Superconvergence Analysis and PPR Recovery of Arbitrary Order Edge Elements for Maxwell’s Equations

Abstract

In this article, we propose a practical scheme for constructing global superconvergent approximations for Maxwell’s equations in both two and three dimensions. Superconvergence of order $O(h^{p+1})$ is established in a discrete norm. This superconvergence result, combined with the polynomial-preserving recovery postprocessing technique, leads to global superconvergence of order $O(h^{p+1})$ for recovered quantities in energy norms. Numerical experiments are provided to confirm our theoretical findings.

Appendix

The other interpolation coefficients in 3D are listed as

$$\begin{aligned} C_{ijk}^2= & {} {\left\{ \begin{array}{ll} \frac{\int _{y_c-h_y}^{y_c+h_y}E_2(x_c+(-1)^{i+1} h_x, y, z_c+(-1)^{j+1}h_z) L_k(y)\text {d}y}{(L_k,L_k)},&{}0\le i,j \le 1,\\ \frac{\int _{y_c-h_y}^{y_c+h_y}\int _{z_c-h_z}^{z_c+h_z}\frac{\partial E_2}{\partial z}(x_c+(-1)^{i+1} h_x,y,z)\phi _j'(z)L_k(y)\text {d}y\text {d}z}{(\phi _j',\phi _j')(L_k,L_k)},&{}0\le i \le 1,\ 2\le j\le p,\\ \frac{\int _{x_c-h_x}^{x_c+h_x}\int _{y_c-h_y}^{y_c+h_y}\frac{\partial E_2}{\partial x}(x,y,z_c+(-1)^{i+1} h_z)\phi _i'(x)L_k(y)\text {d}x\text {d}y}{(\phi _i',\phi _i')(L_k,L_k)},&{}0\le j \le 1,\ 2\le i\le p,\\ \frac{\int _{K}\frac{\partial ^2 E_2}{\partial x \partial z}(x,y,z)\phi _i'(x)\phi _j'(z)L_k(y)\text {d}V}{(\phi _i',\phi _i')(\phi _j',\phi _j')(L_k,L_k)},&{} 2\le i,j\le p, \end{array}\right. }\\ C_{ijk}^3= & {} {\left\{ \begin{array}{ll} \frac{\int _{z_c-h_z}^{z_c+h_z}E_3(x_c+(-1)^{i+1} h_x, y_c+(-1)^{j+1}h_y,z) L_k(z)\text {d}z}{(L_k,L_k)},&{} 0\le i,j \le 1,\\ \frac{\int _{y_c-h_y}^{y_c+h_y}\int _{z_c-h_z}^{z_c+h_z}\frac{\partial E_3}{\partial y}(x_c+(-1)^{i+1} h_x,y,z)\phi _j'(y)L_k(z)\text {d}y\text {d}z}{(\phi _j',\phi _j')(L_k,L_k)},&{}0\le i \le 1,\ 2\le j\le p,\\ \frac{\int _{x_c-h_x}^{x_c+h_x}\int _{z_c-h_z}^{z_c+h_z}\frac{\partial E_3}{\partial x}(x,y_c+(-1)^{i+1} h_y,z)\phi _i'(x)L_k(z)\text {d}x\text {d}z}{(\phi _i',\phi _i')(L_k,L_k)},&{}0\le j \le 1,\ 2\le i\le p,\\ \frac{\int _{K}\frac{\partial ^2 E_3}{\partial x \partial y}(x,y,z)\phi _i'(x)\phi _j'(y)L_k(z)\text {d}V}{(\phi _i',\phi _i')(\phi _j',\phi _j')(L_k,L_k)},&{} 2\le i,j\le p, \end{array}\right. } \end{aligned}$$

with $0\le k\le p-1$.

$$\begin{aligned} D_{ijk}^2= & {} {\left\{ \begin{array}{ll} \frac{\int _{x_c-h_x}^{x_c+h_x}\int _{z_c-h_z}^{z_c+h_z}{H_2}(x, y_c+(-1)^{j+1}h_y, z)L_j(x)L_k(z)\text {d}x\text {d}z}{(L_j,L_j)(L_k,L_k)},\ \ &{}\ 0\le i\le 1,\\ \frac{\int _{K}\frac{\partial H_2}{\partial y}(x,y,z)\phi _i'(y)L_j(x)L_k(z)\text {d}V}{(\phi _i',\phi _i')(L_j,L_j)(L_k,L_k)},\ \ &{} 2\le i\le p, \end{array}\right. }\\ D_{ijk}^3= & {} {\left\{ \begin{array}{ll} \frac{\int _{x_c-h_x}^{x_c+h_x}\int _{y_c-h_y}^{y_c+h_y}{H_3}(x, y, z_c+(-1)^{j+1}h_z)L_j(x)L_k(y)\text {d}x\text {d}y}{(L_j,L_j)(L_k,L_k)},\ \ &{}0\le i\le 1,\\ \frac{\int _{K}\frac{\partial H_3}{\partial z}(x,y,z)\phi _i'(z)L_j(x)L_k(y)\text {d}V}{(\phi _i',\phi _i')(L_j,L_j)(L_k,L_k)},\ \ &{} 2\le i\le p, \end{array}\right. } \end{aligned}$$

with $0\le j, k\le p-1$.

The other interpolation coefficient in 2D is listed as

$$\begin{aligned} C_{ij}^2={\left\{ \begin{array}{ll} \frac{\int _{y_c-h_y}^{y_c+h_y}E_2(x_c+(-1)^{i+1}h_x,y,z_c+(-1)^{i+1}h_z) L_j(y)\text {d}y}{(L_j,L_j)},&{}i=0,1,\ 0\le j\le p-1,\\ \frac{\int _{K}\frac{\partial E_2}{\partial x}(x,y)\phi _i'(x)L_j(y)\text {d}V}{(\phi _i',\phi _i')(L_j,L_j)},&{} 2\le i\le p,\ 0\le j \le p-1.\\ \end{array}\right. } \end{aligned}$$