The Bogner-Fox-Schmit Element Finite Volume Methods on the Shishkin Mesh for Fourth-Order Singularly Perturbed Elliptic Problems

Abstract

In this paper, the Bogner-Fox-Schmit (BFS) element finite volume methods (FVM) on a suitable Shishkin mesh for the fourth-order singular perturbed elliptic problems are constructed and analyzed . Firstly, under the proposed several equivalent discrete semi-norms, we convert the analysis of stability to the proof of positive definite property for several matrices by element analysis and algebraic techniques. Then we obtain the stability of the BFS element finite volume schemes, which is independent of the aspect ratio of rectangular elements. Secondly, with reasonable assumptions about the structure of the solution, we establish the error estimate of a special interpolation on the Shishkin mesh. Furthermore, based on the stability and interpolation error estimate, we analyze the error estimate of the finite volume methods. The optimal convergence rate for the energy norm $N^{-3}+\varepsilon ^{\frac{1}{2}}N^{-2}(\ln N)^2$ is obtained by a particular choice of the transition point for the Shishkin mesh. Finally, numerical experiments are presented to confirm the theoretical results.

Appendices

Appendix A. Some Matrices in the Proof of Lemma 1

The matrices $M_i, i=1\dots 5$ on page 10 are showed as follows.

Appendix B. The Proof of Lemma 2

Proof

According to (2.5) and the definition of $\varPi _h^*u_h$, we get

$$\begin{aligned}&\Vert \varPi _h^*u_h\Vert _{K_{ij}}^2=\sum _{P\in \mathring{K}_{ij}}\int _{K_P^*\cap K_{ij}}(\varPi _h^*u_h)^2dxdy=h_x^ih_y^jU^TW_0U,\\&\Vert u_h\Vert _{K_{ij}}^2=\int _{K_{ij}}(u_h)^2dxdy=h_x^ih_y^jU^TWU, \end{aligned}$$

where U is a 16-dimensional vector as follows

$$\begin{aligned} U=&[u^1,u^2,u^3,u^4,h_x^{i}u^1_x,h_x^{i}u^2_x,h_x^{i}u^3_x,h_x^{i}u^4_x,h_y^{j}u^1_y,h_y^{j}u^2_y,\\&h_y^{j}u^3_y,h_y^{j}u^4_y,h_x^{i}h_y^{j}u^1_{xy},h_x^{i}h_y^{j}u^2_{xy},h_x^{i}h_y^{j}u^3_{xy},h_x^{i}h_y^{j}u^4_{xy}]^T, \end{aligned}$$

and

$$\begin{aligned} W_0=diag\left[ \frac{1}{4}A_0,\frac{1}{48}A_0,\frac{1}{48}A_0,\frac{1}{576}A_0\right] , \quad A_0=diag[1,1,1,1]. \end{aligned}$$

Note that $W_0$ and W are both symmetric positive definite matrixes, we can obtain

$$\begin{aligned} c_5\Vert \varPi _h^*u_h\Vert _{K_{ij}}^2\le \Vert u_h\Vert _{K_{ij}}^2\le c_6\Vert \varPi _h^*u_h\Vert _{K_{ij}}^2, \end{aligned}$$

where $c_5,c_6$ are positive constants independent of $u_h$.

Summing up the above equivalence for all $K_{ij}$, we draw the conclusion. $\square$

Appendix C. Some Proof Details in Theorem 2

Proof of the local stability (3.16)

Recall the definition of $I^1_{K_{ij}}(u_h,\varPi _h^*u_h)$, we have

$$\begin{aligned} I_{K_{ij}}^1(u_h,\varPi _h^*u_h)=&\sum _{P\in \mathring{K}_{ij}} \left[ u_h(P)a_1^K(u_h,\psi _{P}^{(0)})+\left( \frac{\partial u_h}{\partial x}\right) (P)a^K_1(u_h,\psi _{P}^{(1)})\right. \nonumber \\&\left. \quad +\left( \frac{\partial u_h}{\partial y}\right) (P)a^K_1(u_h,\psi _{P}^{(2)})+\left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) (P) a^K_1(u_h,\psi _{P}^{(3)})\right] , \end{aligned}$$

