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Representative meshes in Example 1.
(a) Triangular mesh,
(a) quadtree mesh and
(b) polygonal mesh.

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xmin=10^(1.3), xmax=10^(2.3),
ymin=1e-9, ymax=10,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,
[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_poly_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=4] figures/example1/ex1_poly_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=4] figures/example1/ex1_poly_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=4] figures/example1/ex1_poly_p10.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xmin=10^(1.3), xmax=10^(2.3),
ymin=1e-9, ymax=10,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,
[solid, color=blue, thick, mark=*]
table [x index=3, y index=5] figures/example1/ex1_poly_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=5] figures/example1/ex1_poly_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=5] figures/example1/ex1_poly_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=5] figures/example1/ex1_poly_p10.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xmin=10^(1.3), xmax=10^(2.3),
ymin=1e-9, ymax=10,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,
[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_square_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=4] figures/example1/ex1_square_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=4] figures/example1/ex1_square_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=4] figures/example1/ex1_square_p10.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xmin=10^(1.3), xmax=10^(2.3),
ymin=1e-9, ymax=10,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,
[solid, color=blue, thick, mark=*]
table [x index=3, y index=5] figures/example1/ex1_square_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=5] figures/example1/ex1_square_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=5] figures/example1/ex1_square_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=5] figures/example1/ex1_square_p10.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xmin=10^(1.3), xmax=10^(2.3),
ymin=1e-9, ymax=10,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_triangle_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=4] figures/example1/ex1_triangle_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=4] figures/example1/ex1_triangle_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=4] figures/example1/ex1_triangle_p10.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xmin=10^(1.3), xmax=10^(2.3),
ymin=1e-9, ymax=10,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,
[solid, color=blue, thick, mark=*]
table [x index=3, y index=5] figures/example1/ex1_triangle_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=5] figures/example1/ex1_triangle_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=5] figures/example1/ex1_triangle_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=5] figures/example1/ex1_triangle_p10.txt;

Convergence of the  method in the $L^2$
norm (top) and the energy seminorm (bottom) for Example 1.
Computations are shown for
(a) polygonal, (b) quadrilateral and (c) triangular meshes. Scale is

This example is solved using the  method, and for comparison, using two other DG methods—DGFD method by Jaśkowiec[37] and the version of the penalty-free DG method proposed by Oden at al.[48] In Fig. <ref>, the convergence curves of three DG methods are compared to each other: DG method of Oden et al., [48] DGFD and the  presented in this work. The DGFD and the  have the same convergence rate and the level of accuracy for these two methods is similar. Significantly	worse results in each case are obtained using the DG[48]	method, which is appreciable for $\porder=3$, where additionally, the convergence is nonmonotonic.

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_poly_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=10] figures/example1/ex1_poly_p3.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=8] figures/example1/ex1_poly_p3.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_poly_p5.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=10] figures/example1/ex1_poly_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=8] figures/example1/ex1_poly_p5.txt;

[width=14cm, height=14cm,
legend style=at=(0.03,0.03),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_poly_p10.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=10] figures/example1/ex1_poly_p10.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=8] figures/example1/ex1_poly_p10.txt;

Convergence study for Example 1 on the polygonal meshes using
the three DG methods
(DG,[48] DGFD and ).
The plots are presented for (a) $\porder=3$,
(b) $\porder=5$ and (c) $\porder=10$.
Scale is $\frac{1}{2}\log$–$\log$.

The condition number, which is the ratio of the maximum to minimum singular value of a matrix, is presented for the stiffness matrix in Fig. <ref>. The condition numbers are presented for the three methods , DGFD and DG[48] and for approximation orders $p=3,\, 5,\, 10$. The stiffness matrices for  are much better conditioned in comparison to the other two methods. It confirms that the  method is numerically stable.

[width=14cm, height=14cm,
ylabel=Condition number,
legend style=at=(0.03,0.78),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_sin_poly_conds_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=5] figures/example1/ex1_sin_poly_conds_p3.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=6] figures/example1/ex1_sin_poly_conds_p3.txt;

[width=14cm, height=14cm,
legend style=at=(0.03,0.78),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_sin_poly_conds_p5.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=5] figures/example1/ex1_sin_poly_conds_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=6] figures/example1/ex1_sin_poly_conds_p5.txt;

[width=14cm, height=14cm,
legend style=at=(0.66,0.03),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example1/ex1_sin_poly_conds_p10.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=5] figures/example1/ex1_sin_poly_conds_p10.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=6] figures/example1/ex1_sin_poly_conds_p10.txt;

Condition numbers for the matrices used in Example 1, for
the three DG methods
(DG,[48] DGFD and ).
The plots are presented for (a) $\porder=3$,
(b) $\porder=5$ and (c) $\porder=10$.

