scalars, for which manifest global symmetry is again $\text{U}(1)^{N-1}\times
S_{N}$. The rest of the symmetries are non-linearly realized and not obvious
Coleman:1976uz . More sophisticated tools such as non-abelian bosonization can
make global symmetries manifest.

The vortex (which are instantons in 2d) operators in 2d are given by
$e^{-Y_{i}}$. Integrating out $\Sigma$ at finite $e^{2}$, similar to (167),
one obtains a mass for $Y_{1}+\ldots+Y_{N}$ combination. This mass is given by
$e\sqrt{|r_{0}|}$ where $r_{0}$ is bare FI-parameter. The modes along the
vacuum manifold, which remain massless to all orders in perturbation theory,
acquire masses non-perturbatively, a mass of the order of the strong scale
$\Lambda$ of the sigma model. The precise mechanism of this effect will be
described below, both by using tools of supersymmetry, and as well as by using
cluster expansion.

In the sigma model limit, $e\gg\Lambda$, and because of the suppression of the
kinetic term, $-{1\over 2e^{2}}\overline{\Sigma}\Sigma$, the content of the
$\Sigma$ field becomes non-dynamical. As such, we can integrate out $\Sigma$,
producing the constraint $\sum_{i}Y_{i}=t$. Equivalently, we can say that the
mass term for the $\sum_{i}Y_{i}$ mode forces it to its vacuum value, given by
the constraint. Either way, one obtains the superpotential in terms of $N-$1
dynamical $Y_{i}$ fields:

$\displaystyle\widetilde{W}\,=\,\mu\,\sum_{i=1}^{N}e^{-Y_{i}},$ (195)

This superpotential has $N$ minima, obtained by solving

$\displaystyle\partial_{Y_{i}}\widetilde{W}=0\rightarrow
e^{-Y_{i}^{*}}=e^{-t+Y_{1}^{*}+\ldots+Y_{N-1}^{*}}\Longrightarrow
e^{-Y_{i}^{*}}=e^{-t/N}e^{i\frac{2\pi k}{N}},\qquad k=1,\ldots,N$ (196)

Let $\widetilde{Y}_{i}$ denote the fluctuations around the minima, using
$Y_{i}=\frac{t}{N}+\widetilde{Y}_{i}$. Then, using dimensional transmutation,

$\displaystyle\mu e^{-t(\mu)/N}\equiv\mu
e^{-\frac{4\pi}{g^{2}(\mu)N}+i\theta/N}=\Lambda$ (197)

where $\Lambda$ is the renormalization group invariant strong scale of the
theory, we can rewrite the superpotential as:

$\displaystyle\widetilde{W}\,=\Lambda\;\sum_{i=1}^{N}e^{-\widetilde{Y}_{i}},\qquad\qquad\sum_{i=1}^{N}\widetilde{Y}_{i}=0$
(198)

This is the famous result of , from which some of the most interesting aspects
of non-perturbative physics follows, such as mass gap and chiral symmetry
breaking. In the bulk of the paper, we rederive this result by our refined
instanton analysis of ${\cal N}=(2,2)$ theory.

A nice basis for computations: The superpotential is written in a basis in
terms of $N-1$ complex physical fields,
$(\tilde{y}_{1},\tilde{y}_{2},\ldots,\tilde{y}_{N-1})$, and we denote
$y_{N}=-\sum_{i=1}^{N-1}\tilde{y}_{i}$. Recall that standard irreducible
representation of $S_{N}$ is indeed $N-1$ dimensional.

Below, we will use a parametrization of $Y$ with one (complex) redundant
component, which decouples from the dynamics. We use $y=(y_{1},\ldots,y_{N})$
fields with no restriction on them. The point is the mode associated with
$y_{1}++y_{N}$ totally decouples from our description. In our description, we
can write

$\displaystyle\widetilde{Y}_{i}=\bm{\nu}_{i}\cdot Y,\qquad i=1,\ldots,N$ (199)

where a convenient basis for computations is given by,

$\displaystyle{\bm{\nu}}_{i}={\bm{e}}_{i}-\frac{1}{N}\sum_{j=1}^{N}{\bm{e}}_{j},\qquad({\bm{e}}_{i})_{j}=\delta_{ij},\qquad({\bm{\nu}}_{i})_{j}=\delta_{ij}-\frac{1}{N},\qquad
i,j=1,\ldots,N$ (200)

We can rewrite the superpotential in terms of these variables as:

$\displaystyle\widetilde{W}(Y)\,=\,\Lambda\,\sum_{i=1}^{N}e^{-{\bm{\nu}}_{i}.Y}=\,\Lambda\,\sum_{i=1}^{N}e^{-\left(1-\frac{1}{N}\right)Y_{i}+\frac{1}{N}\sum_{j\neq
j}Y_{j}}$ (201)

##  Appendix B Instanton measure

In this appendix, we determine the measure over the instanton collective
coordinates in 2d bosonic $\mathbf{CP}^{N-1}$ model and its generalization
with the inclusion of $n_{f}$ Dirac fermions. The case $n_{f}=1$ is ${\cal
N}=(2,2)$ supersymmetric theory, and $n_{f}\geq 2$ are multi-flavor models.
Instantons in the $n_{f}$-flavor model have $2N$ bosonic and $2Nn_{f}$
fermionic zero modes.