(6.2)

where

$$\begin{aligned} a_1^K(u_h,\psi _{P}^{(0)})=&\int _{\partial K_P^*\cap K_{ij}}\frac{\partial \varDelta u_h}{\partial {\mathbf {n}}}ds, \\ a_1^K(u_h,\psi _{P}^{(1)})=&\int _{\partial K_P^*\cap K_{ij}}\frac{\partial \varDelta u_h}{\partial {\mathbf {n}}}(x-x_P)ds -\int _{\partial K_P^*\cap K_{ij}}\left( \frac{\partial ^2u_h}{\partial x^2}dy-\frac{\partial ^2u_h}{\partial x\partial y}dx\right) ,\\ a_1^K(u_h,\psi _{P}^{(2)})=&\int _{\partial K_P^*\cap K_{ij}}\frac{\partial \varDelta u_h}{\partial {\mathbf {n}}}(y-y_P)ds +\int _{\partial K_P^*\cap K_{ij}}\left( \frac{\partial ^2u_h}{\partial y^2}dx-\frac{\partial ^2u_h}{\partial x\partial y}dy\right) ,\\ a_1^K(u_h,\psi _{P}^{(3)})=&\int _{\partial K_P^*\cap K_{ij}}\frac{\partial \varDelta u_h}{\partial {\mathbf {n}}}(x-x_P)(y-y_P)ds\\&-\int _{\partial K_P^*\cap K_{ij}}\left( \frac{\partial ^2u_h}{\partial x^2}dy-\frac{\partial ^2u_h}{\partial x\partial y}dx\right) (y-y_P)\\&+\int _{\partial K_P^*\cap K_{ij}}\left( \frac{\partial ^2u_h}{\partial y^2}dx-\frac{\partial ^2u_h}{\partial x\partial y}dy\right) (x-x_P) +2\int _{K^*_P\cap K_{ij}}\frac{\partial ^2u_h}{\partial x\partial y}dxdy. \end{aligned}$$

Utilizing the linear combination of function values and derivative values of $u_h$ and the definition of $|u_h|_{2,h,K_{ij}}$, we have

$$\begin{aligned} I^1_{K_{ij}}(u_h,\varPi _h^*u_h) =&h_x^{i}h_y^{j}\left[ \frac{1}{2}\left( p_1b_{11}+p_2b_{12}+p_3b_{13}+p_4b_{14}+p_5b_{15}\right. \right. \nonumber \\&\left. \left. +p_6b_{16}+p_7b_{17}+p_8b_{18}\right) +t_1b_{19}+t_2b_{20}+t_3b_{21}+t_4b_{22}\right. \nonumber \\&\left. +t_5b_{23}+t_6b_{24}+t_7b_{25}+t_8b_{26}\right] + h_x^{i}h_y^{j}\nonumber \\&\left[ \frac{1}{2}\left( p_1d_{11}+p_3d_{12}+p_5d_{13}+p_7d_{14}+ t_1d_{15}+t_3d_{16}+t_5d_{17}+t_7d_{18}\right) \right. \nonumber \\&\left. + q_2d_{19}+q_3d_{20}+q_4d_{21}+q_7d_{22}+q_5d_{23}+q_6d_{24}+q_8d_{25}+q_9d_{26}\right] , \end{aligned}$$

(6.3)

