_{h})}{\|v_{h}\|_{1,2}\|w_{h}\|_{1,2}}=\alpha_{0}>0.$
(28)

This finding has implications for the well-posedness of the Newton method used
to find solutions. For a more in-depth discussion of using the Newton method
to find multiple solutions, see neuberger2001newton ; rabinowitz1986minimax ;
wang2004local  and the references cited therein. We shall now consider the
solution operator for $A(u;\cdot,\cdot)$ and the error estimates. First we
define the projection operator $\pi_{h}:V\mapsto V_{h}$ as

$A(u;\pi_{h}w,v_{h})=A(u;w,v_{h}),\quad\forall v_{h}\in V_{h}.$ (29)

###### Lemma 4

For the projection operator, we have with $r\geq 1$,

$\|w-\pi_{h}w\|_{0,2}\lesssim h^{r+1}\|w\|_{{r+1},2}\quad\mbox{ and
}\quad\|w-\pi_{h}w\|_{1,2}\lesssim h^{r}\|w\|_{r+1,2}.$ (30)

The following inequality shall be needed for the well-posedness and error
analysis of the discrete solution, which is well-known to be true, for the
Dirichlet boundary condition case, brenner2008mathematical .

$\|\pi_{h}w\|_{0,\infty}\lesssim\|w\|_{0,\infty},\quad\forall w\in
W^{1,2}(\Omega)\cap C^{1}(\Omega).$ (31)

We note that for the mixed boundary condition, it can also be proven to be
valid for the special case in 2D leykekhman2017maximum , while the result in
3D, seems yet to be proven, while it has been proven to be valid for the pure
Neumann bounary case in 3D recently in li2022maximum . We can now establish
that the discrete problem (13) admits a unique solution that can approximate
the fixed isolated solution $u$ with the desired convergence rate in both
$L^{2}$ and $W^{1,2}$ norm, using the argument employed in xu1996two .

###### Theorem 3.1

Suppose the Assumption 3.1 and (31). Then, for $h<h_{0}$, with $h_{0}$
sufficiently small, the finite element equation (13) admits a solution $u_{h}$
satisfying

$\|u-u_{h}\|_{k,2}\lesssim h^{r+1-k},\quad\mbox{ for }k=0,1.$ (32)

Furthermore, $u_{h}$ is the only solution in $N_{\delta}$ for some $\delta>0$.

##  4 Companion-Based Multilevel finite element method (CBMFEM)

In this section, we present a companion-based multilevel finite element method
to solve the nonlinear system (15). Solving this system directly is
challenging due to the presence of multiple solutions. Therefore, we draw
inspiration from the multigrid method discussed in brandt2011multigrid  that
is designed for a single solution. We modify the multilevel finite element
method by introducing a local nonlinear solver that computes the eigenvalues
of the companion matrix. This enables us to generate a set of initial guesses
for Newton’s method, which is used to solve the nonlinear system on the
refined mesh.

Figure 1: Mesh refinement of CBMFEM in 1D (left) and 2D with edge (right). The
square dots are the coarse nodes while filled circles are newly introduced
fine nodes.

We begin by introducing a sequence of nested triangulations, namely,

$\mathcal{T}_{0},\mathcal{T}_{1},\mathcal{T}_{2},\cdots,\mathcal{T}_{N},$ (33)

where $\mathcal{T}_{0}$ and $\mathcal{T}_{N}$ are the coarsest and the finest
triangulations of $\Omega$, respectively. This leads to the construction of a
sequence of nested and conforming finite element spaces
$\\{V_{\ell}\\}_{\ell=0}^{N}\subset V$, given as follows:

$V_{0}\subset\cdots\subset V_{N}.$ (34)

The refinement strategy is shown in Figure 1 for both 1D and 2D cases, where
we introduce new nodes (the filled circles) based on the coarse nodes (the
square dots). Specifically, for a given coarse mesh $\mathcal{T}_{H}$, we
obtain the refined mesh $\mathcal{T}_{h}$ by introducing a new node on
$\mathcal{V}_{H}$. The overall flowchart of CBMFEM is summarized in Figure 2.

Figure 2: A flowchart of the CBMFEM for solving the nonlinear differential
equation. The hierarchical structure of CBMFEM is illustrated on the left,
where for each level, we obtain a solution on the coarse grid, $V_{H}$. We
then solve local nonlinear equations by constructing companion matrices and
generate initial guesses for Newton’s method on the finer level $V_{h}$ on the
right.

Assuming that a solution $u_{H}\in V_{H}$ on a coarse mesh $\mathcal{T}_{H}$
has been well approximated, namely
$\stackunder[1pt]{F}{\scriptscriptstyle\sim}_{H}(\stackunder[1pt]{u}{\scriptscriptstyle\sim}_{H})=\stackunder[1pt]{0}{\scriptscriptstyle\sim}$,
we can refine the triangulation $\mathcal{T}_{H}$ to obtain a finer
triangulation $\mathcal{T}_{h}$. Since
$\mathcal{V}_{H}\subset\mathcal{V}_{h}$, we can find a function
$\widetilde{u}_{h}(x)$ in $\mathcal{V}_{h}$ such that we can express
$u_{H}(x)$ as a linear combination of the basis functions of
$\mathcal{V}_{h}$:

$u_{H}(x)=\widetilde{u}_{h}(x)=\sum_{i=1}^{\dim{\mathcal{V}_{h}}}\widetilde{u}_{h}^{i}\phi_{h}^{i}(x),$
(35)

where $\dim{\mathcal{V}_{h}}$ is the dimension of $\mathcal{V}_{h}$ and
$\phi_{h}^{i}(x)$ are the basis functions of $\mathcal{V}_{h}$. In particular,
for any point $x_{i}\in\mathcal{V}_{H}$, we have
$\widetilde{u}_{h}^{i}=u_{H}(x_{i})$. For points
$x_{i}\in\mathcal{V}_{h}\backslash\mathcal{V}_{H}$, we can calculate
$\widetilde{u}_{h}^{i}$ based on the intrinsic structure of the basis
functions. In other words, we can interpolate $u_{H}(x)$ to $u_{h}(x)$ by
using the basis functions on two levels. Specifically, let’s consider 1D case
and have the relation
$\phi^{i/2}_{H}=\phi^{i/2}_{2h}=\frac{\phi^{i-1}_{h}}{2}+\phi^{i}_{h}+\frac{\phi^{i+1}_{h}}{2}$,
which allows us to calculate the coefficients for (35). More precisely, we can
write $u_{H}(x)$ to $u_{h}(x)$ as follows:

$u_{H}(x)=\sum_{x_{i}\in V_{H}}{u}_{H}^{i/2}\phi_{H}^{i/2}(x)=\sum_{x_{i}\in
V_{H}}{u}_{H}^{i/2}\left(\frac{\phi^{i-1}_{h}}{2}+\phi^{i}_{h}+\frac{\phi^{i+1}_{h}}{2}\right)=\sum_{i=1}^{\dim{\mathcal{V}_{h}}}\widetilde{u}_{h}^{i}\phi_{h}^{i}(x).$
(36)

We next update the value of $u_{h}^{i}$ for
$x_{i}\in\mathcal{V}_{h}\backslash\mathcal{V}_{H}$ on the fine mesh. To do
this, we solve $F_{h}^{i}(u_{h}^{i};(u_{h}^{j}=\widetilde{u}_{h}^{j})_{j\neq
i})=0$ for $u_{h}^{i}$ by fixing the values of other nodes as
$\widetilde{u}_{h}$. Since $f(x,u)$ is a polynomial, we can rewrite
$F_{h}^{i}$ as a single polynomial equation, namely,

$F_{h}^{i}(u_{h}^{i};(u_{h}^{j}=\widetilde{u}_{h}^{j})_{j\neq
i})=P_{h}^{i}(\widehat{u}_{h}^{i})=0,$ (37)

where $\displaystyle P_{h}^{i}(\alpha)=\sum_{n=0}^{m}c_{n}\alpha^{n}$. The
companion matrix of $P_{h}^{i}(\alpha)$ is defined as

$C(P_{h}^{i})=\left[\begin{array}[]{ccccc}0&0&\ldots&0&-c_{0}/c_{m}\\\
1&0&\ldots&0&-c_{1}/c_{m}\\\ 0&1&\ldots&0&-c_{2}/c_{m}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\ldots&1&-c_{m-1}/c_{m}\end{array}\right],$ (38)

where, except for $c_{0}$, the coefficients $c_{n}$ depend only on
$\\{\widetilde{u}_{h}^{j}\\}$ that are near the point
$x_{i}\in\mathcal{V}_{h}\backslash\mathcal{V}_{H}$ , making their computation
local. By denoting the root of Eq. (37 ) as $\widehat{u}^{i}_{h}$, the initial
guess for the solutions on $V_{h}$ is set as

$\widehat{{u}}_{h}=\sum\widehat{{u}}_{h}^{i}\phi^{i}_{h}\hbox{~{}where~{}}\widehat{{u}}_{h}^{i}=\left\\{\begin{array}[]{rl}u^{i}_{H}&x_{i}\in\mathcal{V}_{H},\\\
\widehat{u}^{i}_{h}\hbox{~{}by
solving~{}}(\ref{polynomial})&x_{i}\in\mathcal{V}_{h}\backslash\mathcal{V}_{H}.\end{array}\right.$
(39)

Since all the eigenvalues of $C(P_{h}^{i})$ satisfy the equation
$P_{h}^{i}(y_{i})=0$, there can be up to
$m^{|\mathcal{V}_{h}\backslash\mathcal{V}_{H}|}$ possible initial guesses,
where $|\mathcal{V}_{h}\backslash\mathcal{V}_{H}|$ denotes the number of newly
introduced fine nodes on $\mathcal{V}_{h}$ and $m$ is the degree of the
polynomial (37). However, computing all of these possibilities is
computationally expensive, so we apply the filtering conditions below to
reduce the number of initial guesses and speed up the method:

