A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

Abstract

For two-dimensional (2D) time fractional diffusion equations, we construct a numerical method based on a local discontinuous Galerkin (LDG) method in space and a finite difference scheme in time. We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable. Numerical results indicate the effectiveness and accuracy of the method and confirm the analysis.

Appendix A

Another approximations to the time-fractional derivative (2) are L1-2 and L1-2-3 formulae [16, 32] which can be obtained by using quadratic and cubic interpolation formulae, respectively. The order of convergence with respect to the time variable for L1, L1-2, and L1-2-3 formulas are $2-\alpha$, $3-\alpha$, and $4-\alpha$, respectively. We follow here just L1-2 formula which is

$$\begin{aligned} {\text{D}}^{\alpha }_{t}u(\cdot ,t_n)=\frac{(\Delta t)^{1-\alpha }}{\Gamma (2-\alpha )}\sum _{i=0}^{n-1}c_i\frac{u(\cdot ,t_{n-i})-u(\cdot ,t_{n-i-1})}{\Delta t}+\gamma ^n(\cdot ), \end{aligned}$$

where $c_0=1$ for $n=1$; and for $n\geq 2$,

$$\begin{aligned} c_j=\left\{ \begin{array}{lll} a_0+d_0, &{} &{} j=0,\\ a_j+d_j-d_{j-1}, &{} &{} 1\leq j\leq n-2,\\ a_j-d_{j-1}, &{} &{} j=n-1, \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} a_j=(j+1)^{(1-\alpha )}-j^{(1-\alpha )};~~~~~ 0\leq j\leq n-1, \end{aligned}$$

and

$$\begin{aligned} d_j=[(j+1)^{(2-\alpha )}-j^{(2-\alpha )}]/(2-\alpha )-[(j+1)^{(1-\alpha )}+j^{(1-\alpha )}]/2;~~~~~j\geq 0, \end{aligned}$$

and $\gamma ^n$ is the truncation error with the estimate

$$\begin{aligned} \Vert \gamma ^n\Vert \leq \left\{ \begin{array}{lll} C (\Delta t)^{2-\alpha },&{} &{}n=1,\\ \\ C (\Delta t)^{3-\alpha },&{} &{}n\geq 2. \end{array}\right. \end{aligned}$$

Then, we can define a fully discrete LDG scheme as follows: find $(u_{h}, {\varvec{q}}_{h})$, such that for all test functions $(v, {\varvec{v}})\in V_{h}^{k}\times {\varvec{V}}_{h}^{k}$,

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle (u^m_h, v)+\beta \left( ({\varvec{q}}^m_h, \nabla v)-\sum _{\kappa =1}^{K} \langle {\varvec{n}}\cdot \hat{{\varvec{q}}}^m_h, v\rangle _{\partial {D^\kappa }}\right) \\ =\beta (f^m, v) \displaystyle ~ +\sum _{i=1}^{m-1}(c_{i-1}-c_i)(u_h^{m-i}, v)+c_{m-1}(u_h^0, v),\\ \displaystyle ({\varvec{q}}^m_h, {\varvec{v}})+(u^m_h, \nabla \cdot {\varvec{v}})-\sum _{\kappa =1}^{K}\langle {\hat{u}}^m_h,{\varvec{n}}\cdot {\varvec{v}}\rangle _{\partial {D^\kappa }}=0, \end{array}\right. \end{aligned}$$

(A1)

where $\beta =(\Delta t)^\alpha {\varvec{\Gamma }}(2-\alpha )$. In order to examine the convergence of the scheme (A1), we express the following result.

Theorem A1

Let $u(\cdot ,t_n)$be the exact solution of problem (1) with homogeneous Dirichlet boundary conditions, which is sufficiently smooth with bounded derivatives, and $u_h^n$be the numerical solution of the LDG scheme (23). There holds the following error estimate on Cartesian meshes:

$$\begin{aligned} \Vert u(\cdot ,t_n)-u_h^n\Vert \leq \left\{ \begin{array}{lll} C( h^{k+1}+(\Delta t)^2+(\Delta t)^{\frac{\alpha }{2}} h^{k+\frac{1}{2}}),&{} &{}n=1,\\ \\ C(h^{k+1}+(\Delta t)^2+(\Delta t)^{\frac{\alpha }{2}} h^{k+\frac{1}{2}}),&{} &{}n\geq 2, \end{array}\right. \end{aligned}$$

and on triangular meshes

$$\begin{aligned} \Vert u(\cdot ,t_n)-u_h^n\Vert \leq \left\{ \begin{array}{ll} C( h^{k+1}+(\Delta t)^2+(\Delta t)^{\frac{\alpha }{2}}h^k),&{}n=1,\\ C(h^{k+1}+(\Delta t)^2+(\Delta t)^\frac{\alpha }{2}h^k),&{}n\geq 2, \end{array}\right. \end{aligned}$$

where C is a constant depending on $T, \alpha,$and u.

Proof

It is more or less similar to the proof of Theorem 2.