where

$$\begin{aligned} b_{11}=&\int _0^{\frac{1}{2}}\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) d\eta + \theta _1\int _0^{\frac{1}{2}}\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) d\eta ,\\ b_{13}=&\int _{\frac{1}{2}}^1\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) d\eta + \theta _1\int _{\frac{1}{2}}^1\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) d\eta ,\\ b_{12}=&\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial x^2}d\eta ,~ b_{14}=\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial x^2}d\eta ,~\\ b_{15}=&\int _0^{\frac{1}{2}}\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) \cdot \eta d\eta + \theta _1\int _0^{\frac{1}{2}}\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) \cdot \eta d\eta ,\\ b_{16}=&\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial x^2}\cdot \eta d\eta ,~ b_{18}=\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial x^2}\cdot (\eta -1) d\eta ,~\\ b_{19}=&\int _0^{\frac{1}{2}}\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) d\xi + \theta _1\int _0^{\frac{1}{2}}\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) d\xi ,\\ b_{17}=&\int _{\frac{1}{2}}^1\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) (\eta -1)d\eta + \theta _1\int _{\frac{1}{2}}^1\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) (\eta -1)d\eta ,~\\ b_{20}=&\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial y^2}d\xi ,~ b_{22}=\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial y^2}d\xi ,\\ b_{21}=&\int _{\frac{1}{2}}^1\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) d\xi + \theta _1\int _{\frac{1}{2}}^1\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) d\xi ,\\ b_{23}=&\int _0^{\frac{1}{2}}\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) \cdot \xi d\xi + \theta _1\int _0^{\frac{1}{2}}\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) \cdot \xi d\xi ,\\ b_{24}=&\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial y^2}\xi d\xi ,~ b_{25}=\int _{\frac{1}{2}}^1\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial y^2}\right) (\xi -1)d\xi + \theta _1\int _{\frac{1}{2}}^1\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x^2}\right) (\xi -1) d\xi ,\\ b_{26}=&\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial y^2}\cdot (\xi -1)d\xi ,~ d_{11}=\theta _2\int _0^{\frac{1}{2}}\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) d\eta ,~ d_{12}=\theta _2\int _{\frac{1}{2}}^1\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) d\eta ,\\ d_{13}=&\theta _2\int _0^{\frac{1}{2}}\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) \cdot \eta d\eta ,~ d_{14}=\theta _2\int _{\frac{1}{2}}^1\frac{\partial }{\partial \eta } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) \cdot (\eta -1)d\eta ,~\\ d_{15}=&\theta _2\int _0^{\frac{1}{2}}\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) d\xi ,\\ d_{16}=&\theta _2\int _{\frac{1}{2}}^1\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) d\xi ,~ d_{17}=\theta _2\int _0^{\frac{1}{2}}\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) \cdot \xi d\xi ,~\\ d_{18}=&\theta _2\int _{\frac{1}{2}}^1\frac{\partial }{\partial \xi } \left( \frac{\partial ^2 u_h}{\partial x\partial y}\right) \cdot (\xi -1)d\xi ,\\ d_{19}=&\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial x\partial y}d\xi ,\quad d_{20}=\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial x\partial y}d\xi ,\quad d_{21}=\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial x\partial y}d\eta ,\quad d_{22}=\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial x\partial y}d\eta ,\\ d_{23}=&2\int _0^{\frac{1}{2}}\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial x\partial y}d\xi d\eta -\frac{1}{2}d_{19}-\frac{1}{2}d_{21},~\\ d_{24}=&2\int _0^{\frac{1}{2}}\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial x\partial y}d\xi d\eta -\frac{1}{2}d_{20}-\frac{1}{2}d_{21},\\ d_{25}=&2\int _{\frac{1}{2}}^1\int _0^{\frac{1}{2}}\frac{\partial ^2 u_h}{\partial x\partial y}d\xi d\eta -\frac{1}{2}d_{19}-\frac{1}{2}d_{22},~\\ d_{26}=&2\int _{\frac{1}{2}}^1\int _{\frac{1}{2}}^1\frac{\partial ^2 u_h}{\partial x\partial y}d\xi d\eta -\frac{1}{2}d_{20}-\frac{1}{2}d_{22}. \end{aligned}$$

Here we have noticed that $\frac{\partial ^3u_h}{\partial x\partial y^2}=\theta _1\frac{\partial }{\partial x} \left( \frac{\partial ^2u_h}{\partial y^2}\right) +\theta _2\frac{\partial }{\partial y} \left( \frac{\partial ^2u_h}{\partial x\partial y}\right)$ and $\theta _1+\theta _2=1$, $\frac{\partial ^3u_h}{\partial x^2\partial y}$ similarly. Calculating (6.3) with (3.4)-(3.6) and selecting $\theta _1=\frac{1}{3}, \theta _2=\frac{2}{3}$, we get

$$\begin{aligned} I^1_{K_{ij}}(u_h,\varPi _h^*u_h)=h_x^{i}h_y^{j}\left\{ \left[ \begin{array}{cc} P^T&{}T^T\\ \end{array}\right] B\left[ \begin{array}{c} P\\ T\\ \end{array}\right] +Z^TDZ\right\} , \end{aligned}$$

(6.4)

where P and T are the vectors defined at the beginning of Sect. 3. B is a symmetric positive definite matrix, the minimum eigenvalue is $\lambda _{\min }^1\ge 8\times 10^{-4}$.