  * •

Locality condition: we assume the initial guess is near
$\stackunder[1pt]{\tilde{u}_{h}}{\scriptscriptstyle\sim}$ in term of the
residual, namely,

$\|\stackunder[1pt]{F}{\scriptscriptstyle\sim}_{h}(\stackunder[1pt]{\widehat{u}_{h}}{\scriptscriptstyle\sim})\|_{0,2}<C_{1}\|\stackunder[1pt]{F}{\scriptscriptstyle\sim}_{h}(\stackunder[1pt]{\tilde{u}_{h}}{\scriptscriptstyle\sim})\|_{0,2};$
(40)

  * •

Convergence condition: we apply the convergence estimate to the initial guess,
namely,

$\|\stackunder[1pt]{F}{\scriptscriptstyle\sim}_{h}(\stackunder[1pt]{\widehat{u}_{h}}{\scriptscriptstyle\sim})\|_{0,2}<C_{2}h^{2};$
(41)

  * •

Boundness condition: we assume the initial guess is bounded, namely,

$\|\stackunder[1pt]{\widehat{u}_{h}}{\scriptscriptstyle\sim}\|_{0,\infty}<C_{3}.$
(42)

Finally, we summarize the algorithm of CBMFEM in Algorithm 1.

Algorithm 1 CBMFEM for computing multiple solutions

Given $V_{h},V_{H}$, and solution $u_{H}$ on $V_{H}$.

Interpolate $u_{H}=\sum\widetilde{u}_{h}^{i}\phi_{h}^{i}$ and compute
coefficient $\widetilde{u}_{h}^{i}$.

for $i\in V_{h}\backslash V_{H}$ do

Construct the polynomial equation
$P_{h}^{i}(\widehat{\stackunder[1pt]{u}{\scriptscriptstyle\sim}}_{h}^{i})$

Compute the eigenvalues of the companion matrix, $C(P_{h}^{i})$

end for

Obtain initial guesses in (39) on $V_{h}$ and apply filtering conditions.

Employ Newton method on $V_{h}$ with the obtained initial guesses to solve
$\stackunder[1pt]{F}{\scriptscriptstyle\sim}_{h}(\stackunder[1pt]{u}{\scriptscriptstyle\sim}_{h})=0$

##  5 Numerical Examples

In this section, we present several examples for both 1D and 2D to demonstrate
the effectivity and robustness of CBMFEM with $r=1$ for simplicity. We shall
let $e_{H}^{h}=u_{h}-u_{H}$ where $u_{h}$ and $u_{H}$ are the numerical
solutions with grid step size $h$ and $H$, respectively, and $u$ is the
analytical solution. For error analysis, since most of the examples we
considered do not have analytical solutions, we used asymptotic error analysis
to calculate the convergence rate.

###  5.1 Examples for 1D

####  5.1.1 Example 1

First, we consider the following boundary value problem

$\begin{cases}-u^{\prime\prime}=(1+u^{4})\quad\text{on }\quad[0,1],\\\
u^{\prime}(0)=u(1)=0,\end{cases}$ (43)

which has analytical solutions hao2014bootstrapping . More specifically, by
multiplying both side with $u^{\prime}$ and integrating with respect to $x$,
we obtain

$\frac{(u^{\prime}(x))^{2}}{2}+F(u(x))-F(u_{0})=0,$ (44)

where $F(u)=u+\frac{u^{5}}{5}$ and $u_{0}=u(0)$. Since $u^{\prime}(0)=0$ and
$u^{\prime\prime}(x)<0$, then we have $u^{\prime}<0$ for all $x>0$. Moreover,
$u(x)>u_{0}$ implies $F(u_{0})>F(x)$ for all $x>0$. Therefore

$\frac{u^{\prime}(x)}{\sqrt{F(u_{0})-F(u(x))}}=-\sqrt{2}.$ (45)

By integrating $x$ from $s$ to $1$, we obtain

$\int_{s}^{1}\frac{u^{\prime}(x)}{\sqrt{F(u_{0})-F(u(x))}}dx=-\sqrt{2}(1-s).$
(46)

Due to the boundary condition $u(1)=0$, we have

$\int_{0}^{u(s)}\frac{dx}{\sqrt{F(u_{0})-F(u(x))}}=\sqrt{2}(1-s).$ (47)

By choosing $s=0$, we have the following equation for $u_{0}$,

$\int_{0}^{u_{0}}\frac{dx}{\sqrt{F(u_{0})-F(u(x))}}=\sqrt{2}$ (48)

Then for any given $u_{0}$, the solution of (43) is uniquely determined by the
initial value problem

$\begin{cases}-v^{\prime}=(1+u^{4})\\\
u^{\prime}=v\end{cases}\hbox{~{}with~{}}\begin{cases}v(0)=0\\\
u(0)=u_{0}\end{cases}.$ (49)

By solving (48) with Newton’s method, we get two solutions $u_{0}\approx
0.5227$ and $u_{0}\approx 1.3084$. Then the numerical error is shown in Table
1 for the CBMEFM with Newton’s nonlinear solver.

Figure 3: Numerical solutions of Eq. (43) with $N=1025$ grid points.  h | # 1st Solution | # 2nd solution | CPU(s)
---|---|---|---
$\|e_{h}\|_{0,2}$ | Order | $\|e_{h}\|_{1,2}$ | Order | $\|e_{h}\|_{0,2}$ | Order | $\|e_{h}\|_{1,2}$ | Order | Newton
$2^{-2}$ | 2.6E-04 | x | 1.7E-01 | x | 8.3E-03 | x | 7.0E-02 | x | 0.10
$2^{-3}$ | 6.4E-05 | 2.01 | 8.0E-02 | 1.00 | 2.0E-03 | 2.04 | 4.0E-02 | 1.01 | 0.10
$2^{-4}$ | 1.6E-05 | 2.00 | 4.0E-02 | 1.00 | 5.0E-04 | 2.01 | 2.0E-02 | 1.00 | 0.15
$2^{-5}$ | 4.0E-06 | 2.00 | 2.0E-02 | 1.00 | 1.3E-04 | 2.00 | 9.3E-03 | 1.00 | 0.16
$2^{-6}$ | 1.0E-06 | 2.00 | 1.0E-02 | 1.00 | 3.1E-05 | 2.00 | 4.7E-03 | 1.00 | 0.11
$2^{-7}$ | 2.5E-07 | 2.00 | 5.2E-03 | 1.00 | 7.8E-06 | 2.00 | 2.3E-03 | 1.00 | 0.16
$2^{-8}$ | 6.2E-08 | 2.00 | 2.6E-03 | 1.01 | 1.9E-06 | 2.00 | 1.2E-03 | 1.01 | .25
$2^{-9}$ | 1.6E-08 | 2.00 | 1.3E-03 | 1.03 | 4.9E-07 | 2.00 | 0.6E-03 | 1.04 | 0.64
$2^{-10}$ | 3.9E-09 | 2.00 | 5.6E-04 | 1.16 | 1.2E-07 | 2.00 | 0.3E-03 | 1.16 | 3.55
Table 1: Numerical errors and computing time of CBMEFM with Newton’s nonlinear
solver for solving Eq. (43).

####  5.1.2 Example 2

Next, we consider the following boundary value problem

$\begin{cases}-u^{\prime\prime}=-u^{2}\quad\text{on }\quad[0,1],\\\
u(0)=0\hbox{~{}and~{}}u(1)=1,\end{cases}$ (50)

which has two solutions shown in Fig. 4. We start $N=3$ and compute the
solutions up to $N=1025$ by implementing CBMEFM with nonlinear solver. We
compute the numerical error by using $\|e_{H}^{h}\|_{0,2}$ and summarize the
convergence test and computing time in Table 2.

Figure 4: Numerical solutions of Eq. (50) with $N=1025$ grid points. h | # 1st solution | # 2nd solution | CPUs
---|---|---|---
$\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order | $\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order | Newton
$2^{-2}$ | 2.6E-3 | x | 2.3E-2 | x | 7.9E-1 | x | 5.8E-0 | x | 0.09
$2^{-3}$ | 6.8E-4 | 1.96 | 1.2E-2 | 0.90 | 2.0E-1 | 2.01 | 2.9E-0 | 1.03 | 0.09
$2^{-4}$ | 1.7E-4 | 1.99 | 6.3E-3 | 0.94 | 4.9E-2 | 2.00 | 1.4E-0 | 1.00 | 0.09
$2^{-5}$ | 4.3E-5 | 2.00 | 3.2E-3 | 0.97 | 1.2E-2 | 2.00 | 7.12E-1 | 1.00 | 0.10
$2^{-6}$ | 1.1E-5 | 2.00 | 1.6E-3 | 0.98 | 3.1E-3 | 2.00 | 3.6E-1 | 1.00 | 0.11
$2^{-7}$ | 2.7E-6 | 2.00 | 8.2E-4 | 0.99 | 7.7E-4 | 2.00 | 1.8E-1 | 1.00 | 0.14
$2^{-8}$ | 6.7E-7 | 2.00 | 4.1E-4 | 1.00 | 1.9E-4 | 2.00 | 8.9E-2 | 1.00 | 0.24
$2^{-9}$ | 1.7E-7 | 2.00 | 2.1E-4 | 1.00 | 4.8E-5 | 2.00 | 4.5E-2 | 1.00 | 0.49
$2^{-10}$ | 4.2E-8 | 2.00 | 1.0E-4 | 1.00 | 1.2E-5 | 2.00 | 2.2E-2 | 1.00 | 3.35
Table 2: Numerical errors and computing time of CBMEFM with Newton’s solver
for solving Eq. (50).