The measure of the $k=1$ instanton can be written as a product over the
bosonic and fermionic zero modes as

$\displaystyle e^{-\frac{4\pi}{g^{2}(M_{0})}+i\theta}d\mu_{\rm B}\;d\mu_{\rm
F}\;.$ (202)

Here, $g^{2}(M_{0})$ is the bare coupling at the Pauli–Villars scale $M_{0}$.
The measure over the bosonic and fermionic coordinates are, respectively,

$\displaystyle d\mu_{\rm B}$
$\displaystyle=M_{0}^{2N}\;d^{2N}a\;\left(\sqrt{S_{0}}\right)^{2N}\frac{1}{\sqrt{{\det}^{\prime}(M_{b})}}$
(203) $\displaystyle d\mu_{\rm F}$
$\displaystyle=\prod_{a=1}^{n_{f}}M_{0}^{-N}d^{2N}\xi_{a}\;(\sqrt{S_{0}})^{-2N}{{\det}^{\prime}(M_{f})}$
(204)

Each bosonic zero mode is associated with a factor of $M_{0}$ and
$S_{0}^{1/2}$. $[\det^{\prime}(M_{b})]^{-1/2}$ denotes determinant over the
bosonic fluctuations around the instanton, where prime stands for the removal
of zero modes. Similarly, each fermionic zero mode is associated with
$M_{0}^{-1/2}$ and $S_{0}^{-1/2}$, and determinant denotes fermionic
fluctuations around the instanton.

The $2N$ bosonic moduli parameters of an instanton are usually decomposed into
$x\in\mathbf{R}^{2}$ the center position, $\rho\in\mathbf{R}^{+}$ the size,
and $\Omega\in M_{2N-3}$ the angular moduli, giving $2+1+(2N-3)=2N$ for its
dimension. In this parametrization, the measure is

$\displaystyle d^{2N}a=d^{2}x\;d\rho\rho^{2N-3}\;d\Omega$ (205)

The fluctuation determinants over the bosonic and fermionic non-zero modes are
related and can be combined as:

$\displaystyle[\det(M_{b})]^{-1/2}[\det(M_{f})]^{n_{f}}$
$\displaystyle=[\det(-D^{2}){\bm{1}}_{N}]^{n_{f}-1}=(M_{0}\rho)^{(n_{f}-1)N}$
(206)

The determinants cancel exactly for $n_{f}=1$ theory, which is the
supersymmetric case.

The explicit $M_{0}$ dependence in (204) must cancel by the implicit
dependence that enters through the bare coupling $g_{0}=g(M_{0})$, and turn
into the physical strong scale through dimensional transmutation. To see this,
let us collect powers of the Pauli–Villars scale $M_{0}$. It has contributions
from the bosonic and fermionic measures $M_{0}^{2N-n_{f}N}$, and also from the
bosonic and fermionic fluctuation operators, $(M_{0})^{(n_{f}-1)N}$ that
combines to give

$\displaystyle\underbrace{M_{0}^{2N-n_{f}N}}_{\rm
measure}\cdot\underbrace{M_{0}^{(n_{f}-1)N}}_{\rm
fluc.\;op.}=M_{0}^{N}=M_{0}^{\beta_{0}}$ (207)

where $\beta_{0}=N$ is the renormalization group beta function coefficient of
the $n_{f}$ flavor $\mathbf{CP}^{N-1}$ model. It is well-known from standard
perturbative calculation that the one-loop beta functions of the non-linear
sigma models is independent of the number of fermionic species. This fact also
arises from instanton calculus thanks to the cancellation between the factors
in the measure and the ones in fluctuation determinant. The combination gives

$\displaystyle M_{0}^{\beta_{0}}\times
e^{-\frac{4\pi}{g^{2}(M_{0})}}=\Lambda^{N}$ (208)

where $\Lambda$ is the renormalization group invariant strong scale of the
theory. Combining various factors together we obtain:

$\displaystyle{\Lambda^{N}}\int\prod_{a=1}^{n_{f}}d^{2N}\xi_{a}\;\;\Omega_{2N-3}\int{d^{2}xd\rho}\;\underbrace{\rho^{2N-3}}_{\rm
measure}\;\underbrace{\rho^{(n_{f}-1)N}}_{\rm fluc.\;det.}$ (209)

An immediate physical implication of this result is the non-perturbative
(topological theta angle dependent) contribution to vacuum energy. For the
bosonic case, (209) reduces to

$\displaystyle\Lambda^{N}\int{d^{2}xd\rho}\rho^{N-3},$ (210)

giving rise to the infamous infrared divergent contribution to vacuum energy
density:

$\displaystyle{\cal E}_{\rm vac}\sim-\Lambda^{2}(e^{i\theta}+{\rm c.c.})\int
d(\Lambda\rho)(\Lambda\rho)^{N-3}$ (211)

as $\rho\rightarrow\infty$. Of course, this is unacceptable. Part of our
refined instanton analysis is meant to fix this pathological behavior of the
standard instanton analysis.

##  Appendix C Fluctuation determinant for instantons and quantum interactions

In the bosonic $\mathbf{CP}^{N-1}$ model, the fluctuation determinant can be
written by using the two different parametrizations we described in the paper.
One is to express it as a function of the size modulus $\rho$ and the other is
to use the coordinates $a_{1},\ldots,a_{N}$ for the constituent locations. By
using the mapping between these two (24):

$\displaystyle[\det(M_{b})]^{-1/2}$ $\displaystyle=(M_{0}\rho)^{-N}$ (212)
$\displaystyle=M_{0}^{-N}\Big{(}{\textstyle\frac{1}{N}}\sum_{i}|a_{i}-a_{\rm
c}|^{2}\Big{)}^{-\frac{N}{2}}$ (213)
$\displaystyle=M_{0}^{-N}\Big{(}{\textstyle\frac{1}{2N^{2}}}\sum_{i,j}|a_{i}-a_{j}|^{2}\Big{)}^{-\frac{N}{2}}$
(214)

It is also possible to reinterpret the effect of the fluctuation determinants
in two complementary way. Recall that the classical interaction between the
fictitious constituent in our construction is zero, $V_{\rm int,cl.}=0$,
consistent with the existence of the classical moduli space. However, quantum
mechanically, the moduli space may be lifted. Indeed, the determinant can be
rewritten as $e^{-V_{\rm int,1-loop}}$ where $V_{\rm int,1-loop}$ is 1-loop
induced potential. One can write the instanton measure in the bosonic theory
($n_{f}=0$ in (209)) as:

$\displaystyle\Lambda^{N}\;\int\prod_{i=1}^{N}{d^{2}a_{i}}\;e^{-V_{\rm
int,1-loop}}\qquad{\rm where}$ (215) $\displaystyle V_{\rm int,1-loop}$
$\displaystyle=\frac{N}{2}\log\left[\sum_{i=1}^{N}|a_{i}-a_{\rm
c}|^{2}\right]+c_{1}$ (216)
$\displaystyle=\frac{N}{2}\log\left[\sum_{i,j=1}^{N}|a_{i}-a_{j}|^{2}\right]+c_{2}$
(217)

where $c_{1}=\frac{N}{2}\log N$ and $c_{2}=\frac{N}{2}\log(2N^{2})$ are
$N$-dependent constants.

In the theories with fermions, the nature of the quantum-induced interactions
between constituents change. The net effect of introducing fermions on the
1-loop potential is

$\displaystyle V_{\rm int,1-loop}^{n_{f}}$ $\displaystyle=(1-n_{f})V_{\rm
int,1-loop}.$ (218)

As expected, the one-loop potential vanishes for the supersymmetric theory.
This allows us safely proceed using just classical interactions between
constituents. Furthermore, the sign of the quantum-induced interactions is
flipped compared to that in the bosonic case in the multi-flavor theories.
Note that this is in contrast to the classical interactions between
constituents (47), which are independent of the number of fermion flavors
$n_{f}$.

In the interaction potential (217), the sums over constituent labels are
inside the logarithms, not outside. This tells us that the loop-induced
interaction between the parameters of the moduli, for $N>2$, cannot be written
as a sum of two-body interactions. It is a genuine $N$-body interaction. Only
for $N=2$ can the one-loop potential be written as a two-body potential:

$\displaystyle V_{\rm int,1-loop}$
$\displaystyle=(2-2n_{f})\log|a^{1}-a^{2}|.$ (219)

Some remarks are in order.

First, note that the classical interactions between constituents are robust
and are two-body interactions, while the nature of the quantum one-loop
interactions changes with the number of fermionic flavors. Indeed, for $N>2$,
the quantum interactions are genuine $N$-body interactions not reducible to
two-body interactions.

Another observation is that Berg:1979uq  starts with vacuum energy density
${\cal E}_{\rm vac}\sim\int d\rho\rho^{N-3}$ which derives from one-loop
fluctuation determinant $[\det(M_{b})]^{-1/2}=(M_{0}\rho)^{-N}$ (their
equation (1)). The fluctuation determinant can also be viewed as a one-loop
potential for $\rho$ moduli. It is $e^{-V_{\rm int}}=e^{-N\log(M_{0}\rho)}$.
Ref. Berg:1979uq  ends, after tedious calculation, with a fluctuation
determinant for a general instanton in terms of parameters
$c_{\alpha},a_{\alpha},\alpha=1,\ldots N$ (their equation (94)). But in fact,
$c_{\alpha}$ there are not moduli but encodes the $v$ vector which correspond
to the boundary condition of the field at spacetime infinity,
$|z|\rightarrow\infty$, and in particular, one does not integrate over them.
In our construction, we set $c_{\alpha}=1$ in order to make invariance under
the permutation group $S_{N}$ (which is the Weyl group of $\text{SU}(N)$)
global symmetry manifest. With this, eq.(94) for the one-loop potential in
Berg:1979uq  reduces to our (217). In other words, the simple formula
Berg:1979uq  starts with (their equation (1)) and the complicated-looking
expression it ends with (their equation (94)), are actually the same
expressions in two different parametrizations of the moduli space, related via
the mapping (24).

As a final remark, we note that although the fluctuation determinant (and
hence one-loop potential) evaluated in Fateev:1979dc  and Berg:1979uq  is
correct, it is incorrect to built an effective field theory just based on it.
Refs.Fateev:1979dc  and Berg:1979uq  argue to do so because the _classical_
interactions between instantons (and hence instanton constituents) are zero
and the leading interactions seem to be _quantum_ , starting at one-loop
level. However, the classical Coulomb interaction between constituents of
${\cal I}$ and $\overline{\cal I}$ are non-zero, and these interactions
($O(1/g^{2})$) are parametrically stronger than the quantum ones ($O(g^{0})$).
One can in principle just describe a gas of instantons with quantum
interactions, but this has nothing to do with the $\mathbf{CP}^{N-1}$ model.
As we showed explicitly, and as the mirror symmetry construction of the
Landau-Ginzburg dual formulation shows, the statistical field theory is based
on the classical interactions between the constituents of ${\cal I}$ and
$\overline{\cal I}$.

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# A convection-diffusion problem with a large shift on Durán meshes

Mirjana Brdar<EMAIL_ADDRESS>Sebastian Franz  sebastian.franz@tu-
dresden.de Hans-Görg Roos<EMAIL_ADDRESS>Faculty of Technology
Novi Sad, University of Novi Sad, Bulevar cara Lazara 1, 21000 Novi Sad,
Serbia  Institute of Scientific Computing, Technische Universität Dresden,
Germany  Institute of Numerical Mathematics, Technische Universität Dresden,
Germany

###### Abstract

A convection-diffusion problem with a large shift in space is considered.
Numerical analysis of high order finite element methods on layer-adapted Durán
type meshes, as well as on coarser Durán type meshes in places where weak
layers appear, is provided. The theoretical results are confirmed by numerical
experiments.