$$\begin{aligned}&\widetilde{B_1} =\frac{1}{320}\left[ \begin{array}{rrrr} 390&{}90&{}71 &{} -29\\ 90 &{} 390&{}29&{} -71 \\ 71&{}29&{} 16&{} -9\\ -29&{}-71&{} -9&{} 16\\ \end{array} \right] , ~ \widetilde{B_2} =\frac{1}{96}\left[ \begin{array}{rrrr} 36&{}-36 &{}6 &{} 6\\ -36 &{} 36&{}-6&{} -6 \\ 6&{}-6&{} 1&{} 1\\ 6&{}-6&{}1&{} 1\\ \end{array} \right] , ~\\&\widetilde{B_3} =\frac{1}{192}\left[ \begin{array}{rrrr} -24&{}-24 &{}-6 &{} 6\\ 24 &{} 24&{} 6&{} -6 \\ -4&{}-4&{}-1&{}1\\ -4&{}-4&{}-1&{}1\\ \end{array} \right] , \\&\widetilde{B_4} =\frac{1}{960}\left[ \begin{array}{rrrr} 390&{}90 &{}71 &{}-29\\ 90&{}390&{}29&{} -71 \\ 71&{}29&{}16&{}-9\\ -29&{}-71&{}-9&{} 16\\ \end{array} \right] , ~ \widetilde{B_5} =\left[ \begin{array}{rrrr} 0&{}0 &{}0 &{}0\\ 0 &{}0&{}0&{} 0 \\ 0&{}0&{} 0&{} 0\\ 0&{}0&{} 0&{} 0\\ \end{array} \right] , \quad B=\left[ \begin{array}{llll} \widetilde{B_1} &{}\widetilde{B_2} &{}\widetilde{B_5} &{} \widetilde{B_3} ^T\\ \widetilde{B_2} &{}\widetilde{B_1} &{} \widetilde{B_3} ^T&{} \widetilde{B_5} \\ \widetilde{B_5} &{} \widetilde{B_3} &{}\widetilde{B_4} &{} \widetilde{B_5} \\ \widetilde{B_3} &{} \widetilde{B_5} &{}\widetilde{B_5} &{}\widetilde{B_4} \\ \end{array} \right] . \end{aligned}$$

$Z=\frac{1}{h_x^{i}h_y^{j}}U$ is a vector,where U is defined in the proof of Lemma 2 and D is a following 16th order real matrix,

$$\begin{aligned} \begin{matrix} D=\frac{1}{48}\left[ \begin{array}{rrrrrrrrrrrrrrrr} 72&{}-72&{}-72&{}72&{}36&{}12 &{}-36&{}-12&{}36&{}-36&{}12&{}-12&{}10&{}6&{}6&{}2\\ -72&{}72&{}72&{}-72&{}-12&{}-36&{}12&{}36&{}-36&{}36&{}-12&{}12&{}-6 &{}-10 &{}-2 &{}-6\\ -72&{}72&{}72&{}-72&{}-36&{}-12&{}36&{}12&{}-12&{}12&{}-36&{}36&{} -6&{} -2&{}-10 &{}-6\\ 72&{}-72&{}-72&{}72&{}12&{}36&{}-12&{}-36&{}12&{}-12&{}36&{}-36&{} 2&{} 6&{}6 &{}10\\ -12&{}12&{}12&{}-12&{}18&{}-6&{}-18&{}6&{}10&{}-10&{}-2&{}2&{}5 &{}1 &{}3 &{}-1\\ -12&{}12&{}12&{}-12&{}-6&{}18&{}6&{}-18&{}10&{}-10&{}-2&{}2&{}1 &{}5 &{}-1 &{}3\\ 12&{}-12&{}-12&{}12&{}-18&{}6&{}18&{}-6&{}2&{}-2&{}-10&{}10&{}-3 &{} 1&{}-5 &{}-1\\ 12&{}-12&{}-12&{}12&{}6&{}-18&{}-6&{}18&{}2&{}-2&{}-10&{}10&{}1&{}-3 &{} -1&{}-5\\ -12&{}12&{}12&{}-12&{}10&{}-2&{}-10&{}2&{}18&{}-18&{}-6&{}6&{}5 &{} 3&{}1 &{}-1\\ 12&{}-12&{}-12&{}12&{}2&{}-10&{}-2&{}10&{}-18&{}18&{}6&{}-6&{}-3 &{}-5 &{}1 &{}-1\\ -12&{}12&{}12&{}-12&{}10&{}-2&{}-10&{}2&{}-6&{}6&{}18&{}-18&{}1&{}-1 &{} 5&{}3\\ 12&{}-12&{}-12&{}12&{}2&{}-10&{}-2&{}10&{}6&{}-6&{}-18&{}18&{}1 &{}-1 &{}-3 &{}-5\\ -6&{}6&{}6&{}-6&{}-1&{}-1&{}1&{}1&{}-1&{}1&{}-1&{}1&{} 4&{}-1 &{}-1 &{}0\\ -6&{}6&{}6&{}-6&{}-1&{}-1&{}1&{}1&{}-1&{}1&{}-1&{}1&{}-1 &{} 4&{}0 &{}-1\\ -6&{}6&{}6&{}-6&{}-1&{}-1&{}1&{}1&{}-1&{}1&{}-1&{}1&{}-1 &{}0 &{}4 &{}-1\\ -6&{}6&{}6&{}-6&{}-1&{}-1&{}1&{}1&{}-1&{}1&{}-1&{}1&{}0 &{}-1 &{}-1 &{}4\\ \end{array} \right] . \end{matrix} \end{aligned}$$