####  5.1.3 Example 3

Thirdly, we consider the following nonlinear parametric differential equation

$\begin{cases}-u^{\prime\prime}=u^{2}(p-u^{2})\quad\text{on }\quad[0,1],\\\
u^{\prime}(0)=u(1)=0,\end{cases}$ (51)

where $p$ is a parameter. The number of solutions increases as $p$ gets larger
hao2020adaptive . We compute the numerical solutions for $p=1,7$, and $18$
using CBMEFM with Newton’s solver and show the solutions in Fig. 5. he
computation time and the number of solutions for different values of $p$ and
step sizes are summarized in Table 3. As $p$ increases, the number of
solutions increases, and hence the computation time becomes longer.

Figure 5: Numerical solutions of Eq. (51) with $1025$ grid points for $p=1$, $p=7$, and $p=18$, respectively.  h | p=1 | p=7 | p=18
---|---|---|---
CPUs | # of sols | CPUs | # of sols | CPUs | # of sols
$2^{-2}$ | 0.17 | 2 | 0.15 | 5 | 0.29 | 19
$2^{-3}$ | 0.12 | 2 | 0.35 | 9 | 3.01 | 37
$2^{-4}$ | 0.11 | 2 | 0.20 | 4 | 28.17 | 10
$2^{-5}$ | 0.11 | 2 | 0.11 | 4 | 0.38 | 8
$2^{-6}$ | 0.11 | 2 | 0.14 | 4 | 0.18 | 8
$2^{-7}$ | 0.14 | 2 | 0.23 | 4 | 0.33 | 8
$2^{-8}$ | 0.26 | 2 | 0.35 | 4 | 0.66 | 8
$2^{-9}$ | 0.57 | 2 | 0.95 | 4 | 1.93 | 8
$2^{-10}$ | 3.08 | 2 | 5.55 | 4 | 11.54 | 8
Table 3: Computing time (in seconds) and the number of solutions of Eq. (51) for different $p$ by CBMEFM with Newton’s solver.  h | # 1st solution | # 2nd solution | # 3rd solution
---|---|---|---
$\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order | $\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order | $\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order
$2^{-2}$ | 1.6E-1 | x | 1.43 | x | 1.6E-1 | x | 9.0E-1 | x | 2.7E-2 | x | 1.1E-1 | x
$2^{-3}$ | 2.4E-1 | -0.54 | 1.22 | 0.23 | 3.3E-2 | 2.30 | 4.7E-1 | 0.93 | 6.8E-3 | 2.00 | 5.1E-2 | 1.08
$2^{-4}$ | 4.2E-2 | 2.52 | 7.4E-1 | 0.73 | 7.4E-3 | 2.13 | 2.2E-1 | 1.07 | 1.7E-3 | 2.00 | 2.5E-2 | 1.06
$2^{-5}$ | 8.4E-3 | 2.31 | 3.0E-1 | 1.28 | 1.9E-3 | 2.01 | 1.1E-1 | 1.01 | 4.3E-4 | 2.00 | 1.2E-2 | 1.03
$2^{-6}$ | 2.1E-3 | 2.04 | 1.4E-1 | 1.08 | 4.6E-4 | 2.00 | 5.5E-2 | 1.00 | 1.1E-4 | 2.00 | 5.9E-3 | 1.02
$2^{-7}$ | 5.1E-4 | 2.01 | 7.0E-2 | 1.03 | 1.2E-4 | 2.00 | 2.8E-2 | 1.00 | 2.7E-5 | 2.00 | 3.0E-3 | 1.01
$2^{-8}$ | 1.3E-4 | 2.00 | 3.5E-2 | 1.02 | 2.9E-5 | 2.00 | 1.4E-2 | 1.00 | 6.7E-6 | 2.00 | 1.5E-3 | 1.00
$2^{-9}$ | 3.2E-5 | 2.00 | 1.7E-2 | 1.01 | 7.2E-6 | 2.00 | 6.9E-3 | 1.00 | 1.7E-6 | 2.00 | 7.4E-4 | 1.00
$2^{-10}$ | 8.0E-6 | 2.00 | 8.6E-3 | 1.00 | 1.8E-6 | 2.00 | 3.5E-3 | 1.00 | 4.2E-7 | 2.00 | 3.7E-4 | 1.00
Table 4: Numerical errors only nontrivial solutions for Eq. (51) when $p=7$ by
CBMEFM with Newton’s solver.

####  5.1.4 Example 4

We consider the following semi-linear elliptic boundary value problem

$\begin{cases}-u^{\prime\prime}(x)=d^{r-2}|x|^{r}u^{3}(x)\quad\text{on}\quad[-1,1],\\\
u(-1)=u(1)=0,\end{cases}$ (52)

where $d$ is the scaling coefficient corresponding to the domain. This example
is based on a problem considered in xie2012finding ; xie2022solving . We have
re-scaled the domain from $[-1,1]$ to $[-d,d]$ and formulated the problem
accordingly.

We start with $r=3$ and $d=1$ and find $4$ non-negative solutions with
$N=1025$ using CBMEFM. Using the homotopy method with respect to both $r$ and
$d$, we also discover the same number of solutions for $r=3,d=10$ and
$r=12,d=1$. The numerical solutions with different parameters are shown in
Fig. 6. Then, we also create a bifurcation diagram for the numerical solutions
with respect to $r$ by choosing $d=1$ and using $1025$ grid points, as shown
in Fig. 7. We only display the non-trivial non-negative solutions on this
diagram. When $r$ is small, we have only one non-trivial solution. As $r$
increases, the number of non-trivial solutions increases and bifurcates to
three solutions.

Figure 6: Numerical solutions of Eq. (52) with $N=1025$ grid points for different $r$ and $d$. We have symmetric solutions so we only concern with one of them. Figure 7: Bifurcation diagram of Eq. (52) with respect to $r$ with $N=1025$ grid points and $d=1$. h | # 1st solution | # 2nd solution
---|---|---
$\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order | $\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order
$2^{-3}$ | 1.3E-0 | x | 7.9E-0 | x | 1.0E-0 | x | 9.2E-0 | x
$2^{-4}$ | 3.2E-1 | 2.04 | 4.2E-0 | 0.90 | 5.7E-1 | 0.87 | 5.4E-0 | 0.76
$2^{-5}$ | 8.0E-2 | 2.03 | 2.0E-0 | 1.06 | 1.1E-1 | 2.33 | 2.3E-0 | 1.22
$2^{-6}$ | 2.0E-2 | 2.03 | 1.0E-0 | 1.01 | 2.7E-2 | 2.03 | 1.2E-0 | 1.02
$2^{-7}$ | 4.9E-3 | 2.01 | 5.0E-1 | 1.00 | 6.7E-3 | 2.00 | 5.7E-1 | 1.00
$2^{-8}$ | 1.2E-3 | 2.00 | 2.5E-1 | 1.00 | 1.7E-3 | 2.00 | 2.9E-1 | 1.00
$2^{-9}$ | 3.1E-4 | 2.00 | 1.3E-1 | 1.00 | 4.2E-4 | 2.00 | 1.4E-1 | 1.00
$2^{-10}$ | 7.6E-5 | 2.00 | 6.3E-2 | 1.00 | 1.0E-4 | 2.00 | 7.2E-2 | 1.00
Table 5: Numerical errors only nontrivial solutions for Eq. (52) when
$r=3,d=1$ by CBMEFM with Newton’s solver.

####  5.1.5 The Schnakenberg model

We consider the steady-state system of the Schnakenberg model in 1D with no-
flux boundary conditions

$\begin{cases}u^{\prime\prime}+\eta(a-u+u^{2}v)=0,\quad\text{on }\quad[0,1]\\\
\ dv^{\prime\prime}+\eta(b-u^{2}v)=0,\quad\quad\text{on }\quad[0,1]\\\ \
u^{\prime}(0)=u^{\prime}(1)=v^{\prime}(0)=v^{\prime}(1)=0.\end{cases}$ (53)

This model exhibits complex solution patterns for different parameters
$\eta,a,b,$ and $d$ hao2020spatial . Since there is only one nonlinear term
$u^{2}v$, we rewrite the steady-state system as follows

$\begin{cases}u^{\prime\prime}+dv^{\prime\prime}+\eta(a+b-u)=0,\\\
dv^{\prime\prime}+\eta(b-u^{2}v)=0.\end{cases}$ (54)

In this case, we only have one nonlinear equation in the system (54). After
discretization, we can solve
${\stackunder[1pt]{v}{\scriptscriptstyle\sim}}_{h}$ in terms of
${\stackunder[1pt]{u}{\scriptscriptstyle\sim}}_{h}$ using the first equation
and plug it into the second equation. We obtain a single polynomial equation
of ${\stackunder[1pt]{u}{\scriptscriptstyle\sim}}_{h}$, which allows us to use
the companion matrix to solve for the roots. Then we use CBMEFM to solve the
Schakenberg model in 1D with the parameters $a=1/3,b=2/3,d=50,\eta=50$ up to
$h=2^{-9}$. There are 3 solutions computed (8) and we show the computing time
in Table (6).

h | # 1st solution | CPUs
---|---|---
$\|e_{H}^{h}\|_{0,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order |
$2^{-3}$ | 2.9E-2 | x | 1.9E-1 | x | 0.58
$2^{-4}$ | 6.3E-3 | 2.20 | 6.0E-2 | 1.63 | 0.42
$2^{-5}$ | 1.6E-3 | 2.02 | 2.7E-2 | 1.18 | 0.55
$2^{-6}$ | 3.9E-4 | 2.00 | 1.3E-2 | 1.06 | 0.84
$2^{-7}$ | 9.6E-5 | 2.00 | 6.3E-3 | 1.02 | 1.96
$2^{-8}$ | 2.4E-5 | 2.00 | 3.1E-3 | 1.01 | 6.14
$2^{-9}$ | 6.0E-6 | 2.00 | 1.6E-3 | 1.00 | 22.14
$2^{-10}$ | 1.5E-6 | 2.00 | 7.8E-4 | 1.00 | 157.88
Table 6: Numerical errors and computing time of CBMEFM with Newton’s solvers
for solving Eq. (53). We have symmetric solutions so we only concern with one
of them.