###### keywords:

spatial large shift , singularly perturbed , Galerkin method , Durán mesh AMS
Mathematics Subject Classification (2010): 65M12, 65M15, 65M60

##  1 Introduction

Singularly perturbed problems with delay are differential equations that allow
past actions to be included in mathematical models, bringing the model closer
to real-world phenomena. Their solution depends not only on on the solution at
a current stage, but also on the solution at some past stages. This type of
problem occurs in the control theory and biosciences [3, 4, 12], and in the
study of chemostat models, circadian rhythms, epidemiology, the respiratory
system, tumour growth and neural networks. Although singularly perturbed
problems have been studied intensively in recent years, there are not many
papers on these problems with a on these problems with large displacements.

Here we consider the following convection-diffusion problem with a large
shift: Find $u$ such that

$\displaystyle-\varepsilon
u^{\prime\prime}(x)-b(x)u^{\prime}(x)+c(x)u(x)+d(x)u(x-1)$
$\displaystyle=f(x),\quad x\in\Omega=(0,2),$ (1a) $\displaystyle u(2)$
$\displaystyle=0,$ (1b) $\displaystyle u(x)$ $\displaystyle=\Phi(x),\quad
x\in(-1,0],$ (1c)

where $b\geq\beta>0,$ $d\geq 0$ and $0<\varepsilon\ll 1$ is a small
perturbation parameter. We assume $\displaystyle
c-\frac{b^{\prime}}{2}-\frac{\|d\|_{L^{\infty}(1,2)}}{2}\geq\gamma>0,$ as well
as $\Phi(0)=0$, which is not a restriction for the function $\Phi,$ since this
condition can always be guaranteed by a simple transformation. Thus, it holds
$u\in H_{0}^{1}(\Omega).$

For convection dominated singularly perturbed problems with an additional
shift you can find only a few paper, see [8, 10, 11], where authors consider a
negative coefficient $d$, which supports a maximum principle. Then they use
the finite-difference method on layer-adapted meshes. Here we consider finite
element methods of arbitrary order for positive coefficients $d$. The solution
decomposition for this kind of problem we proved in [1], as well as the error
analysis on a standard S-type mesh. Our problem has an exponential layer near
$x=0$ and one weak layer near $x=1$ whose appearance is caused by the delay.

Numerical methods for singularly perturbed problems typically use some known
behavior of the exact solution to design a priori adapted meshes in order to
approximate boundary layers well. Probably the most well-known approximations
of this kind are those based on the so-called Shishkin meshes (see [9] and
references therein). Durán and Lombardi in [2] introduced a new graded mesh in
order to solve a convection-diffusion model problem by standard bilinear
finite elements. They proved an (almost) optimal error estimate. Up to the
logarithmic factor, those estimates are valid uniformly in the perturbation
parameter. Hence, the graded meshes could be an excellent replacement for the
well-known piecewise uniform Shishkin mesh which has transition point(s)
dividing a domain on coarse and fine mesh regions. Indeed, according to some
numerical experiments, the graded mesh procedure seems to be more robust in
the sense that the numerical results are not strongly affected by variations
of parameters which define the mesh. This is the reason why we used the Durán
mesh in the paper. Since the problem has a weak layer, we constructed a
coarser Durán mesh based on [1, 7].

The outline of the paper is as follows. In Section 2, we give the solution
decomposition and introduce the recursively defined graded mesh. The following
section contains a description of the finite element method and error analysis
for the stationary singularly perturbed shift problem and the main convergence
result. In Section 4 we present a coarser mesh of Durán-type and numerical
analysis on this mesh for problem (1). Finally, Section 5 provides some
numerical results supporting our theoretical analysis.

######  Notation.

We use the standard notation of Sobolev spaces, where, for a set $D$,
$\|\cdot\|_{L^{2}(D)}$ is the $L^{2}-$norm and
$\left\langle\cdot,\cdot\right\rangle_{D}$ is the standard scalar product in
$L^{2}(D).$ Also, we write $A\lesssim B$ if there exists a generic positive
constant $C$ independent of the perturbation parameter $\varepsilon$ and the
mesh, such that $A\leq CB$.

##  2 Solution decomposition and mesh

The solution decomposition of this problem proved in [1] we give in the
following theorem.

######  Theorem 2.1.

Let $k\geq 0$ be a given integer and the data of (1) smooth enough. Then it
holds

$u=S+E+W,$

where for any $l\in\\{0,1,\dots,k\\}$ it holds

$\displaystyle\|{S^{(\ell)}}\|_{L^{2}(0,1)}+\|{S^{(\ell)}}\|_{L^{2}(1,2)}$
$\displaystyle\lesssim 1,$ $\displaystyle|E^{(\ell)}(x)|$
$\displaystyle\lesssim\varepsilon^{-\ell}\mathrm{e}^{-\beta\frac{x}{\varepsilon}},\quad
x\in[0,2],$ $\displaystyle|W^{(\ell)}(x)|$
$\displaystyle\lesssim\begin{cases}0,&x\in(0,1),\\\
\varepsilon^{1-\ell}\mathrm{e}^{-\beta\frac{(x-1)}{\varepsilon}},&x\in(1,2).\end{cases}$

Thus, $E$ is the layer function corresponding to the left boundary, while $W$
is an interior layer function, and $S$ represents the smooth part.

Knowing how the layers are structured allows us to create a layer-adapted mesh
that resolves the layers. We will modify the recursively defined graded mesh
of the Durán type to our problem. Its advantages are the simple construction
and generation of mesh points (without transition point(s)) and a certain
robustness property. More precisely, if we are approximating a singularly
perturbed problem with a mesh that has been adapted a priori, we can expect
that a mesh designed for one value of the small parameter will perform well
for larger values of the small parameter as well. In this respect, the
recursively graded meshes have better behavior in numerical experiments.