The same example is solved on the quadrilateral mesh that is randomly refined and the approximation orders in the elements $\porder\in[3,10]$ are set randomly (see Fig. <ref>). The error distributions for the three versions of the DG method are shown in Fig. <ref>. The errors for the  and the DGFD are comparable to each other. Much larger errors appear in the DG method of Oden et al., [48] which reveals that it is not well-suited for such meshes.

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[line width=0.6pt] (0.777778, -0.333333) – (1.000000, -0.333333) – (1.000000, -0.111111) – (0.888889, -0.111111) – (0.833333, -0.111111) – (0.777778, -0.111111) – (0.777778, -0.222222) – (0.777778, -0.333333);
[line width=0.6pt] (0.777778, -0.111111) – (0.833333, -0.111111) – (0.833333, -0.055556) – (0.777778, -0.055556) – (0.777778, -0.111111);
[line width=0.6pt] (0.777778, 0.111111) – (0.888889, 0.111111) – (1.000000, 0.111111) – (1.000000, 0.333333) – (0.777778, 0.333333) – (0.777778, 0.111111);
[line width=0.6pt] (0.777778, 0.333333) – (1.000000, 0.333333) – (1.000000, 0.555556) – (0.777778, 0.555556) – (0.777778, 0.527778) – (0.777778, 0.500000) – (0.777778, 0.444444) – (0.777778, 0.333333);
[line width=0.6pt] (0.777778, 0.555556) – (1.000000, 0.555556) – (1.000000, 0.777778) – (0.777778, 0.777778) – (0.777778, 0.666667) – (0.777778, 0.555556);
[line width=0.6pt] (0.777778, 0.777778) – (1.000000, 0.777778) – (1.000000, 1.000000) – (0.777778, 1.000000) – (0.777778, 0.888889) – (0.777778, 0.777778);
[line width=0.6pt] (-0.666667, 0.111111) – (-0.555556, 0.111111) – (-0.555556, 0.222222) – (-0.666667, 0.222222) – (-0.666667, 0.166667) – (-0.666667, 0.111111);
[line width=0.6pt] (-0.666667, 0.222222) – (-0.555556, 0.222222) – (-0.555556, 0.333333) – (-0.666667, 0.333333) – (-0.666667, 0.222222);
[line width=0.6pt] (-0.777778, 0.222222) – (-0.722222, 0.222222) – (-0.666667, 0.222222) – (-0.666667, 0.333333) – (-0.777778, 0.333333) – (-0.777778, 0.222222);
[line width=0.6pt] (-0.222222, 0.777778) – (-0.111111, 0.777778) – (-0.111111, 0.833333) – (-0.111111, 0.888889) – (-0.222222, 0.888889) – (-0.222222, 0.861111) – (-0.222222, 0.833333) – (-0.222222, 0.777778);
[line width=0.6pt] (-0.222222, 0.888889) – (-0.111111, 0.888889) – (-0.111111, 1.000000) – (-0.222222, 1.000000) – (-0.222222, 0.944444) – (-0.222222, 0.888889);
[line width=0.6pt] (-0.333333, 0.888889) – (-0.305556, 0.888889) – (-0.305556, 0.916667) – (-0.319444, 0.916667) – (-0.333333, 0.916667) – (-0.333333, 0.888889);
[line width=0.6pt] (-0.888889, 0.777778) – (-0.777778, 0.777778) – (-0.777778, 0.888889) – (-0.888889, 0.888889) – (-0.888889, 0.777778);
[line width=0.6pt] (-0.888889, 0.888889) – (-0.777778, 0.888889) – (-0.777778, 1.000000) – (-0.888889, 1.000000) – (-0.888889, 0.888889);
[line width=0.6pt] (-1.000000, 0.888889) – (-0.888889, 0.888889) – (-0.888889, 1.000000) – (-1.000000, 1.000000) – (-1.000000, 0.888889);
[line width=0.6pt] (0.222222, 0.777778) – (0.250000, 0.777778) – (0.250000, 0.805556) – (0.222222, 0.805556) – (0.222222, 0.777778);