Figure 8: $3$ different solutions for equation (53) with $N=1025$ grid points.
The same colors are paired solutions.

###  5.2 Examples in 2D

In this section, we discuss a couple of two dimensional examples.

####  5.2.1 Example 1

$\begin{cases}\Delta u(x,y)+u^{2}(x,y)=s\sin(\pi x)\sin(\pi
y)\quad\text{on}\quad\Omega,\\\
u(x,y)=0,\quad\text{on}\quad\partial\Omega\end{cases}$ (55)

where $\Omega=(0,1)\times(0,1)$ breuer2003multiple .

In this example, we utilized edge refinement to generate a multi-level grid
and provide multiple initial guesses for the next level, as shown in Fig. 9.
The number of nodes and triangles on each level is summarized in Table 7.

Figure 9: Multi-level grid based on the edge refinement with a rectangular domain. Step size | $2^{0}$ | $2^{-1}$ | $2^{-2}$ | $2^{-3}$ | $2^{-4}$ | $2^{-5}$ | $2^{-6}$ | $2^{-7}$
---|---|---|---|---|---|---|---|---
# of Nodes | $5$ | $13$ | $41$ | $145$ | $545$ | $2113$ | $8321$ | $33025$
# of Triangles | $4$ | $16$ | $64$ | $256$ | $1024$ | $4096$ | $16384$ | $65536$
Table 7: The number of nodes and triangles of the multigrid with the edge
refinement for each level.

First, we compute the solutions with $s=1600$. We only applied the CBMEFM for
the first three multi-level grids until $\ell=2$ due to the extensive
computation caused by a large number of solution combinations on the higher
level. Starting $\ell=3$, we used Newton refinement with an interpolation
initial guess from the coarse grid. We found 10 solutions and computed them
until $h=2^{-7}$. Since some solutions are the same up to the rotation, we
plot only four solutions in Fig. 11. It is worth noting that Eq. (55) remains
unchanged even when $x$ is replaced by $1-x$, and similarly for $y$.
Therefore, rotating solutions 3 and 4 from Fig. 10 would also yield valid
solutions. Therefore, although there are 10 solutions in total, we only plot
the 4 solutions in Fig. 10. up to the rotation. We also computed the numerical
error and convergence order in both $L^{2}$ and $H^{1}$ norms for these
solutions and summarized them in Table 9.

Method | Step size | # of solutions | Comp. Time
---|---|---|---
CBMEFM | $2^{0}$ | 2 | 0.2s
$2^{-1}$ | 10 | 1.4s
$2^{-2}$ | 10 | 142.7s
Newton’s refinement | $2^{-3}$ | 10 | 7.1s
$2^{-4}$ | 10 | 20.6s
$2^{-5}$ | 10 | 76.8s
$2^{-6}$ | 10 | 540.4s
$2^{-7}$ | 10 | 5521.3s
Table 8: The numerical performance summary of solving (55) with $s=1600$. Until the level $\ell=2$ we used CBMEFM and applied Newton’s refinement starting from $\ell=3$. Figure 10: Multiple solutions of Eq. (55) with $s=1600$ and a step size of $2^{-7}$. h | # 1st solution | # 2nd solution | # 3rd solution | # 4th solution
---|---|---|---|---
$\|e_{{}_{H}}^{h}\|_{0,2}$ | Order | $\|e_{{}_{H}}^{h}\|_{0,2}$ | Order | $\|e_{{}_{H}}^{h}\|_{0,2}$ | Order | $\|e_{{}_{H}}^{h}\|_{0,2}$ | Order
$2^{-1}$ | 2.1E-0 | x | 1.2E-0 | x | x | x | x | x
$2^{-2}$ | 5.1E-0 | -1.26 | 2.8E-0 | -1.26 | 2.3E+1 | x | 5.4E-0 | x
$2^{-3}$ | 3.9E-0 | 0.37 | 0.7E-0 | 1.94 | 9.5E-0 | 1.28 | 6.9E-0 | -0.36
$2^{-4}$ | 1.6E-0 | 1.30 | 0.2E-0 | 1.98 | 3.2E-0 | 1.56 | 2.3E-0 | 1.58
$2^{-5}$ | 0.5E-0 | 1.70 | 0.5E-1 | 1.99 | 0.9E-0 | 1.85 | 0.7E-0 | 1.83
$2^{-6}$ | 0.1E-0 | 1.90 | 0.1E-1 | 2.00 | 0.2E-0 | 1.95 | 0.2E-0 | 1.95
$2^{-7}$ | 3.3 E-3 | 1.97 | 2.9E-03 | 2.00 | 0.6E-1 | 1.99 | 0.4E-1 | 1.99
h | $\|e_{H}^{h}\|_{1,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order | $\|e_{H}^{h}\|_{1,2}$ | Order
$2^{-1}$ | 2.6E+1 | x | 1.4E+1 | x | x | x | x | x
$2^{-2}$ | 6.8E+1 | -1.40 | 2.7E+1 | -0.93 | 1.9E+2 | x | 7.2E+1 | x
$2^{-3}$ | 5.6E+1 | 0.29 | 1.5E+1 | 0.91 | 9.9E+1 | 0.92 | 7.9E+1 | -0.13
$2^{-4}$ | 3.1E+1 | 0.86 | 7.4E-0 | 0.97 | 5.2E+1 | 0.92 | 4.2E+1 | 0.92
$2^{-5}$ | 1.6E+1 | 0.97 | 3.7E-0 | 0.99 | 2.6E+1 | 1.02 | 2.1E+1 | 1.01
$2^{-6}$ | 7.8E-0 | 1.00 | 1.9E-0 | 1.00 | 1.3E+1 | 1.01 | 1.0E+1 | 1.01
$2^{-7}$ | 3.9E-0 | 1.00 | 0.9E-0 | 1.00 | 6.3E-0 | 1.00 | 5.1E-0 | 1.00
Table 9: Numerical $L^{2}$ and $H^{1}$ errors for multiple solutions of Eq.
(55) with $s=1600$ shown in Fig. 10.

Finally, we also explored the solution structure with respect to For small
values of $s$, only two solutions are observed. However, as $s$ increases, the
number of solutions also increases, as demonstrated in Fig. 11.

Figure 11: Bifurcation diagram of solutions of Eq. (55) with respect to $s$.

####  5.2.2 The Gray–Scott model in 2D

The last example is the steady-state Gray-Scott model, given by

$\begin{cases}D_{A}\Delta A=-SA^{2}+(\mu+\rho)A,\quad\text{on}\quad\Omega\\\
D_{S}\Delta S=SA^{2}-\rho(1-S),\quad\text{on}\quad\Omega\\\
\displaystyle\frac{\partial u}{\partial\textbf{n}}=\frac{\partial
v}{\partial\textbf{n}}=0\quad\text{on}\quad\partial\Omega\par\end{cases}$ (56)

$\Omega=(0,1)\times(0,1)$ and n is normal vector hao2020spatial . The
solutions of this model depend on the constants $D_{A}$, $D_{S}$, $\mu$, and
$\rho$. In this example, we choose $D_{A}=2.5\times 10^{-4}$, $D_{S}=5\times
10^{-5}$, $\rho=0.04$, and $\mu=0.065$. Similar to the example of the
Schnakenberg model in 1D (54), we can modify the system as follows:

$\begin{cases}D_{A}\Delta A+D_{S}\Delta S=(\mu+\rho)A-\rho(1-S),\\\
D_{S}\Delta S=SA^{2}-\rho(1-S),\end{cases}$ (57)

As the first equation in the Gray-Scott model is linear, we can solve for $A$
in terms of $S$ and substitute into the second equation to obtain a single
polynomial equation. When using the companion matrix on the coarsest grid
($\ell=0$) to solve the polynomial equation, we obtain $3^{5}$ complex
solutions, leading to a large number of possible combinations on finer grids.
To address this, we applied two approaches. The first approach involved
keeping the real solutions and real parts of complex solutions, and using
linear interpolation and Newton’s method to refine them on finer grids. This
approach yielded 24 solutions on a step size of $2^{-6}$, up to rotation shown
in Fig. 12. The second approach involved keeping only the real solutions,
resulting in $1$ solution on $\ell=0$ and $3^{13}$ initial guesses on
$\ell=1$. Using interpolation and Newton’s method, we obtained 88 solutions on
a step size of $2^{-6}$, up to rotation shown in Fig. 13. If we use more large
coefficient for filtering conditions we can get 187 solutions.

Figure 12: We have 24 solutions and plot only $A(x,y)$ from (57) with a step
size of $2^{-6}$. The initial guesses were refined by considering both real
solutions and real parts of complex solutions on the coarsest grid ($\ell=0$).

Figure 13:  We have 88 solutions and plot only $A(x,y)$ from (57) with a step
size of $2^{-6}$. The initial guesses were refined by considering only the
real solutions on the $\ell=1$ grid.