For the second term of (6.4), using the congruent transformation $Z=M_0Q$, we have

$$\begin{aligned} Z^TDZ=Q^T{\widetilde{D}}Q, \end{aligned}$$

where ${\widetilde{D}}=M_0^TDM_0$ is a symmetric positive definite matrix, and $\lambda _{\min }^2\ge 3.65\times 10^{-2}$.

Therefore, we obtain

$$\begin{aligned} I^1_{K_{ij}}(u_h,\varPi _h^*u_h)= & {} h_x^{i}h_y^{j}\min \left\{ \lambda _{\min }^1,\lambda _{\min }^2\right\} (P^2+T^2+Q^2)\\\ge & {} C|u_h|_{2,h,K_{ij}}^2. \end{aligned}$$

Distinctly, the stability of $I_{K_{ij}}^1(u_h,\varPi _h^*u_h)$ is independent of the length-width ratio of rectangular elements.

Proof of the local stability (3.19).

The proof is similar with the proof of (3.16). We have

$$\begin{aligned} I^2_{K_{ij}}(u_h,\varPi _h^*u_h) =&\sum _{P\in \mathring{K}_{ij}} \left[ u_h(P)a^K_2(u_h,\psi _{P}^{(0)})+\left( \frac{\partial u_h}{\partial x}\right) (P)a^K_2(u_h,\psi _{P}^{(1)})\right. \nonumber \\&\left. \quad + \left( \frac{\partial u_h}{\partial y}\right) (P)a^K_2(u_h,\psi _{P}^{(2)})+\left( \frac{\partial u_h}{\partial x\partial y}\right) (P) a^K_2(u_h,\psi _{P}^{(3)})\right] , \end{aligned}$$

(6.5)

where

$$\begin{aligned}&a^K_2(u_h,\psi _{P}^{(0)})= -\int _{\partial K^*_P\cap K_{ij}}\frac{\partial u_h}{\partial {\mathbf {n}}}\psi _{P}^{(0)}ds,\\&a^K_2(u_h,\psi _{P}^{(1)})=-\int _{\partial K^*_P\cap K_{ij}}\frac{\partial u_h}{\partial {\mathbf {n}}}\psi _{P}^{(1)}ds +\int _{K^*_{P}\cap K_{ij}}\frac{\partial u_h}{\partial x}dxdy,\\&a^K_2(u_h,\psi _{P}^{(2)})=-\int _{\partial K^*_P\cap K_{ij}}\frac{\partial u_h}{\partial {\mathbf {n}}}\psi _{P}^{(3)}ds +\int _{K^*_{P}\cap K_{ij}}\frac{\partial u_h}{\partial y}dxdy,\\&a^K_2(u_h,\psi _{P}^{(3)})=-\int _{\partial K^*_P\cap K_{ij}}\frac{\partial u_h}{\partial {\mathbf {n}}}\psi _{P}^{(4)}ds +\int _{K^*_{P}\cap K_{ij}}\frac{\partial u_h}{\partial x}(y-y_P)+\frac{\partial u_h}{\partial y}(x-x_P)dxdy, \end{aligned}$$