[line width=0.6pt] (0.222222, 0.888889) – (0.277778, 0.888889) – (0.305556, 0.888889) – (0.333333, 0.888889) – (0.333333, 0.944444) – (0.333333, 1.000000) – (0.222222, 1.000000) – (0.222222, 0.888889);
[line width=0.6pt] (0.111111, 0.888889) – (0.222222, 0.888889) – (0.222222, 1.000000) – (0.111111, 1.000000) – (0.111111, 0.888889);
[line width=0.6pt] (0.000000, 0.555556) – (0.111111, 0.555556) – (0.111111, 0.666667) – (0.000000, 0.666667) – (0.000000, 0.611111) – (0.000000, 0.555556);
[line width=0.6pt] (0.000000, 0.666667) – (0.111111, 0.666667) – (0.111111, 0.777778) – (0.000000, 0.777778) – (0.000000, 0.666667);
[line width=0.6pt] (-0.111111, 0.666667) – (-0.055556, 0.666667) – (0.000000, 0.666667) – (0.000000, 0.777778) – (-0.055556, 0.777778) – (-0.111111, 0.777778) – (-0.111111, 0.666667);
[line width=0.6pt] (0.666667, -0.333333) – (0.777778, -0.333333) – (0.777778, -0.222222) – (0.666667, -0.222222) – (0.666667, -0.333333);
[line width=0.6pt] (0.666667, -0.222222) – (0.777778, -0.222222) – (0.777778, -0.111111) – (0.666667, -0.111111) – (0.666667, -0.222222);
[line width=0.6pt] (0.555556, -0.222222) – (0.666667, -0.222222) – (0.666667, -0.111111) – (0.555556, -0.111111) – (0.555556, -0.222222);
[line width=0.6pt] (0.666667, 0.333333) – (0.777778, 0.333333) – (0.777778, 0.444444) – (0.722222, 0.444444) – (0.666667, 0.444444) – (0.666667, 0.333333);
[line width=0.6pt] (0.666667, 0.444444) – (0.722222, 0.444444) – (0.722222, 0.500000) – (0.666667, 0.500000) – (0.666667, 0.444444);
[line width=0.6pt] (0.555556, 0.444444) – (0.666667, 0.444444) – (0.666667, 0.500000) – (0.666667, 0.555556) – (0.555556, 0.555556) – (0.555556, 0.444444);
[line width=0.6pt] (0.666667, -0.555556) – (0.777778, -0.555556) – (0.777778, -0.444444) – (0.666667, -0.444444) – (0.666667, -0.555556);
[line width=0.6pt] (0.666667, -0.444444) – (0.777778, -0.444444) – (0.777778, -0.333333) – (0.666667, -0.333333) – (0.666667, -0.444444);
[line width=0.6pt] (0.555556, -0.444444) – (0.666667, -0.444444) – (0.666667, -0.333333) – (0.555556, -0.333333) – (0.555556, -0.444444);
[line width=0.6pt] (-0.888889, -0.333333) – (-0.777778, -0.333333) – (-0.777778, -0.222222) – (-0.888889, -0.222222) – (-0.888889, -0.333333);
[line width=0.6pt] (-0.888889, -0.222222) – (-0.777778, -0.222222) – (-0.777778, -0.111111) – (-0.888889, -0.111111) – (-0.888889, -0.222222);
[line width=0.6pt] (-1.000000, -0.222222) – (-0.888889, -0.222222) – (-0.888889, -0.111111) – (-1.000000, -0.111111) – (-1.000000, -0.222222);
[line width=0.6pt] (-0.888889, 0.555556) – (-0.777778, 0.555556) – (-0.777778, 0.666667) – (-0.888889, 0.666667) – (-0.888889, 0.555556);
[line width=0.6pt] (-0.888889, 0.666667) – (-0.777778, 0.666667) – (-0.777778, 0.777778) – (-0.888889, 0.777778) – (-0.888889, 0.666667);
[line width=0.6pt] (-1.000000, 0.666667) – (-0.888889, 0.666667) – (-0.888889, 0.777778) – (-1.000000, 0.777778) – (-1.000000, 0.666667);