##  6 Conclusion

In this paper, we have presented a novel approach, the Companion-Based
Multilevel finite element method (CBMFEM), which efficiently and accurately
generates multiple initial guesses for solving nonlinear elliptic semi-linear
equations with polynomial nonlinear terms. Our numerical results demonstrate
the consistency of the method with theoretical analysis, and we have shown
that CBMFEM outperforms existing methods for problems with multiple solutions.

Furthermore, CBMFEM has potential applications in more complex PDEs with
polynomial nonlinear terms. To generalize our approach, we need to conduct
further investigations to identify better filtering condition constants and
better nonlinear solvers. In our future work, we shall incorporate the
multigrid method to speed up the Newton method, which should further improve
the efficiency of the method. Overall, CBMFEM is a promising approach for
solving elliptic PDEs with multiple solutions, and we will apply it to
widespread applications in various scientific and engineering fields.

Data availability Data sharing is not applicable to this article as no
datasets were generated or analyzed during the current study.

Declarations WH and SL is supported by NIH via 1R35GM146894. YL is supported
by NSF via DMS 2208499 There is no conflict of interest.

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Dr. Nagwa Abu-Elsaad

# Exploring the Magnetic Behavior of Ferrites: From Diamagnetism to
Superparamagnetism

Maryam M. Ahmed  Physics Department, Faculty of Science, Zagazig University

###### Abstract

Ferrites, magnetic materials primarily composed of iron oxides, exhibit
diverse magnetic behaviors, including diamagnetism, paramagnetism,
ferromagnetism, antiferromagnetism, ferrimagnetism, and superparamagnetism.
This paper explores the fundamental principles governing these magnetic
phenomena, with a focus on the relationship between magnetic field strength
($H$), magnetic induction ($B$), and magnetic susceptibility ($\chi$). The
study delves into the structural and electronic origins of magnetism in
ferrites, highlighting the contributions of electron spin and orbital motions.
The technological significance of ferrites, particularly in high-frequency
applications, is examined through their classification based on crystal
structures, including spinel, garnet, hexagonal, and orthoferrites. Special
attention is given to the manufacturing processes and applications of ferrites
in modern technology, such as their use in magnetic cores, sensors, and memory
devices. The paper concludes with a discussion on the future directions in
ferrite research and potential innovations in their applications.

##  1 Introduction

As early as 800 BC, magnetism was discovered in load stone, a naturally
occurring substance that had been used for navigation. According to
contemporary theory, magnetism exists within all materials, including
semiconductors, insulators, and metals, though in a variety of ways.
Additionally, the magnetizing properties of solids are essential, and
investigation into them has provided profound insights into the underlying
structure of many solids, both metallic and non-metallic. The magnetic field
strength, also known as the externally applied field, is denoted by H. The
magnetic induction or magnetic flux density denoted by B reflects the amount
of internal field strength inside a material as a consequence of the H field.
Tesla units are used for B. The relationship between flux density and magnetic
field strength is given by

$\bf{B}=\mu\bf{H}$ (1)

where $\mu$ indicates the permeability. The dimensions of permeability are
measured in Henries per meter (H/m) or Weber per ampere meter $(Wb/Am)$. In
vacuum

$\bf{B}=\mu_{0}\bf{H}$ (2)

where $\mu_{0}$ is the permeability of vacuum and it has a value $4\pi\times
10^{-7}(1.257\times 10^{-6})\;H/m$ When a solid is subjected to a magnetic
field and generated magnetic moment, it becomes magnetized. The magnetic
dipole moment per unit volume is the magnetization vector M. Within the
medium, there is magnetic induction as :

$\bf{B}=\mu_{0}\bf{H}+\mu_{0}\bf{M}$ (3)

where $\mu_{0}\bf{M}$ is the result of medium magnetization and
$\mu_{0}\bf{H}$ is caused by an external field. Since the field induces
magnetization, we can say that $\bf{M}$ is proportional to $\bf{H}$, hence,

$\bf{M}=\chi\bf{H}$ (4)

The magnetic susceptibility of the medium is denoted by $\chi$ the
proportionality constant $\chi$ which is defined as the amount of
magnetization produced per unit applied magnetic
field,i.e.$\chi=\bf{M}[Am^{-1}]/\bf{H}[Am^{-1}]$[unitless], is a parameter
used to measure the state of magnetic materials.

In this case we have assumed $\chi$ to be isotropic. Nevertheless actual
crystals are anisotropic therefore $\chi$ is a tensor. If Equation (4)
substituted in Equation (3), we get:

$\bf{B}=\mu_{0}\bf{H}+\mu_{0}\chi\bf{H}=\mu_{0}(1+\chi)\bf{H}=\mu\bf{H}$ (5)

Where

$\bf{\mu}=\mu_{0}(1+\chi)$

and

$\mu_{r}=1+\chi$

Based on modern theories, electron orbital and spin motions as well as
nuclei’s spins are responsible for the creation of magnetism in solids. The
magnetic effects are created by electron motion, which is analogous to an
electric current. Permanent electronic magnetic moments are created by the
spin of unpaired valence electrons, which makes a significant contribution. A
net non-zero magnetic moment can be produced by many of these magnetic moments
aligning themselves in various directions. Consequently the nature of
magnetization generated depends on the number of unpaired valence electrons
present in the atoms of the solid and on the relative orientations of
surrounding magnetic moments. The following six categories describe the
magnetism inherent in solids:

  1. i.

Diamagnetism

  2. ii.

Paramagnetism

  3. iii.

Ferromagnetism

  4. iv.

Antiferromagnetism

  5. v.

Ferrimagnetismit

  6. vi.

superparamagnetism

##  2 DIA, PARA, FERRI, ANTIFERRO, FERRO AND SUPERPARAMAGNETIC MATERIALS

###  2.1 Diamagnetism

Diamagnetism is very lacking form of magnetism and is non-permanent and
persists only in the event an external magnetic field is applied. It arises
from an alteration in the electron’s orbital motion when an external magnetic
field is applied. The induced moment has a very little magnitude. The applied
field’s direction and the induced magnetic moment’s direction are opposing.
The relative permeability is somewhat smaller than unity. The magnitude of $B$
within the diamagnetic material is smaller than that in a vacuum,
demonstrating a negative magnetic susceptibility. In cases where a diamagnetic
material is set between the poles of a powerful electromagnet it is attracted
towards regions at which field is weakened. Solid substances have a
susceptibility of around $-10^{-5}$. All materials contain diamagnetism. It
can only be seen when all other forms of magnetism are completely gone due to
its extreme weakness. It is noticed that ionic and covalent crystals exhibit
diamagnetism. Other examples include metals like copper, gold, silver, and
bismuth, organic solids like benzene and naphthalene, and atoms with rare gas
configuration like argon, neon, and helium.

Figure 1: Atomic dipole configuration for a diamagnetic material without and
with magnetic field

###  2.2 Paramagnetism

When a substance experiences a permanent dipole moment due to partial
cancellation of its electron spin and/or orbital magnetic moments, it is said
to be paramagnetized. In the absence of a magnetic field, the orientation of
these magnetic dipole moments is random, leading to no net magnetization. Once
the magnetic field is applied, the dipole moments try to align with the
magnetic field and the induced magnetization tends to increase the magnetic
field . Because the alignment forces are smaller than the forces from thermal
motion that attempt to disrupt the alignment, paramagnetism is often rather
weak. The paramagnetism is sensitive to the temperature, the lower the
temperature the stronger the effect. It seems that there is greater alignment
at low temperatures since the influence of thermal motion is reduced Since
ions and atoms with an odd number of electrons must be paramagnetic due to
their angular momentum, paramagnetic states are seen in these
materials.paramagnetic has positive and very little susceptibility on a range
of $10^{-3}$to $10^{-5}$. This behavior in the presence of a very weak
magnetic field is explained by Curie’s law

$\chi=M/H=C/T$

Here $C$ is known as the Curie constant.

Figure 2: Schematic of the alignment of magnetic moments in a paramagnetic
material: (a) shows the disordered arrangement in the absence of an external
field, and (b) shows the response when a field of moderate strength is
applied.

###  2.3 Ferromagnetism

In the absence of an external magnetic field, some metallic materials such as
Fe, Co, and Ni possess a permanent magnetic moment and show extremely strong
magnetizations. In ferromagnetic materials, atomic magnetic moments resulting
from uncancelled electron spins as a consequence of electron structure give
rise to permanent magnetic moments. Above a specifically temperature known as
Curie temperature $(Tc)$ the ferromagnetic material demonstrate paramagetism

Figure 3: The orientation of atomic dipoles for a ferromagnetic material

Weiss’ theory of ferromagnetism states that a sample of ferromagnetic material
has several tiny, spontaneously magnetic areas termed domains. The vector sum
of the magnetic moments of the individual domains determines the overall
amount of spontaneous magnetization of the specimen. The exchange field,
$B_{E}$, which tends to generate a parallel alignment of the atomic dipoles,
is what causes the spontaneous magnetization of each domain. It is assumed
that the field $B_{E}$ is proportionate to each domain’s magnetization M, thus
that

$B_{E}=\lambda{M}$

where $\lambda$ is a temperature-independent constant known as the Weiss-field
constant. In general, the applied field B is not as strong as the field
$B_{E}$, which is also referred to as the molecular field. For iron, $B_{E}$
equals 1000 Tesla. thereby, the effective magnetic field on atom transforms
into

$B_{eff}=B+B_{E}=B+\lambda M$

When an external magnetic field is present, the ferromagnetic material becomes
magnetized. This is caused by two factors:

  1. 1.

The size of the domains with favorable orientations grows, while the size of
the domains with unfavorable orientations decreases in response to the applied
field.