We first define the points on the interval $[0,1]$ and then the rest of the
domain to construct this mesh. Let $H\in(0,1)$ be arbitrary. Define the number
M

by

$M=\left\lceil\left\lceil\frac{1}{H}\right\rceil-\frac{\ln\left(H\varepsilon\left\lceil\frac{1}{H}\right\rceil\right)}{\ln(1+H)}\right\rceil.$
(2)

We define mesh points recursively in the following way:

$x_{i}=\begin{cases}0,&i=0,\\\ iH\varepsilon,&i=1\leq
i\leq\left\lceil\frac{1}{H}\right\rceil,\\\
(1+H)x_{i-1},&\left\lceil\frac{1}{H}\right\rceil<i\leq M-1,\\\ 1,&i=M,\\\
1+x_{i-M},&M\leq i\leq 2M.\\\ \end{cases}$ (3)

If the interval $(x_{M-1},1)$ is too small in relation to $(x_{M-2},x_{M-1})$,
we simply omit the mesh point $x_{M-1}$. The total number of mesh subintervals
is $N=2M.$ It depends on the parameters $H$ and conditions (2), see [2].
Furthermore, the inequality

$H\lesssim N^{-1}\ln(1/\varepsilon)$ (4)

holds.

For mesh step sizes $h_{i}=x_{i}-x_{i-1}$, $1\leq i\leq 2M$, is valid
$CH\varepsilon\leq h_{i}\leq H,$ where $C$ is a constant independent of
$\varepsilon$ and the number of mesh points. It also applies

$\begin{array}[]{lll}h_{i}=H\varepsilon,&i\in\\{1,\dots,\left\lceil\frac{1}{H}\right\rceil\\}\cup\\{M+1,\dots,M+1+\left\lceil\frac{1}{H}\right\rceil\\},\\\\[2.15277pt]
h_{i}\leq
Hx,&i\in\\{\left\lceil\frac{1}{H}\right\rceil+1,\dots,M\\}\cup\\{M+2+\left\lceil\frac{1}{H}\right\rceil,\dots,2M\\},\\\\[2.15277pt]
\end{array}$ (5)

where $x\in[x_{i-1},x_{i}].$

######  Remark 2.2.

For an arbitrarily chosen parameter H in the Durán mesh, $2M$ steps of the
mesh are obtained, which is generally not comparable to the number of steps in
other papers that deal with the same or similar issues. An approach that
eliminates this shortcoming is given in the paper [5].

##  3 Finite element method and error estimates

The bilinear and linear form for problem (1) are given by

$\displaystyle B(u,v)$ $\displaystyle:=\varepsilon\left\langle
u^{\prime},v^{\prime}\right\rangle_{\Omega}+\left\langle cu-
bu^{\prime},v\right\rangle_{\Omega}+\left\langle
du(\cdot-1),v\right\rangle_{(1,2)}$ $\displaystyle=\left\langle
f,v\right\rangle_{\Omega}-\left\langle
d\phi(\cdot-1),v\right\rangle_{(0,1)}=:F(v)$ (6)

for $u,v\in\mathcal{U}$.

We use $\mathcal{U}_{N}:=\\{v\in
H_{0}^{1}(\Omega):v|_{\tau}\in\mathcal{P}_{k}(\tau)\\}$ for discrete space,
where $\mathcal{P}_{k}(\tau)$ is the space of polynomials of degree $k$ at
most on a cell $\tau$ of the mesh. Let $I$ be the standard Lagrange-
interpolation operator into $\mathcal{U}_{N}$ using equidistant points or any
other suitable distribution of points.

The finite element method is given by: Find $u_{N}\in\mathcal{U}_{N}$ such
that for all $v\in\mathcal{U}_{N}$ it holds

$B(u_{N},v)=F(v).$ (7)

The bilinear form is coercive in the energy norm
$\left|\\!\\!\;\left|\\!\\!\;\left|{u}\right|\\!\\!\;\right|\\!\\!\;\right|^{2}:=\varepsilon\|{u^{\prime}}\|_{L^{2}(\Omega)}^{2}+\gamma\|{u}\|_{L^{2}(\Omega)}^{2}$
and satisfies the Galerkin orthogonality $B(u-u_{N},v)=0$ for all
$v\in\mathcal{U}_{N}.$

######  Lemma 3.1.

For the standard piecewise Lagrange interpolation operator on the graded mesh
(3) we have

$\displaystyle\|{u-Iu}\|_{L^{2}(\Omega)}$ $\displaystyle\lesssim H^{k+1},$ (8)
$\displaystyle\|{(u-Iu)^{\prime}}\|_{L^{2}(\Omega)}$
$\displaystyle\lesssim{\varepsilon}^{-1/2}H^{k}.$ (9)

###### Proof.

In the proof we use the norm definitions, the solution decomposition and
standard interpolation error estimates, given on any cell $\tau_{i}$ with
width $h_{i}$ and $1\leq s\leq k+1$ and $1\leq t\leq k$ by

$\displaystyle\|v-Iv\|_{L^{2}(\tau_{i})}$ $\displaystyle\lesssim
h_{i}^{s}\|v^{(s)}\|_{L^{2}(\tau_{i})},$ (10a)
$\displaystyle\|(v-Iv)^{\prime}\|_{L^{2}(\tau_{i})}$ $\displaystyle\lesssim
h_{i}^{t}\|v^{(t+1)}\|_{L^{2}(\tau_{i})},$ (10b)

for $v$ smooth enough. We obtain

$\displaystyle\|S-IS\|_{L^{2}(0,1)}^{2}=\sum\limits_{i=1}^{M}\|S-IS\|_{L^{2}(I_{i})}^{2}$
$\displaystyle\lesssim\sum\limits_{i=1}^{\left\lceil\frac{1}{H}\right\rceil}h_{i}^{2(k+1)}\|S^{(k+1)}\|_{L^{2}(I_{i})}^{2}+\sum\limits_{i=\left\lceil\frac{1}{H}\right\rceil+1}^{M}h_{i}^{2(k+1)}\|S^{(k+1)}\|_{L^{2}(I_{i})}^{2}$
$\displaystyle\lesssim\sum\limits_{i=1}^{\left\lceil\frac{1}{H}\right\rceil}(H\varepsilon)^{2(k+1)}\|S^{(k+1)}\|_{L^{2}(I_{i})}^{2}+\sum\limits_{i=\left\lceil\frac{1}{H}\right\rceil+1}^{M}H^{2(k+1)}\|x^{k+1}S^{(k+1)}\|_{L^{2}(I_{i})}^{2}\lesssim
H^{2(k+1)}$

where $I_{i}=(x_{i-1},x_{i}).$ The same estimate we get on $[1,2].$ In the
same way we get