[line width=0.6pt] (-0.277778, 0.777778) – (-0.222222, 0.777778) – (-0.222222, 0.833333) – (-0.250000, 0.833333) – (-0.277778, 0.833333) – (-0.277778, 0.777778);
[line width=0.6pt] (-0.277778, 0.833333) – (-0.250000, 0.833333) – (-0.250000, 0.861111) – (-0.277778, 0.861111) – (-0.277778, 0.833333);
[line width=0.6pt] (-0.333333, 0.833333) – (-0.277778, 0.833333) – (-0.277778, 0.861111) – (-0.277778, 0.888889) – (-0.305556, 0.888889) – (-0.333333, 0.888889) – (-0.333333, 0.833333);
[line width=0.6pt] (0.444444, 0.333333) – (0.500000, 0.333333) – (0.500000, 0.388889) – (0.444444, 0.388889) – (0.444444, 0.333333);
[line width=0.6pt] (0.444444, 0.444444) – (0.500000, 0.444444) – (0.555556, 0.444444) – (0.555556, 0.555556) – (0.444444, 0.555556) – (0.444444, 0.444444);
[line width=0.6pt] (0.333333, 0.444444) – (0.444444, 0.444444) – (0.444444, 0.555556) – (0.333333, 0.555556) – (0.333333, 0.444444);
[line width=0.6pt] (0.888889, -0.111111) – (1.000000, -0.111111) – (1.000000, 0.000000) – (0.888889, 0.000000) – (0.888889, -0.055556) – (0.888889, -0.111111);
[line width=0.6pt] (0.888889, 0.000000) – (1.000000, 0.000000) – (1.000000, 0.111111) – (0.888889, 0.111111) – (0.888889, 0.000000);
[line width=0.6pt] (0.777778, 0.000000) – (0.833333, 0.000000) – (0.888889, 0.000000) – (0.888889, 0.111111) – (0.777778, 0.111111) – (0.777778, 0.000000);
[line width=0.6pt] (0.222222, -0.333333) – (0.333333, -0.333333) – (0.333333, -0.222222) – (0.222222, -0.222222) – (0.222222, -0.333333);
[line width=0.6pt] (0.222222, -0.222222) – (0.333333, -0.222222) – (0.333333, -0.111111) – (0.222222, -0.111111) – (0.222222, -0.222222);
[line width=0.6pt] (0.111111, -0.222222) – (0.222222, -0.222222) – (0.222222, -0.111111) – (0.111111, -0.111111) – (0.111111, -0.222222);
[line width=0.6pt] (0.722222, 0.444444) – (0.777778, 0.444444) – (0.777778, 0.500000) – (0.750000, 0.500000) – (0.722222, 0.500000) – (0.722222, 0.444444);
[line width=0.6pt] (0.722222, 0.500000) – (0.750000, 0.500000) – (0.750000, 0.527778) – (0.736111, 0.527778) – (0.722222, 0.527778) – (0.722222, 0.500000);
[line width=0.6pt] (0.666667, 0.500000) – (0.722222, 0.500000) – (0.722222, 0.527778) – (0.722222, 0.541667) – (0.722222, 0.555556) – (0.666667, 0.555556) – (0.666667, 0.500000);
[line width=0.6pt] (0.277778, 0.777778) – (0.333333, 0.777778) – (0.333333, 0.833333) – (0.305556, 0.833333) – (0.277778, 0.833333) – (0.277778, 0.819444) – (0.277778, 0.805556) – (0.277778, 0.777778);
[line width=0.6pt] (0.277778, 0.833333) – (0.305556, 0.833333) – (0.305556, 0.861111) – (0.277778, 0.861111) – (0.277778, 0.833333);
[line width=0.6pt] (0.222222, 0.833333) – (0.250000, 0.833333) – (0.263889, 0.833333) – (0.277778, 0.833333) – (0.277778, 0.861111) – (0.277778, 0.888889) – (0.222222, 0.888889) – (0.222222, 0.833333);
[line width=0.6pt] (0.750000, 0.500000) – (0.777778, 0.500000) – (0.777778, 0.527778) – (0.750000, 0.527778) – (0.750000, 0.500000);
[line width=0.6pt] (0.750000, 0.527778) – (0.777778, 0.527778) – (0.777778, 0.555556) – (0.750000, 0.555556) – (0.750000, 0.541667) – (0.750000, 0.527778);