  2. 2.

The rotation of several domains’ magnetization orientations along the applied
field’s direction.

These two magnetization processes are illustrated in figure. 4

Figure 4: Two fundamental processes of magnetization in ferromagnetic
materials

Typically, in weak fields, domain boundary displacement causes the
magnetization, increasing the size of the favorably oriented domains. In
strong field the magnetization takes place by the rotation of domains. When
the field is removed the domain boundary do not move totally back to their
initial position, and the material stays still magnetized. At this point the
material seems permanent magnet. The ferromagnetic material becomes
paramagnetic with individual atomic dipoles when the domains split apart at
high temperatures. The susceptibility of a ferromagnetic material follows the
Curie-Weiss equation above the Curie temperature, $T_{C}$.

$\chi=\frac{C}{T-T_{c}}$

Here C is the Curie- Weiss constant

###  2.4 Antiferromagnetism:

Beyond ferromagnetic materials, other materials also exhibit this phenomenon
of magnetic moment coupling between neighboring atoms or ions. The name
”antiferromagnetism” refers to a group in which this coupling produces an
antiparallel alignment, or the alignment of nearby atoms’ or ions’ spin
moments in completely opposite directions.Manganese oxide [MnO] crystals were
the first materials that showed this type of magnetism. The material does not
show any magnetization when there is no external magnetic field because the
magnetic moments of the adjacent particles cancel each other out.
Nevertheless, when a field is applied, little magnetization becomes apparent
in the direction of the field and gets stronger further with temperature .
Below what is known as the Neel temperature $T_{N}$, antiferromagnetism takes
place.Thermal energy is enough above the Neel temperature $T_{N}$ to cause the
atomic moments oriented in the opposite direction to become randomly
distributed, which eliminates their long-range order. In this situation, the
material exhibits paramagnetic behavior.

Figure 5: The arrangement of the atomic magnets in an antiferromagnetic
material

###  2.5 Ferrimagnetism

Similar to ferromagnets, ferrimagnets display spontaneous magnetization even
in the absence of an applied field below a certain temperature, Tc.
Nonetheless, a typical ferrimagnetic magnetization curve differs greatly from
a ferromagnetic curve in shape, as seen in Fig. 6. In contrast to most
ferromagnets, which are metals, ferrimagnets are electrically insulators
simply because they are ionic solids. With the exception of the two
sublattices’ varying magnetization magnitudes, which provide a non-zero net
magnetization value, ferrimagnetism and antiferromagnetism are actually the
same

Figure 6: Comparison of magnetization and inverse susceptibility in typical
ferri and ferromagnets.

Materials like ferrites, which are fundamentally the oxides of several metal
elements, exhibit this type of magnetism and these materials may be described
using the chemical formula $MFe_{2}O_{4}$, since M can be any one of different
metallic elements .The most prevalent example is that of ferrous ferrite,
frequently referred to as magnetite $Fe_{3}O_{4}$ or $FeO$.$Fe_{2}O_{3}$.

Figure 7: Ordering of magnetic ions in a ferrimagnetic lattice

###  2.6 Ferrites

The materials referred to as ferrites are the most significant ferrimagnets in
terms of technology. Ferrites are electrically insulators and ferrimagnetic
transition-metal oxides. They therefore find use in circumstances where the
electrical conductivity exhibited by the majority of ferromagnetic materials
would be harmful. Because an ac field does not create undesired eddy currents
in an insulating material, they are frequently utilized in high-frequency
applications Ceramic processing procedures are mostly used in the
manufacturing of ferrites. For instance, powdered NiO and $Fe_{2}O_{3}$ are
combined, shaped, and heated to create $NiO.Fe_{2}O_{3}$. One benefit of this
technology is that it’s simple to regulate the magnet’s form by selecting the
appropriate mold. There are two primary families of magnetic ferrites, each
having a distinct crystal structure:

1\. Cubic. The usual formula for these is MO.$Fe_{2}O_{3}$, where M is an ion
of a divalent metal, such as Mn, Ni, Fe, Co, or Mg. While the other cubic
ferrites are magnetically soft, cobalt ferrite, $CoO.Fe_{2}O_{3}$, is
magnetically hard. These ferrites are both ancient and modern magnetic
materials as magnetite $Fe_{3}O_{4}$ , also known as iron ferrite, is the
earliest magnetic substance ever discovered by humans and is referred to as
the ”lodestone” of antiquity.

2\. Hexagonal. The most significant members of this category are the
magnetically hard barium and strontium ferrites, BaO.6$Fe_{2}O_{3}$ and
SrO.6$Fe_{2}O_{3}$.

###  2.7 superparamagnetism

Small ferromagnetic or ferrimagnetic nanoparticles exhibit a form of magnetism
known as superparamagnetism (SPM). This suggests sizes that, depending on the
substance, range from a few nanometers to a few tenths of a nanometer. These
nanoparticles are single-domain particles as well. The overall magnetic moment
of the nanoparticle may be roughly expressed as one large magnetic moment made
up of all the individual magnetic moments of the constituent atoms.
Nanoparticles frequently exhibit a preference for the direction that their
magnetization aligns with. It is claimed that these nanoparticles exhibit an
anisotropy in these directions. Uniaxial anisotropy is used to describe it
when there is just one primary direction. Uniaxially anisotropic nanoparticles
randomly reverse the direction of their magnetization. Thermal energy is what
causes this phenomenon. The relaxation time indicates the typical amount of
time needed to complete such a reverse:

$\tau={\tau_{0}}exp\left(\frac{\Delta E}{KT}\right)$

where : $\tau$is the investigated material’s characteristic length of time.
$\Delta{E}$ is the energy barrier the magnetization flip has to overcome by
thermal energy. K is the Boltzmann constant and T is the temperature However,
the energy barrier $\Delta{E}$ and temperature T are not the only factors that
affect the detection of nanoparticles in a superparamagnetic state: Every
experimental method has a unique measurement time $\tau_{m}$. Two possible
outcomes can arise based on measurement time :

$\tau_{m}<<\tau$ : The measurement time is significantly smaller than the
average time -between flips. This places the particles in a clearly defined
state that is commonly known as the system’s blocked state.

$\tau_{m}>>\tau$ : On the other hand, the measurement time may be
significantly longer than the average time between flips. This suggests that a
fluctuating condition with many unresolved magnetization spin orientations is
really observed by the experiment. In the absence of any external field, a
time-averaged net moment of zero is measured. A system is said to be in its
superparamagnetic condition in this circumstance. The temperature that
separates the blocked state from the superparamagnetic state is known as the
blocking temperature, or $T_{B}$. This indicates that at the blocking
temperature, $\tau_{m}=\tau$

$T_{B}=\frac{\Delta E}{KLn(\frac{\tau_{m}}{\tau_{0}})}$

As a result, the following characteristics allow us to differentiate the two
states:

$\tau_{m}<<\tau$ or $T_{B}>T$ is the blocked state.

$\tau_{M}>>\tau$ or $T_{B}<T$ is the superparamagnetic state.

Figure 8: Case (a): The relaxation time is significantly longer than the
measurement time, One observes a clearly defined state. Case (b): The relaxing
time is significantly smaller than the measurement time .The superparamagnetic
condition,characterized by a time-averaged net moment of zero, is caused by
the magnetization’s fluctuating state.

What consequences arise from such superparamagnetic states? The net moment is
0 in the absence of an external magnetic field. The nanoparticles behave
similarly to a paramagnet when an external field is applied, which is why the
word ”paramagnetism” is used. The only difference is that the nanoparticles’
magnetic susceptibility is significantly greater, which is why the name
”super” is used. To be clear: Any ferromagnetic or ferrimagnetic substance may
often exhibit paramagnetically. This is starting at what is known as the Curie
temperature, or $T_{C}$, and going higher. Nevertheless when superparamagnetic
behavior is seen below the Cure temperature, an other explanation is required.

##  3 FERRITES

Fundamentally, ferrites are extremely hard and brittle ceramic materials with
a dark grey or black appearance. Ferrites are magnetic materials made of
oxides with ferric ions as the primary constituent component (the term
”ferrite” is derived from the Latin ”ferrum,” which means iron) and are
categorized as magnetic materials due to their ferrimagnetic properties
.Ferrites have exceptional magnetic and electrical properties. Among their
many advantages are their wide frequency range, form flexibility,
affordability, high electrical resistivity, minimal eddy current loss, high
permeability, and time-temperature stability. The high-temperature solid-state
reaction method, sol-gel method, coprecipitation, pulsed laser deposition,
high-energy ball milling, and hydrothermal process can all be used to create
ferrites in powder or thin film form. In order to create a ferrite core, a
combination of powders comprising the component raw materials is pressed into
the desired form before being sintered into a ceramic component. The
interactions between metallic ions that occupy certain places in relation to
the oxygen ions in the oxide’s crystal structure give birth to the magnetic
characteristics.

###  3.1 Classification of Ferrites

Ferrites are categorized in the following ways based on their magnetic
characteristics and crystal structures: Fig. 9

Figure 9: Classification of ferrites

###  3.2 Based on Crystal Structure

The crystal structures of ferrites vary from one another. Ferrites may be
divided into four categories according to their crystal structures: spinel,
garnet,hexagonal, and Orthoferrites ferrite.