$\|(S-IS)^{\prime}\|_{L^{2}(\Omega)}^{2}\lesssim H^{2k}.$

For the boundary layer part we obtain

$\displaystyle\|$ $\displaystyle
E-IE\|_{L^{2}(0,1)}^{2}\lesssim\sum\limits_{i=1}^{\left\lceil\frac{1}{H}\right\rceil}(H\varepsilon)^{2(k+1)}\|E^{(k+1)}\|_{L^{2}(I_{i})}^{2}+\sum\limits_{i=\left\lceil\frac{1}{H}\right\rceil+1}^{M}H^{2(k+1)}\|x^{k+1}E^{(k+1)}\|_{L^{2}(I_{i})}^{2}$
$\displaystyle\lesssim(H\varepsilon)^{2(k+1)}\int\limits_{0}^{\varepsilon}\varepsilon^{-2(k+1)}e^{-\frac{2\beta
x}{\varepsilon}}dx+H^{2(k+1)}\int\limits_{\varepsilon}^{1}x^{2(k+1)}\varepsilon^{-2(k+1)}e^{-\frac{2\beta
x}{\varepsilon}}dx\lesssim\varepsilon H^{2(k+1)}.$

and the same estimate on $[1,2]$. Also, we get

$\displaystyle\|(E-IE)^{\prime}\|_{L^{2}(0,\varepsilon)}^{2}\lesssim$
$\displaystyle\sum\limits_{i=1}^{\left\lceil\frac{1}{H}\right\rceil}(H\varepsilon)^{2k}\|E^{(k+1)}\|_{L^{2}(I_{i})}^{2}\lesssim\varepsilon^{-1}H^{2k},$
$\displaystyle\|(E-IE)^{\prime}\|_{L^{2}(\varepsilon,1)}^{2}\lesssim$
$\displaystyle\sum\limits_{i=\left\lceil\frac{1}{H}\right\rceil+1}^{M}H^{2k}\|x^{k+1}E^{(k+1)}\|_{L^{2}(I_{i})}^{2}\lesssim\varepsilon
H^{2k}.$

For interior layer function we have following estimates

$\displaystyle\|$ $\displaystyle W-IW\|_{L^{2}(1,2)}^{2}$
$\displaystyle\lesssim\sum\limits_{i=M}^{M+\left\lceil\frac{1}{H}\right\rceil}(H\varepsilon)^{2(k+1)}\|W^{(k+1)}\|_{L^{2}(I_{i})}^{2}+\sum\limits_{i=M+\left\lceil\frac{1}{H}\right\rceil+1}^{2M}H^{2(k+1)}\|x^{k+1}W^{(k+1)}\|_{L^{2}(I_{i})}^{2}$
$\displaystyle\lesssim(H\varepsilon)^{2(k+1)}\int\limits_{1}^{1+\varepsilon}(\varepsilon^{1-(k+1)})^{2}e^{-\frac{2\beta(x-1)}{\varepsilon}}dx+H^{2(k+1)}\int\limits_{1+\varepsilon}^{2}x^{2(k+1)}(\varepsilon^{1-(k+1)})^{2}e^{-\frac{2\beta(x-1)}{\varepsilon}}dx$
$\displaystyle\lesssim\varepsilon^{3}H^{2(k+1)},$ $\displaystyle\|$
$\displaystyle(W-IW)^{\prime}\|_{L^{2}(1,2)}^{2}\lesssim(H\varepsilon)^{2k}\int\limits_{1}^{1+\varepsilon}\varepsilon^{-2k}e^{-\frac{2\beta(x-1)}{\varepsilon}}dx+H^{2k}\int\limits_{1+\varepsilon}^{2}\varepsilon^{-2k}x^{2k}e^{-\frac{2\beta(x-1)}{\varepsilon}}dx\lesssim\varepsilon
H^{2k}.$

Using Theorem 2.1 and above estimates the statement of the lemma follows. ∎

######  Theorem 3.2.

For the solution $u$ of problem (1) and the numerical solution $u_{N}$ of (7)
on a Durán mesh (3) it holds

$\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|\lesssim
H^{k}.$

###### Proof.

Based on the triangle inequality
$\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|\leq\left|\\!\\!\;\left|\\!\\!\;\left|{u-Iu}\right|\\!\\!\;\right|\\!\\!\;\right|+\left|\\!\\!\;\left|\\!\\!\;\left|{Iu-
u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|$, and using the evaluation of
Lemma 3.1 for the first term, it is sufficient to evaluate the second term.
Let $\eta:=u-Iu$ and $\chi:=Iu-u_{N}\in\mathcal{U}_{N}.$ Coercivity and
Galerkin orthogonality with parts of the proof of Lemma 3.1 yield