[line width=0.6pt] (0.722222, 0.527778) – (0.736111, 0.527778) – (0.736111, 0.541667) – (0.722222, 0.541667) – (0.722222, 0.527778);
[line width=0.6pt] (0.666667, 0.555556) – (0.722222, 0.555556) – (0.736111, 0.555556) – (0.750000, 0.555556) – (0.777778, 0.555556) – (0.777778, 0.666667) – (0.666667, 0.666667) – (0.666667, 0.555556);
[line width=0.6pt] (0.666667, 0.666667) – (0.777778, 0.666667) – (0.777778, 0.777778) – (0.666667, 0.777778) – (0.666667, 0.666667);
[line width=0.6pt] (0.555556, 0.666667) – (0.666667, 0.666667) – (0.666667, 0.777778) – (0.555556, 0.777778) – (0.555556, 0.666667);
[line width=0.6pt] (-0.055556, 0.555556) – (0.000000, 0.555556) – (0.000000, 0.611111) – (-0.055556, 0.611111) – (-0.055556, 0.555556);
[line width=0.6pt] (-0.055556, 0.611111) – (0.000000, 0.611111) – (0.000000, 0.666667) – (-0.055556, 0.666667) – (-0.055556, 0.611111);
[line width=0.6pt] (-0.111111, 0.611111) – (-0.055556, 0.611111) – (-0.055556, 0.666667) – (-0.111111, 0.666667) – (-0.111111, 0.611111);
[line width=0.6pt] (-0.722222, 0.111111) – (-0.666667, 0.111111) – (-0.666667, 0.166667) – (-0.722222, 0.166667) – (-0.722222, 0.111111);
[line width=0.6pt] (-0.722222, 0.166667) – (-0.666667, 0.166667) – (-0.666667, 0.222222) – (-0.722222, 0.222222) – (-0.722222, 0.166667);
[line width=0.6pt] (-0.777778, 0.166667) – (-0.722222, 0.166667) – (-0.722222, 0.222222) – (-0.777778, 0.222222) – (-0.777778, 0.166667);
[line width=0.6pt] (-0.277778, 0.888889) – (-0.250000, 0.888889) – (-0.222222, 0.888889) – (-0.222222, 0.944444) – (-0.277778, 0.944444) – (-0.277778, 0.916667) – (-0.277778, 0.888889);
[line width=0.6pt] (-0.277778, 0.944444) – (-0.222222, 0.944444) – (-0.222222, 1.000000) – (-0.277778, 1.000000) – (-0.277778, 0.944444);
[line width=0.6pt] (-0.333333, 0.944444) – (-0.319444, 0.944444) – (-0.305556, 0.944444) – (-0.277778, 0.944444) – (-0.277778, 1.000000) – (-0.333333, 1.000000) – (-0.333333, 0.944444);
[line width=0.6pt] (-0.250000, 0.833333) – (-0.222222, 0.833333) – (-0.222222, 0.861111) – (-0.250000, 0.861111) – (-0.250000, 0.833333);
[line width=0.6pt] (-0.250000, 0.861111) – (-0.222222, 0.861111) – (-0.222222, 0.888889) – (-0.250000, 0.888889) – (-0.250000, 0.861111);
[line width=0.6pt] (-0.277778, 0.861111) – (-0.250000, 0.861111) – (-0.250000, 0.888889) – (-0.277778, 0.888889) – (-0.277778, 0.861111);
[line width=0.6pt] (0.444444, 0.777778) – (0.555556, 0.777778) – (0.555556, 0.888889) – (0.444444, 0.888889) – (0.444444, 0.777778);
[line width=0.6pt] (0.444444, 0.888889) – (0.555556, 0.888889) – (0.555556, 1.000000) – (0.444444, 1.000000) – (0.444444, 0.944444) – (0.444444, 0.888889);
[line width=0.6pt] (0.333333, 0.888889) – (0.388889, 0.888889) – (0.388889, 0.944444) – (0.333333, 0.944444) – (0.333333, 0.888889);
[line width=0.6pt] (-0.666667, -0.111111) – (-0.555556, -0.111111) – (-0.555556, 0.000000) – (-0.666667, 0.000000) – (-0.666667, -0.111111);
[line width=0.6pt] (-0.666667, 0.000000) – (-0.555556, 0.000000) – (-0.555556, 0.111111) – (-0.666667, 0.111111) – (-0.666667, 0.000000);