####  3.2.1 Spinel Ferrites [Cubic Ferrites]

A spinel is a cubic structure with the theoretical equation A$B_{2}O_{4}$,
where A denotes a divalent cation and B denotes a trivalent cation.On the
other hand, the formula $MFe_{2}O_{4}$, where M denotes divalent metal ions
such as Fe, Cu, Ni, Mg, Mn, Co, Zn, Cd, etc., describes Spinel ferrites.Al,
Cr, Ga, In, and other trivalent ions can be used in place of the $Fe^{3+}$.The
word ”ferrites” was applied to these minerals because of their structural
similarity to the naturally occurring mineral spinel, $MgAl_{2}O_{4}$. The
most significant class of magnetic materials, spinel ferrites, have several
intriguing uses. Due to their exceptional magnetic characteristics, spinel
ferrites, which are magnetically soft and a better option than metallic
magnets like Fe and layered Fe-Si alloys, perform better . They can also offer
minimal magnetic losses as well as excellent electrical resistance. The two
most common types of ceramic magnets in the spinel ferrite family are
manganese-zinc ferrites and nickel-zinc ferrites. Their high magnetic
permeability, high electrical resistivity, and potential for wide-spectrum
alteration of inherent characteristics make them attractive ceramic materials.
Spinel ferrite crystallizes in the cubic structure and is formed of a close-
packed oxygen anions organization in which 32 oxygen ions form the unit cell.
Two types of spaces are left between the anions as a result of the packing of
these anions in a face-centered cubic (FCC) arrangement: tetrahedrally
coordinated sites (A), which are surrounded by the four nearest oxygen atoms,
and octahedrally coordinated sites (B), which are surrounded by the six
nearest neighbor oxygen atoms.with two interstitial sites, tetrahedral (A) and
octahedral (B), the crystal structure is cubic with spinel-type, or
$MgAl_{2}O_{4}$. Since the unit cell consists of eight units (cubes), it may
be expressed as $M_{8}Fe_{16}O_{32}$,which was initially created by Nishikawa
and Bragg The spinel ferrite structure is shown in Fig. 10

Figure 10: Crystal structure of a spinel ferrites

(a) Classification:

Spinel ferrites are divided into three forms according to the distribution of
cations in the two main sites, tetrahedral site (A) and octahedral site (B).

Figure 11: Types of spinel ferrites

(i) Normal spinel ferrites:

When $2Fe^{3+}$ ions are at the octahedral (B) site and divalent metal ions
are at the tetrahedral (A) site, the ferrite is referred to as normal spinel.
Commonly, cubic close-packed oxides with four octahedral and eight tetrahedral
sites per formula unit make up normal spinel structures. Zinc
(Zn$Fe_{2}O_{4}$) and cadmium (Cd$Fe_{2}O_{4}$) ferrites, where the divalent
metallic ions $Zn^{2+}$ or $Cd^{2+}$are at the (A) site and $Fe^{3+}$ ions are
at the (B) site, are the greatest examples of typical spinel ferrites. In
general,

$(M)_{A}[Fe_{2}]_{B}O_{4}$

can be employed to represent the cation distribution.

(ii) Inverse spinel ferrites:

One trivalent ferric ion, $Fe^{3+}$, is located at the tetrahedral (A) site in
inverse spinel ferrite, whilst the remaining trivalent ferric ions, $Fe^{3+}$,
and the divalent metallic ions, $M^{2+}$, are located at the (B) site. This
group really includes the majority of simple ferrites, such as cobalt ferrite
(Co$Fe_{2}O_{4}$), nickel ferrite (Ni$Fe_{2}O_{4}$), and ferric oxide
($Fe_{3}O_{4}$) . In inverse spinel ferrite, the cation distribution is
represented as

$(Fe)_{A}[MFe]_{B}O_{4}$

.

(iii) Random spinel ferrites:

The ferrite is known as random spinel ferrite when the divalent metal ions
$M^{2+}$ and trivalent $Fe^{3+}$ ions are dispersed at both the tetrahedral
(A) and octahedral (B) sites. Additionally, of the random spinel ferrites,
copper ferrite ($CuFe_{2}O_{4}$) is the most well-known example.Ionic radii,
their electronic configuration, and the electrostatic energy of the lattice
all play a part in the delicate balance of contributions that determine the
distribution of ions between two types of sites . Random spinel’s cation
distribution is often represented by

$(M_{1-x}Fe_{x})_{A}[M_{x}Fe_{2-x}]_{B}O_{4}$

.

####  3.2.2 Hexagonal Ferrites

Went, Rathenau, Gorter, and Van Oostershout (1952) and Jonker, Wijn, and Braun
(1956) made the initial identification of hexaferrite. Rhombohedral
ferromagnetic oxides are another name for hexagonal ferrites. The formula
$MFe_{12}O_{19}$, where M is an element such as barium (Ba), strontium (Sr),
calcium (Ca), or lead (Pb), is often used to identify hexagonal ferrites.These
ferrites have a strong coercive permanent (hard) magnet-i.e.with typical
coercivities of around 200 kA/m- characteristic with oxygen ions having a
closely packed hexagonal crystal structure. Because it is difficult to readily
switch the axis of magnetization in hexagonal ferrites, they are known as hard
ferrites.In addition, unlike cubic ferrites,hexagonal ferrites have several
types, including M-type hexaferrite, W-type hexaferrite, Y-type hexaferrite,
Z-type hexaferrite,and U-type hexaferrite,as shown in Table. 1.

Sr. no. |  Hexaferrite |  Chemical formula
---|---|---
1 |  M-type |  $AFe_{12}O_{19}$
2 |  Y-type |  $A_{2}Me_{2}Fe_{12}O_{22}$
3 |  W-type |  $AMe_{2}Fe_{16}O_{27}$
4 |  X-type |  $A_{2}Me_{2}Fe_{28}O_{46}$
5 |  U-type |  $A_{4}Me_{2}Fe_{36}O_{60}$
6 |  Z-type |  $A_{3}Me_{2}Fe_{24}O_{41}$
Table 1: Different types of hexaferrites.

A few of instances of hexaferrite include strontium ferrite
($SrFe_{12}O_{19}$) and barium ferrite ($BaFe_{12}O_{19}$). In terms of
technology and commerce, hexagonal ferrite is now a very significant material.
Of the hexagonal ferrites, barium ferrite ($BaO.6Fe_{2}O_{3}$) is the most
significant. The crystal structure of barium ferrite is hexagonal
magnetoplumbite (Fig.12). The magnetoplumbite structure is made up of four
building blocks, denoted S, $S^{*}$, R, and $R^{*}$ in the figure, and has ten
oxygen layers in its primary unit cell. The S and$S^{*}$ blocks are spinels
with two oxygen layers and six $Fe^{3+}$ ions. Two of the $Fe^{3+}$ ions are
in tetrahedral sites, having opposite spin direction to the octahedral iron
ions, while the remaining four are in octahedral sites with their spins
aligned parallel to each other (say up-spin).

Figure 12:  Structure of barium ferrite.

The S and $S^{*}$ blocks are equivalent nevertheless rotated $180^{\circ}$
with respect to each other. The R and $R^{*}$ blocks are made up of three
oxygen layers, with a barium ion taking the place of one of the oxygen anions
in the middle layer. Each R block consists of six $Fe^{3+}$ ions, five of
which are in octahedral sites with three up-spin and two down-spin, and one of
which is coordinated by five $O^{2-}$ anions and has up-spin. Each unit cell
has a net magnetic moment of 20$\mu$B. Similar to cubic ferrites, they are
inexpensive to make using ceramic processing techniques and are easily
powdered and shaped into whatever shape one wants.

####  3.2.3 Garnet Ferrites:

Researchers’ attention has recently been drawn to garnet-based nanoferrites
due to their wide range of applications. In simple terms, garnets are minerals
with the general formula $X_{3}Fe_{5}O_{12}$, which contains two magnetic
ions: an iron ion and a rare earth element (where ”X” is an element such as
Sm, Eu, Gd, Tb, Dy, Er, Tm, Lu, and Y). Garnets have dodecahedral
(12-coordinated) sites in addition to tetrahedral and octahedral sites which
are magnetically hard Their unit cell has a cubic form with edges that are
around 12.5 Å long. One well-known garnet is yttrium iron garnet, or
$Y_{3}Fe_{5}O_{12}$.The garnets’ crystal structure is orthorhombic, with
oxygen polyhedral surrounding the cations. However, trivalent cations, such as
Fe3+ and rare earth, occupy tetrahedral (d), octahedral (a), or dodecahedral—a
12-sided distorted polyhedral (c) site. In particular, the net magnetic moment
is antiparallel to the rare earth ions on the ”c” sites, as well as the
interaction between the tetrahedral and octahedral sites is antiparallel.
Among all crystal structures, garnet structure is one of the most complex,
making it challenging to depict in two dimensions while include all 160 ions
in the unit cell. Fig.13 only displays an octant of a garnet structure, which
just displays the cation locations, for the reason of simplicity. The garnet
structure is consists of a combination of octahedral (trivalent cation
surrounded by six oxygen ions), tetrahedral (trivalent cations surrounded by
four oxygen ions), and 12-sided polyhedral dodecahedral (trivalent cations
surrounded by 8 oxygen atoms) sites, the orientations of which are shown in
Fig.12. The finding of a complicated crystal structure is noteworthy due to
its applicability in memory devices.