Through some linear combinations of function values and derivative values of $u_h$, we get

$$\begin{aligned}&I^2_{K_{ij}}(u_h,\varPi _h^*u_h)\\&\quad =h_x^{i}h_y^{j}\left[ e_1c_{11}+ e_2\left( c_{18}-\frac{1}{2}c_{11}\right) +e_3(c_{17}-c_{18}) +e_4c_{12}+e_5\left( c_{20}-\frac{1}{2}c_{12}\right) +e_6(c_{19}-c_{20})\right. \\&\qquad +\left. e_7c_{21}+e_8\left( c_{29}-\frac{1}{2}c_{21}\right) +e_9(c_{27}-c_{29})+e_{10}c_{23} +e_{11}\left( c_{33}-\frac{1}{2}c_{23}\right) +e_{12}(c_{31}-c_{33})\right. \\&\qquad +\left. f_1c_{13}+f_2\left( c_{22}-\frac{1}{2}c_{13}\right) +f_3(c_{24}-c_{22})+f_4c_{14}+f_5(c_{26}-\frac{1}{2}c_{14}) +f_6(c_{25}-c_{26})\right. \\&\qquad \left. +f_7c_{15}+f_8\left( c_{32}-\frac{1}{2}c_{15}\right) + f_9(c_{28}-c_{32})+f_{10}c_{16}+f_{11} \left( c_{34}-\frac{1}{2}c_{16}\right) + f_{12}(c_{30}-c_{34}) \right] \\&\quad =h_x^{i}h_y^{j}\left[ E^TJE+F^TJF\right] , \end{aligned}$$

where

$$\begin{aligned}&c_{11}=\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial x}d\eta ,~ c_{12}=\int _0^{\frac{1}{2}}\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial x}d\xi d\eta ,~\\&c_{13}=\int _0^{\frac{1}{2}}\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial x}d\xi d\eta ,~ c_{14}=\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial x}d\eta ,\\&c_{15}=\int _{\frac{1}{2}}^1\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial x}d\xi d\eta ,~ c_{16}=\int _{\frac{1}{2}}^1\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial x}d\xi d\eta ,~ c_{17}=\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial x}\cdot \eta d\eta , \\&c_{18}=\int _0^{\frac{1}{2}}\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial x}\cdot \eta d\xi d\eta ,~ c_{19}=\int _0^{\frac{1}{2}}\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial x}\cdot \eta d\xi d\eta ,~ c_{20}=\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial x}\cdot (\eta -1) d\eta ,\\&c_{21}=\int _{\frac{1}{2}}^1\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial x}\cdot (\eta -1)d\xi d\eta ,~ c_{22}=\int _0^{\frac{1}{2}}\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial y}d\xi d\eta ,~\\&c_{23}=\int _{\frac{1}{2}}^1\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial x}\cdot (\eta -1)d\xi d\eta ,\\&c_{24}=\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial y}d\xi ,~ c_{25}=\int _{\frac{1}{2}}^1\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial y}d\xi d\eta ,~\\&c_{26}=\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial y}d\xi ,~ c_{27}=\int _{\frac{1}{2}}^1\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial y}d\xi d\eta ,\\&c_{28}=\int _0^{\frac{1}{2}}\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial y}d\xi d\eta ,~ c_{29}=\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial y}\cdot \xi d\xi ,~ \\&c_{30}=\int _{\frac{1}{2}}^1\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial y}\cdot \xi d\xi d\eta ,~ c_{31}=\int _0^{\frac{1}{2}}\int _0^{\frac{1}{2}}\frac{\partial u_h}{\partial y}\cdot \xi d\xi d\eta , \\&c_{32}=\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial y}\cdot (\xi -1) d\xi ,~ c_{33}=\int _{\frac{1}{2}}^1\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial y}\cdot (\xi -1) d\xi d\eta ,~\\&c_{34}=\int _0^{\frac{1}{2}}\int _{\frac{1}{2}}^1\frac{\partial u_h}{\partial y}\cdot (\xi -1) d\xi d\eta , \end{aligned}$$

and J is a real symmetric positive definite matrix and minimum eigenvalue is $\lambda _{\min }^3\ge 1\times 10^{-4}$.