[line width=0.6pt] (-0.777778, 0.000000) – (-0.666667, 0.000000) – (-0.666667, 0.111111) – (-0.722222, 0.111111) – (-0.777778, 0.111111) – (-0.777778, 0.000000);
[line width=0.6pt] (0.000000, -0.111111) – (0.111111, -0.111111) – (0.111111, 0.000000) – (0.000000, 0.000000) – (0.000000, -0.111111);
[line width=0.6pt] (0.000000, 0.000000) – (0.111111, 0.000000) – (0.111111, 0.111111) – (0.000000, 0.111111) – (0.000000, 0.000000);
[line width=0.6pt] (-0.111111, 0.000000) – (0.000000, 0.000000) – (0.000000, 0.111111) – (-0.111111, 0.111111) – (-0.111111, 0.000000);
[line width=0.6pt] (-0.444444, 0.111111) – (-0.333333, 0.111111) – (-0.333333, 0.222222) – (-0.444444, 0.222222) – (-0.444444, 0.111111);
[line width=0.6pt] (-0.444444, 0.222222) – (-0.333333, 0.222222) – (-0.333333, 0.333333) – (-0.444444, 0.333333) – (-0.444444, 0.222222);
[line width=0.6pt] (-0.555556, 0.222222) – (-0.444444, 0.222222) – (-0.444444, 0.333333) – (-0.555556, 0.333333) – (-0.555556, 0.222222);
[line width=0.6pt] (0.000000, 0.777778) – (0.111111, 0.777778) – (0.111111, 0.888889) – (0.000000, 0.888889) – (0.000000, 0.833333) – (0.000000, 0.777778);
[line width=0.6pt] (0.000000, 0.888889) – (0.111111, 0.888889) – (0.111111, 1.000000) – (0.000000, 1.000000) – (0.000000, 0.888889);
[line width=0.6pt] (-0.111111, 0.888889) – (-0.055556, 0.888889) – (0.000000, 0.888889) – (0.000000, 1.000000) – (-0.111111, 1.000000) – (-0.111111, 0.888889);
[line width=0.6pt] (0.666667, 0.777778) – (0.777778, 0.777778) – (0.777778, 0.888889) – (0.666667, 0.888889) – (0.666667, 0.777778);
[line width=0.6pt] (0.666667, 0.888889) – (0.777778, 0.888889) – (0.777778, 1.000000) – (0.666667, 1.000000) – (0.666667, 0.888889);
[line width=0.6pt] (0.555556, 0.888889) – (0.666667, 0.888889) – (0.666667, 1.000000) – (0.555556, 1.000000) – (0.555556, 0.888889);
[line width=0.6pt] (-0.305556, 0.888889) – (-0.277778, 0.888889) – (-0.277778, 0.916667) – (-0.305556, 0.916667) – (-0.305556, 0.888889);
[line width=0.6pt] (-0.305556, 0.916667) – (-0.277778, 0.916667) – (-0.277778, 0.944444) – (-0.305556, 0.944444) – (-0.305556, 0.930556) – (-0.305556, 0.916667);
[line width=0.6pt] (-0.333333, 0.916667) – (-0.319444, 0.916667) – (-0.319444, 0.930556) – (-0.333333, 0.930556) – (-0.333333, 0.916667);
[line width=0.6pt] (0.833333, -0.111111) – (0.888889, -0.111111) – (0.888889, -0.055556) – (0.833333, -0.055556) – (0.833333, -0.111111);
[line width=0.6pt] (0.833333, -0.055556) – (0.888889, -0.055556) – (0.888889, 0.000000) – (0.833333, 0.000000) – (0.833333, -0.055556);
[line width=0.6pt] (0.777778, -0.055556) – (0.833333, -0.055556) – (0.833333, 0.000000) – (0.777778, 0.000000) – (0.777778, -0.055556);
[line width=0.6pt] (0.388889, 0.888889) – (0.444444, 0.888889) – (0.444444, 0.944444) – (0.388889, 0.944444) – (0.388889, 0.888889);
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Randomly refined quadrilateral mesh for Example 1.
(a) Quadtree mesh with hanging nodes and
(b) approximation order in each element.