$\displaystyle\left|\\!\\!\;\left|\\!\\!\;\left|{\chi}\right|\\!\\!\;\right|\\!\\!\;\right|$
$\displaystyle\leq
B(\chi,\chi)=B(\eta,\chi)=\varepsilon\left\langle\eta^{\prime},\chi^{\prime}\right\rangle_{\Omega}+\left\langle
c\eta-b\eta^{\prime},\chi\right\rangle_{\Omega}+\left\langle
d\eta(\cdot-1),\chi\right\rangle_{(1,2)}$ $\displaystyle\lesssim
H^{k}\left|\\!\\!\;\left|\\!\\!\;\left|{\chi}\right|\\!\\!\;\right|\\!\\!\;\right|+\left\langle
b(E-IE),\chi^{\prime}\right\rangle_{\Omega}\lesssim H^{k}.\qed$

##  4 On a coarser mesh

Following the idea from [7] for the weak layer, we use a mesh that is sparser
than the one defined in (3). This is called a coarser mesh. Numerical results
show that for small $k$ and reasonably small $\varepsilon$ it is possible to
use coarser meshes. For higher polynomial degrees, the weak layer should be
resolved by a classical layer-adapted mesh.

Here, we construct the mesh in the following way. Let $M$ be defined as in (2)
and analogously

$M_{2}=\left\lceil\left\lceil\frac{1}{H}\right\rceil-\frac{\ln\left(H\varepsilon^{\frac{k-1}{k}}\left\lceil\frac{1}{H}\right\rceil\right)}{\ln(1+H)}\right\rceil.$
(11)

Then the mesh nodes are given by

$x_{i}=\begin{cases}0,&i=0,\\\ iH\varepsilon,&1\leq
i\leq\left\lceil\frac{1}{H}\right\rceil,\\\
(1+H)x_{i-1},&\left\lceil\frac{1}{H}\right\rceil<i\leq M-1,\\\ 1,&i=M,\\\
1+(i-M)H\varepsilon^{\frac{k-1}{k}},&M+1\leq i\leq
M+\left\lceil\frac{1}{H}\right\rceil,\\\
1+(1+H)(x_{i-1}-1),&M+\left\lceil\frac{1}{H}\right\rceil<i\leq M+M_{2}-1,\\\
2,&i=M+M_{2}.\end{cases}$ (12)

The mesh step sizes $h_{i}=x_{i}-x_{i-1}$ satisfy

$\begin{array}[]{lll}h_{i}=H\varepsilon,&i\in\\{1,\dots,\left\lceil\frac{1}{H}\right\rceil\\}\\\\[2.15277pt]
h_{i}\leq
Hx,&i\in\\{\left\lceil\frac{1}{H}\right\rceil+1,\dots,M\\}\\\\[2.15277pt]
h_{i}=H\varepsilon^{\frac{k-1}{k}},&i\in\\{M+1,\dots,M+\left\lceil\frac{1}{H}\right\rceil\\},\\\\[2.15277pt]
h_{i}\leq
Hx,&i\in\\{M+\left\lceil\frac{1}{H}\right\rceil+1,\dots,M+M_{2}\\}.\end{array}$
(13)

where $x\in[x_{i-1},x_{i}].$

######  Lemma 4.1.

Let us assume $e^{-\varepsilon^{-1/k}}\leq H^{k-1}.$ Then for the standard
piecewise Lagrange interpolation operator on the graded mesh (12) we have

$\displaystyle\|{u-Iu}\|_{L^{2}(\Omega)}$ $\displaystyle\lesssim
H^{k+\frac{1}{2}},$ (14)
$\displaystyle\left|\\!\\!\;\left|\\!\\!\;\left|{u-Iu}\right|\\!\\!\;\right|\\!\\!\;\right|$
$\displaystyle\lesssim H^{k}.$ (15)

###### Proof.

Similarly to the proof of the previous lemma, using solution decomposition and
estimates (10a) and (10b), we obtain same estimates for
$\|S-IS\|_{L^{2}(0,2)},$ $\|E-IE\|_{L^{2}(0,2)}.$ Using the assumption
$e^{-\varepsilon^{-1/k}}\leq H^{k-1}$ we get

$\displaystyle\|(E-IE)^{\prime}\|_{L^{2}(1,1+\varepsilon^{\frac{k-1}{k}})}\lesssim\varepsilon^{-\frac{1}{2}}H^{k}.$

For the estimation of $W$ we follow the idea given in [6]. From

$\displaystyle\|W-IW\|_{L^{2}(1+\varepsilon^{\frac{k-1}{k}},2)}\lesssim$
$\displaystyle
H\|xW^{\prime}\|_{L^{2}(1+\varepsilon^{\frac{k-1}{k}},2)}\lesssim\varepsilon^{\frac{1}{2}}H^{k},$

where we use assumption $e^{-\varepsilon^{-1/k}}\leq H^{k-1},$ and from

$\displaystyle\|W-IW\|_{L^{2}(1+\varepsilon^{\frac{k-1}{k}},2)}\lesssim$
$\displaystyle
H^{2}\|x^{2}W^{\prime\prime}\|_{L^{2}(1+\varepsilon^{\frac{k-1}{k}},2)}\lesssim\varepsilon^{-\frac{1}{2}}H^{k+1},$

we obtain

$\displaystyle\|W-IW\|_{L^{2}(1+\varepsilon^{\frac{k-1}{k}},2)}\lesssim
H^{k+\frac{1}{2}}.$

For the derivative we obtain

$\displaystyle\|(W-IW)^{\prime}\|_{L^{2}(1,1+\varepsilon^{\frac{k-1}{k}})}\lesssim$
$\displaystyle
h_{i}^{k}\|W^{(k+1)}\|_{L^{2}(1,1+\varepsilon^{\frac{k-1}{k}})}\lesssim\varepsilon^{-\frac{1}{2}}H^{k},$

and

$\displaystyle\|(W-IW)^{\prime}\|_{L^{2}(1+\varepsilon^{\frac{k-1}{k}},2)}\lesssim$
$\displaystyle
h_{i}\|W^{\prime\prime}\|_{L^{2}(1+\varepsilon^{\frac{k-1}{k}},2)}\lesssim\varepsilon^{-\frac{1}{2}}H^{k}.$