By (3.9), we get

$$\begin{aligned} I^2_{K_{ij}}(u_h,\varPi _h^*u_h)\ge C|u_h|_{1,h,K_{ij}}^2. \end{aligned}$$

Appendix D. The Specific Representation of $\left| I_{K_{ij}}^1\right|$ and $\left| I^2_{K_{ij}}\right|$

$$\begin{aligned}&|I^1_{K_{ij}}(u-\varPi _hu,\varPi _h^*w_h)|\\&\quad =\varepsilon ^2\left| \sum _{l=1,3}\frac{h_x^i}{2}\left[ w^l_x+w^{l+1}_x-\frac{2}{h_x^i}(w^{l+1}-w^l)\right] \int _{\overline{M_lQ}} \sum _{m=0,2}\frac{\partial ^3(\varPi _hu-u)}{\partial x^{3-m}\partial y^m}dy\right. \\&\qquad +\left. \sum _{l=1,2}\frac{h_y^j}{2}\left[ w^{l}_y+w^{l+2}_y-\frac{2}{h_y^j}(w^{l+2}-w^{l})\right] \int _{\overline{M_{2l}Q}} \sum _{m=0,2}\frac{\partial ^3(\varPi _hu-u)}{\partial x^m\partial y^{3-m}}dx\right. \\&\qquad +\left. \sum _{l=1,3}\frac{h_x^i}{2}\left[ w^l_{xy}+w^{l+1}_{xy}-\frac{2}{h_x^i}(w^{l+1}_y-w^l_y)\right] \int _{\overline{M_lQ}} \sum _{m=0,2}\frac{\partial ^3(\varPi _hu-u)}{\partial x^{3-m}\partial y^m}(y-y_l)dy\right. \\&\qquad +\left. \sum _{l=1,2}\frac{h_y^j}{2}\left[ w^{l}_{xy}+w^{l+2}_{xy} -\frac{2}{h_y^j}(w^{l+2}_x-w^{l}_x)\right] \int _{\overline{M_{2l}Q}}\sum _{m=0,2}\frac{\partial ^3(\varPi _hu-u)}{\partial x^{m}\partial y^{3-m}}(x-x_{l})dx\right. \\&\qquad +\left. \sum _{l=1,3}(w^{l+1}_x-w^l_x)\int _{\overline{M_lQ}}\frac{\partial ^2(\varPi _hu-u)}{\partial x^2}dy+ \sum _{l=1,2}(w^{l+2}_y-w^l_y)\int _{\overline{M_{2l}Q}}\frac{\partial ^2(\varPi _hu-u)}{\partial y^2}dx\right. \\&\qquad +\left. \sum _{l=1,3}(w^{l+1}_y-w^l_y)\int _{\overline{M_lQ}}\frac{\partial ^2(\varPi _hu-u)}{\partial x\partial y}dy+ \sum _{l=1,2}(w^{l+2}_x-w^l_x)\int _{\overline{M_{2l}Q}}\frac{\partial ^2(\varPi _hu-u)}{\partial x\partial y}dx \right| \\&|I^2_{K_{ij}}(u-\varPi _hu,\varPi _h^*w_h)|\\&\quad =\left| \sum _{l=1,3}\left[ w^{l+1}-w^l-(w_x^{l+1}+w_x^l)h_x^i/2\right] \int _{\overline{M_lQ}}\frac{\partial (\varPi _hu-u)}{\partial x}dy\right. \\&\qquad +\left. \sum _{l=1,2}\left[ w^{l+2}-w^l-(w_y^{l+2}+w_y^l)h_y^j/2\right] \int _{\overline{M_lQ}}\frac{\partial (\varPi _hu-u)}{\partial y}dx\right. \\&\qquad +\left. \sum _{l=1,3}\left[ w_y^{l+1}-w_y^l-(w_{xy}^{l+1}+w_{xy}^l)h_x^i/2\right] \int _{\overline{M_lQ}}\frac{\partial (\varPi _hu-u)}{\partial x}(y-y_l)dy\right. \\&\qquad +\left. \sum _{l=1,2}\left[ w_x^{l+2}-w_x^l-(w_{xy}^{l+2}+w_{xy}^l)h_y^j/2\right] \int _{\overline{M_lQ}}\frac{\partial (\varPi _hu-u)}{\partial y}(x-x_l)dy\right. \\&\qquad +\left. \sum _{l=1}^4w_{xy}^l\int _{K^l_{ij}}\frac{\partial (\varPi _hu-u)}{\partial x}(y-y_l)+ \frac{\partial (\varPi _hu-u)}{\partial y}(x-x_l)dxdy\right| . \end{aligned}$$