Error distribution on the randomly refined quadrilateral
mesh for Example 1. (a) , (b) DGFD
and (c) DG method.[48]

§.§ Example 2: Poisson problem with exponential solution

In this example the Poisson problem in $\Omega = (-1,1)^2$ is considered with the exact solution of the following exponential form:

\begin{align}\label{eq:exp2d_example}
u(x,y) = \sum_{i=1}^{50}  \exp - \left(a_i (x - x_i)^2 + b_i (y-y_i)^2\right) \eqcomma
\end{align}

where $a_i$ and $b_i$ are randomly generated positive values, $(x_i,y_i)$ are randomly generated points in the domain. The plots showing the function $u$ 			in (<ref>) are depicted in Fig. <ref>. This function is irregular with many hills randomly distributed in the domain. This example has been solved using polygonal meshes for uniform approximation orders in the elements, $p=$ 3, 5, 7, and 10.

Surface and contour plots of the
exact solution of the exact solution (<ref>) for Example 2.

Convergence plots for the  method are presented in Fig. <ref> on the whole domain and also along the mesh skeleton and outer boundary. The rates of the convergences are as expected for the particular orders. The computed residuals confirm  the correctness of the   method.
[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,

xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=4] figures/example2/ex1b_poly_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=4] figures/example2/ex1b_poly_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=4] figures/example2/ex1b_poly_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=4] figures/example2/ex1b_poly_p10.txt;

[width=14cm, height=14cm,
legend style=at=(0.03,0.03),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=6] figures/example2/ex1b_poly_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=6] figures/example2/ex1b_poly_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=6] figures/example2/ex1b_poly_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=6] figures/example2/ex1b_poly_p10.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=7] figures/example2/ex1b_poly_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=7] figures/example2/ex1b_poly_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=7] figures/example2/ex1b_poly_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=7] figures/example2/ex1b_poly_p10.txt;

[width=14cm, height=14cm,

legend style=at=(0.03,0.03),anchor=south west,
xtick = 24.372115, 39.166312, 62.928531, 101.118742, 162.480768,
xticklabels = 594, 1534, 3960, 10225, 26400,

[solid, color=blue, thick, mark=*]
table [x index=3, y index=5] figures/example2/ex1b_poly_p3.txt;

[dashed, color=red, thick, mark=square*]
table [x index=3, y index=5] figures/example2/ex1b_poly_p5.txt;

[dashdotted, color=brown, thick, mark=triangle*, mark size=3pt]
table [x index=3, y index=5] figures/example2/ex1b_poly_p7.txt;

[dotted, color=black, thick, mark=diamond*, mark size=3pt]
table [x index=3, y index=5] figures/example2/ex1b_poly_p10.txt;

Convergence study of the  method
for Example 2 on polygonal meshes.
Study is conducted in the $L^2$ norm
on the (a) whole domain,
(b) mesh skeleton and (c) outer boundary, and
(d) energy seminorm over the whole domain.
Scale is $ \frac{1}{2}\log $–$\log$.

§.§ Example 3: Poisson problem on L-shaped domain

We consider the standard  benchmark Poisson problem on the L-shaped domain, with the exact solution

\begin{align}
\temperature(r,\theta) = r^{\frac{2}{3}} \sin\left( \frac{2}{3}\theta \right) .
\end{align}

This function is harmonic, so the right-hand side of (<ref>) is zero. The exact solution has  a derivative singularity at the origin, and hence the error in the numerical solution concentrates at the origin. In this example, the vertices of the L-shaped domain are chosen as $(-1, -1)$, $(0, -1)$, $(0, 0)$, $(1, 0)$, $(1, 1)$, $(-1, 1)$ and the domain is meshed with structured quadrilaterals and then refined several times in the vicinity of the origin (see Fig. <ref>). The approximation orders in the elements are homogeneous in the entire mesh and calculations are performed on elements with orders $p=$ 3, 5, 7, and 10.