Figure 13: Crystal structure of Pyrope (Garnet) in three dimensions Figure 14:
Schematic octant of a garnet crystal structure with only cation positions
Figure 15: An octant of a garnet crystal structure

####  3.2.4 Orthoferrites:

Orthoferrites are denoted by the symbol MFeO3, in which M stands for one or
more rare earth elements. Orthoferrites are lightly ferromagnetic and have an
orthorhombic crystal structure. Examples of orthoferrite materials are
dysprosium orthoferrite ($DyFeO_{3}$), lanthanum orthoferrite (LaFe$O_{3}$),
and praseodymium orthoferrite (PrFe$O_{3}$). Orthoferrites have application in
gas separators, catalysts, spin valves, magneto-optic materials, optical
Internet, communication systems, cathodes in solid oxide fuel cells, and
electrical circuits. In Fig. 17, the perovskite structure is illustrated. Tiny
trivalent or tetravalent metal ions (B) occupy the cube’s core, whereas large
divalent or trivalent metal ions (A) occupy the cube’s corners. On the cube’s
faces, the oxygen ions are positioned in the center

Figure 16: Structure of an orthoferrite perovskites

###  3.3 According to Magnetic Properties

Ferrites demonstrating spontaneous magnetization are classified into two
groups based on their magnetic properties: soft ferrites and hard ferrites.

####  3.3.1 Soft Ferrites:

Ferrimagnetic materials that lose their magnetism after being magnetized are
known as soft ferrites. Soft ferrites have little energy losses during
magnetization, a short and tiny hysteresis loop, and a sharply rising
magnetization curve. These materials are easily magnetized and demagnetized
and may be created by heating and slowly cooling. Soft ferrites are ceramic
insulators that have the cubic crystal structure $MFe_{2}O_{4}$, where M
stands for transition metal ions like Zn, Ni, and Mn. Low eddy current losses,
low retentivity and coercivity, and high susceptibility and permeability
(Nickel Zinc ferrite, for instance, has 1.26 ×$10^{-5}$-2.89 × $10^{-3}$ H/m)
are characteristics of soft ferrites .Moreover, soft ferrites with cubic
spinel structures known as NiZn, MnZn, while MgMnZn ferrites, and soft
ferrites with garnet structures known as YIG. Soft ferrites are therefore
utilized in electromagnets, computer data storage, transformer cores,
telephone signal transmitters and receivers, and other devices.

####  3.3.2 hard ferrites:

Because hard ferrites retain their magnetism even after being magnetized, they
are also referred to as permanent magnetic materials. Hard ferrites are
ferrimagnetic materials characterized by a large hysteresis loop, a gradually
increasing magnetization curve, as well as large energy losses during the
magnetization process. Hard ferrites are difficult to magnetize and
demagnetize and can be created by heating and abruptly cooling. Iron and
barium or strontium oxides combine to generate hard ferrites.Furthermore, the
structure of hard ferrite is hexagonal. Hard ferrites have strong retentivity
and coercivity, high eddy current losses, low susceptibility and permeability,
and high saturation flux density. Hard ferrites are therefore utilized in loud
speakers, DC magnets, permanent magnets, etc.

Figure 17: shows the hysteresis loop[M-H]of (a) soft ferrite,( b) hard ferrite

####  3.3.3 Fundamental Applications Based on the Properties

As a result of their higher resistivity, reduced cost, ease of production, and
superior magnetization properties, ferrites are thought to be more effectively
magnetic materials than pure metals. Ferrites are widely employed in satellite
communication, radar, bubble devices, computer memory cores, audio-video and
digital recording, and microwave devices. Applications for ferrite are
several, ranging from radio frequencies to microwaves. It is utilized in radio
reception antenna cores, TV image tube fly back transformers, broad band
transformers, mechanical filters, ultrasonic generators, and isolators,
moderators, and phase shifters. Ferrite is employed in computers, control
devices, and telephone exchanges these days. Additional applications include
magnetrons, couplings, and zero friction bearings in aviation and space
technology; speakers and motors for household appliances; sensors, routers,
and converters in electrotechnology; and driving engines, igniters,
ventilation systems, speed indicators, and existing power stations in electric
drive trains. Clocks, ear buds, and magnetic locks are used in medical; in
manufacturing, there are magnetic prostheses, cancer cell artificial organs,
nuclear magnetic resonance apparatus, separators, and audiovisual analogue
devices.

##  4 Nature Of Domains

The theory of magnetic domains inside solids was first presented by Weiss in
1907.Any ferromagnetic material, according to Weiss, is made up of tiny volume
regions where all magnetic dipole moments are mutually aligned in the same
direction. Every one of these areas is magnetized to its saturation
magnetization and is typically referred to as a domain.The domain walls or
boundaries that divide adjacent domains cause the magnetization to
progressively change to a different direction. Bloch wall is the term for this
wall or boundary. A portion of an unmagnetized domain contains several domains
that are all spontaneously magnetized, but the orientation of the
magnetization in each domain varies randomly. Each domain has around
$10^{17}$–$10^{21}$ atoms. In the event of an external magnetic field, these
domains align. When an external magnetic field is present, the energy of the
domains with net magnetic moments parallel to the field’s direction decreases
while the energy of the domains without such moments increases. If every
domain is parallel to the applied magnetic field, the energy of the crystal
can be reduced. Here are the methods via which this can be acquired:

(i) A domain’s magnetization direction can change all at once.

(ii) A favorably oriented domain expands at the expense of less favorably
oriented domains. (Fig.18)

(iii) The magnetization rotates in the direction of the applied magnetic field
under strong magnetic fields (Fig.18 )

Figure 18: Domain structure

Single crystals’ magnetization in a certain direction is always followed by a
change in their physical dimensions. Depending on the crystal, it can stretch
or contract along the magnetization direction. It contracts for Ni and expands
for iron. We refer to this phenomena as magnetostriction. Bloch wall thickness
is limited, i.e., spin orientation progressively changes in the transition
region (Fig.19). It is not endlessly tiny. Because the spin reversal is
achieved in various steps, there is not much difference in spin orientation
between two neighboring moments. As a result, the exchange energy associated
to the wall decreases.

Figure 19: Bloch wall

It has been noted that in a ferromagnet, there are two crystallographic
directions: one for easy magnetization,i.e.soft ferromagnetic materials, where
saturation is obtained spontaneously, and another for hard
magnetization,i.e.hard ferromagnetic materials, where saturation requires the
application of an external field. The directions of easy and hard
magnetization are [111] and [100] for nickel, respectively, whereas the
opposite is true for iron. Magnetic anisotropy is the name given to this
phenomenon. The magnetic anisotropy energy is the difference in energy between
the easy and hard directions. Because more dipoles point in the hard direction
when a wall is thicker, this energy has the effect of decreasing the thickness
of the Bloch wall. Anisotropic energy favors a thin wall whereas exchange
energy favors a thick wall. Little transverse domains are present at the end
of the sample, as can be seen by closely examining the domain structure
(Fig.20). These are known as closure domains because they further reduce
magnetostatic energy by effectively completing the magnetic loop between two
neighboring domains. ”Bitter powder patterns,” which are created by placing a
drop of a colloidal solution of ferromagnetic particles on the specimen’s
meticulously prepared surface, serve as experimental proof that domains exist.
Local magnetic fields close to the domain borders are quite strong. There, the
particles are gathered, and a microscope may be used to view the domain.

Figure 20: Closure domains

###  4.1 Magnetism and hysteresis

The spontaneous magnetization of ferromagnetic materials, like iron or
magnetite, is the most amazing demonstration of magnetism in a solid.
Typically, spontaneous magnetism is linked to hysteresis, a phenomena that
James Ewing first observed and termed in 1881.For ferromagnets and
ferrimagnets, flux density B and the field density H are not proportional. B
changes as a function of H if the substance is originally unmagnetized. The
Curie point—a point at which a material’s properties rapidly change—is used to
describe ferromagnetic substances. The susceptibility follows a Curie-Weiss
equation (Eq. ), roughly, and is independent of field strength above the Curie
temperature. The behavior differs greatly below the Curie temperature. Very
small fields can produce very large amounts of magnetization, and the
relationship between magnetization and field strength is quite nonlinear. A
typical plot of magnetic induction B as a function of field H in an iron
sample is displayed in Figure 18.

Figure 21: B versus H hysteresis curve

When a slowly rising field is given to initially unmagnetized iron, B travels
along the OAS magnetization curve.The value of B is nearly constant at 1.5
Wb/m2 in a field of a few hundred ampere/meter. Reducing the magnetic field H
causes the induction B to follow the path SR rather than the magnetization
curve. B remains close to the saturation value even at H = 0, which
corresponds to point R in Fig. 18. The retention of magnetization in zero
field is referred to as remanence, and the value of B at this point is known
as the residual induction.The value of B decreases and eventually becomes zero
(point C) when a reverse field is applied; the field reverse induction value
is then set up and rapidly approaches the saturation value. We refer to
coercivity as the reverse field H at which B = 0. When a positive field is
introduced and the reverse field is gradually eliminated, the induction
eventually draws out the curve in the direction of S’BS. The hysteresis curve
is the name given to this RCS’BSR curve. It demonstrates that variations in
the applied magnetic field are always preceded by variations in the magnetic
induction (B).Based on Weiss’s domain theory, the B-H curve makes sense. The
domains in an unmagnetized polycrystalline sample are randomly orientated,
resulting in the absence of any magnetic moment in any direction. When a field
is applied, domains with magnetization that is parallel to the field or at a
slight angle to it grow at the expense of domains with antiparallel or nearly
parallel magnetization, shifting the boundary between the two domains. At the
beginning (OA in the B-H curve), the material’s magnetization advances through
minor (reversible) boundary displacements; however, the larger (irreversible)
displacements are responsible for the steeper portion AB of the magnetization
curve.Magnetization above the curve’s knee occurs by rotation of the
magnetization direction of overall domains; this is a challenging process with
a comparatively sluggish rate of increase in magnetization. The magnetization
stays relatively high until reverse fields are applied because there is
limited change in the domain structure when the applied field reduces, which
results in the hysteresis previously mentioned.The amountof work required to
remagnetize a ferromagnet with a unit volume is proportional to the area of
the B-H curve. During the remagnetization process, all of the work is
converted to heat.

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