Convergence Analysis of a LDG Method for Time–Space Tempered Fractional Diffusion Equations with Weakly Singular Solutions

Abstract

A class of time–space tempered fractional diffusion equations is considered in this paper. The solution of these problems generally have a weak singularity near the initial time $t = 0$. To solve the time–space tempered fractional diffusion equations, a fully discrete local discontinuous Galerkin (LDG) method is proposed. The basic idea is to apply LDG method in the space on uniform meshes and a finite difference method in the time on graded meshes to deal with the weak singularity at initial time $t = 0$. The discrete fractional Grönwall inequality is used to analyze the stability and convergence of the method. Numerical results show that the proposed method for time–space tempered fractional diffusion equation is accurate and reliable.

Appendix

Recently, Stynes et al. [50] proved that the following fractional subdiffusion differential equation has a unique solution under proper regularity and compatibility assumption,

$$\begin{aligned} {}_0^C\mathbb {D} _t^{\beta } u(x,t) - d\, \frac{\partial ^2 u(x,t)}{\partial x^2} = f(x,t),\,\, on \,\,\,\,{\varOmega } \times (0,T], \end{aligned}$$

(56)

and there exist a constant $C_u$ such that

$$\begin{aligned} \left| \frac{\partial ^{\ell } u(x,t)}{\partial t^\ell } \right| \le C_u(1+t^{\beta -\ell }), \quad \ell =0,1,2, \quad 0<t\le T. \end{aligned}$$

Liao et al. [32] presented the sharp error estimate of the nonuniform L1 formula for linear subdiffusion equations by assuming

$$\begin{aligned} \left| \frac{\partial u^{\ell }(x,t)}{\partial t^\ell } \right| \le C_u(1+t^{\sigma -\ell }), \quad \ell =0,1,2, \quad 0<t\le T, \end{aligned}$$

(57)

where the parameter $\sigma \in (0,1) \cup (1,2)$ reflects the regularity of the solution.

Time–space tempered fractional diffusion equation (1) can be written as follows by considering definition 2,

$$\begin{aligned} {}_0^C\mathbb {D} _t^{\beta } v(x,t) -d\,{}_a{D}_x^{\alpha , \lambda } v(x,t) = g(x,t),\,\, on \,\,\,\,{\varOmega } \times (0,T], \end{aligned}$$

(58)

where $v(x,t)=e^{\gamma t}u(x,t)$ and $g(x,t)= e^{\gamma t}f(x,t)$. Fractional differential equation (58) is a version of fractional differential equation (56). Hence, the solution of (58) must be satisfied the following conditions,

$$\begin{aligned} \left| \frac{\partial ^{\ell } v(x,t)}{\partial t^\ell } \right| \le C_v(1+t^{\beta -\ell }), \quad \ell =0,1,2, \quad 0<t\le T. \end{aligned}$$

(59)

Note that, conditions (56) are equivalent to the following conditions

$$\begin{aligned} \left| \frac{\partial u^{\ell }(x,t)}{\partial t^\ell } \right| \le C_u(1+t^{\beta -\ell }), \quad \ell =0,1,2, \quad 0<t\le T. \end{aligned}$$

(60)

To present the error estimate, we assume the following condition similarly to Liao et al. [32, 34], Ren et al. [48] and Cao et al. [5]

$$\begin{aligned} \left| \frac{\partial ^{\ell } u(x,t)}{\partial t^\ell } \right| \le C_u(1+t^{\sigma -\ell }), \quad \ell =0,1,2, \quad 0<t\le T. \end{aligned}$$

where the parameter $\sigma \in (0,1) \cup (1,2)$ reflects the regularity of the solution.