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[line width=0.6pt] (0.062500, 0.125000) – (0.093750, 0.125000) – (0.093750, 0.156250) – (0.062500, 0.156250) – (0.062500, 0.125000);
[line width=0.6pt] (0.062500, 0.187500) – (0.093750, 0.187500) – (0.125000, 0.187500) – (0.125000, 0.250000) – (0.062500, 0.250000) – (0.062500, 0.187500);
[line width=0.6pt] (0.000000, 0.187500) – (0.031250, 0.187500) – (0.062500, 0.187500) – (0.062500, 0.250000) – (0.000000, 0.250000) – (0.000000, 0.187500);
[line width=0.6pt] (0.187500, 0.250000) – (0.250000, 0.250000) – (0.250000, 0.312500) – (0.187500, 0.312500) – (0.187500, 0.250000);
[line width=0.6pt] (0.187500, 0.312500) – (0.250000, 0.312500) – (0.250000, 0.375000) – (0.187500, 0.375000) – (0.187500, 0.312500);
[line width=0.6pt] (0.125000, 0.312500) – (0.187500, 0.312500) – (0.187500, 0.375000) – (0.125000, 0.375000) – (0.125000, 0.312500);
[line width=0.6pt] (0.187500, 0.375000) – (0.250000, 0.375000) – (0.250000, 0.437500) – (0.187500, 0.437500) – (0.187500, 0.375000);
[line width=0.6pt] (0.187500, 0.437500) – (0.250000, 0.437500) – (0.250000, 0.500000) – (0.187500, 0.500000) – (0.187500, 0.437500);
[line width=0.6pt] (0.125000, 0.437500) – (0.187500, 0.437500) – (0.187500, 0.500000) – (0.125000, 0.500000) – (0.125000, 0.437500);
[line width=0.6pt] (0.062500, 0.375000) – (0.125000, 0.375000) – (0.125000, 0.437500) – (0.062500, 0.437500) – (0.062500, 0.375000);
[line width=0.6pt] (0.062500, 0.437500) – (0.125000, 0.437500) – (0.125000, 0.500000) – (0.062500, 0.500000) – (0.062500, 0.437500);
[line width=0.6pt] (0.000000, 0.437500) – (0.062500, 0.437500) – (0.062500, 0.500000) – (0.000000, 0.500000) – (0.000000, 0.437500);
[line width=0.6pt] (0.437500, 0.000000) – (0.500000, 0.000000) – (0.500000, 0.062500) – (0.437500, 0.062500) – (0.437500, 0.000000);
[line width=0.6pt] (0.437500, 0.062500) – (0.500000, 0.062500) – (0.500000, 0.125000) – (0.437500, 0.125000) – (0.437500, 0.062500);
[line width=0.6pt] (0.375000, 0.062500) – (0.437500, 0.062500) – (0.437500, 0.125000) – (0.375000, 0.125000) – (0.375000, 0.062500);
[line width=0.6pt] (0.437500, 0.125000) – (0.500000, 0.125000) – (0.500000, 0.187500) – (0.437500, 0.187500) – (0.437500, 0.125000);
[line width=0.6pt] (0.437500, 0.187500) – (0.500000, 0.187500) – (0.500000, 0.250000) – (0.437500, 0.250000) – (0.437500, 0.187500);
[line width=0.6pt] (0.375000, 0.187500) – (0.437500, 0.187500) – (0.437500, 0.250000) – (0.375000, 0.250000) – (0.375000, 0.187500);
[line width=0.6pt] (0.312500, 0.125000) – (0.375000, 0.125000) – (0.375000, 0.187500) – (0.312500, 0.187500) – (0.312500, 0.125000);
[line width=0.6pt] (0.312500, 0.187500) – (0.375000, 0.187500) – (0.375000, 0.250000) – (0.312500, 0.250000) – (0.312500, 0.187500);
[line width=0.6pt] (0.250000, 0.187500) – (0.312500, 0.187500) – (0.312500, 0.250000) – (0.250000, 0.250000) – (0.250000, 0.187500);
[line width=0.6pt] (0.031250, 0.000000) – (0.062500, 0.000000) – (0.062500, 0.031250) – (0.031250, 0.031250) – (0.031250, 0.000000);
[line width=0.6pt] (0.031250, 0.031250) – (0.062500, 0.031250) – (0.062500, 0.062500) – (0.031250, 0.062500) – (0.031250, 0.031250);
[line width=0.6pt] (0.000000, 0.031250) – (0.031250, 0.031250) – (0.031250, 0.062500) – (0.000000, 0.062500) – (0.000000, 0.031250);
[line width=0.6pt] (-0.093750, -0.125000) – (-0.062500, -0.125000) – (-0.062500, -0.093750) – (-0.093750, -0.093750) – (-0.093750, -0.125000);
[line width=0.6pt] (-0.093750, -0.093750) – (-0.062500, -0.093750) – (-0.062500, -0.062500) – (-0.093750, -0.062500) – (-0.093750, -0.093750);
[line width=0.6pt] (-0.125000, -0.093750) – (-0.093750, -0.093750) – (-0.093750, -0.062500) – (-0.125000, -0.062500) – (-0.125000, -0.093750);
[line width=0.6pt] (-0.093750, 0.000000) – (-0.062500, 0.000000) – (-0.062500, 0.031250) – (-0.093750, 0.031250) – (-0.093750, 0.000000);
[line width=0.6pt] (-0.093750, 0.031250) – (-0.062500, 0.031250) – (-0.062500, 0.062500) – (-0.093750, 0.062500) – (-0.093750, 0.031250);