Collecting all these estimates, we get the statement of lemma. ∎

######  Theorem 4.2.

For the solution $u$ of problem (1) and the numerical solution $u_{N}$ of (7)
on a coarser Durán mesh it holds

$\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|\lesssim
H^{k}.$

###### Proof.

Similar to the proof of the Theorem 3.2 using the result of the Lemma 4.1. ∎

##  5 Numerical results

As an example, let us look at the following problem taken from [1]

$\displaystyle-\varepsilon
u^{\prime\prime}(x)-(2+x)u^{\prime}(x)+(3+x)u(x)-d(x)u(x-1)$
$\displaystyle=3,\,x\in(0,2),$ $\displaystyle u(2)$ $\displaystyle=0,$
$\displaystyle u(x)$ $\displaystyle=x^{2},\,x\in(-1,0],$

where

$d(x)=\begin{cases}1-x,&x<1,\\\ 2+\sin(4\pi x),&x\geq 1.\end{cases}$

In this case the exact solution is not known. Our numerical simulations are
performed using the finite-element framework $\mathbb{SOFE}$ developed by L.
Ludwig, see
github.com/SOFE-Developers/SOFE. We compare the results both on a Durán mesh
and on a coarse Durán mesh. Since the exact solution to this example is
unknown, we use a reference solution as a proxy. This is computed on a
standard or coarsened Durán mesh with $k=4$ and $H=0.05$. Table 1

Table 1: Errors for various polynomial degrees on standard Durán meshes and $\varepsilon=10^{-6}$ |  | $k=1$ | $k=2$ | $k=3$
---|---|---|---|---
$H$ | $\tilde{M}$ | $\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|$ | $\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|$ | $\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|$
0.90 | 46 | 3.14e-01 | 1.20 | 6.83e-02 | 2.44 | 1.04e-02 | 3.67
0.80 | 50 | 2.85e-01 | 1.05 | 5.57e-02 | 2.09 | 7.63e-03 | 3.39
0.70 | 56 | 2.52e-01 | 0.99 | 4.40e-02 | 2.08 | 5.19e-03 | 2.27
0.60 | 64 | 2.21e-01 | 1.19 | 3.33e-02 | 2.39 | 3.84e-03 | 3.64
0.50 | 74 | 1.86e-01 | 1.24 | 2.36e-02 | 2.49 | 2.26e-03 | 3.37
0.40 | 88 | 1.50e-01 | 1.17 | 1.53e-02 | 2.31 | 1.26e-03 | 3.40
0.30 | 112 | 1.13e-01 | 1.08 | 8.77e-03 | 2.17 | 5.56e-04 | 3.64
0.20 | 162 | 7.59e-02 | 1.06 | 3.94e-03 | 2.12 | 1.45e-04 | 3.23
0.10 | 310 | 3.81e-02 |  | 9.96e-04 |  | 1.78e-05 |

shows the results for different polynomial degrees $k$ on the standard Durán
mesh for $\varepsilon=10^{-6}$. Given are, for different values of $H$, the
corresponding number of cells $\tilde{M}=2M$, the error measured in the energy
norm and the numerical rate of convergence. Obviously we have a convergence of
order $k$. Table 2

Table 2: Errors for various polynomial degrees on coarse Durán meshes and $\varepsilon=10^{-6}$ | $k=1$ | $k=2$ | $k=3$
---|---|---|---
$H$ | $\tilde{M}$ | $\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|$ | $\tilde{M}$ | $\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|$ | $\tilde{M}$ | $\left|\\!\\!\;\left|\\!\\!\;\left|{u-u_{N}}\right|\\!\\!\;\right|\\!\\!\;\right|$
0.90 | 25 | 3.14e-01 | 1.31 | 35 | 6.85e-02 | 2.49 | 39 | 1.02e-02 | 3.43
0.80 | 27 | 2.84e-01 | 1.11 | 38 | 5.58e-02 | 1.93 | 42 | 7.91e-03 | 3.64
0.70 | 30 | 2.53e-01 | 1.11 | 43 | 4.40e-02 | 2.19 | 47 | 5.25e-03 | 2.06
0.60 | 34 | 2.20e-01 | 1.23 | 49 | 3.31e-02 | 2.23 | 54 | 3.94e-03 | 4.89
0.50 | 39 | 1.86e-01 | 1.07 | 57 | 2.36e-02 | 2.64 | 62 | 2.01e-03 | 3.31
0.40 | 47 | 1.52e-01 | 1.20 | 67 | 1.54e-02 | 2.26 | 74 | 1.12e-03 | 3.42
0.30 | 60 | 1.13e-01 | 1.03 | 86 | 8.75e-03 | 2.18 | 95 | 4.76e-04 | 3.41
0.20 | 86 | 7.81e-02 | 1.10 | 124 | 3.94e-03 | 2.11 | 137 | 1.37e-04 | 3.16
0.10 | 165 | 3.81e-02 |  | 238 | 9.96e-04 |  | 262 | 1.76e-05 |

shows the results on the coarse version of the Durán mesh with
$\tilde{M}=M+M_{2}$ cells and $\varepsilon=10^{-6}$. Again we observe a
convergence of order $k$. Note that in the case of $k=1$ we essentially have
an equidistant mesh in $[1,2]$ with $M_{2}=\left\lceil\frac{1}{H}\right\rceil$
cells.

Comparing the results of the two tables, there is little difference in the
calculated errors, but a large reduction in the number of cells. Thus, using
the coarsened mesh gives equally good results with less computational cost.

Acknowledgment. The first author is supported by the Ministry of Science,
Technological Development and Innovation of the Republic of Serbia under grant
no. 451-03-47/2023-01/200134.

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