Fast calculation based on a spatial two‐grid finite element algorithm for a nonlinear space–time fractional diffusion model

Abstract

In this article, a spatial two-grid finite element (TGFE) algorithm is used to solve a two-dimensional nonlinear space-time fractional diffusion model and improve the computational efficiency. First, the second-order backward difference scheme is used to formulate the time approximation, where the time-fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In order to reduce the computation time of the standard FE method, a TGFE algorithm is developed. The specific algorithm is to iteratively solve a nonlinear system on the coarse grid and then to solve a linear system on the fine grid. We prove the scheme stability of the TGFE algorithm and derive a priori error estimate with the convergence result O(Δt 2 + h r + 1 − + H 2r + 2 − 2). Finally, through a two-dimensional numerical calculation, we improve the computational efficiency and reduce the computation time by the TGFE algorithm. KEYWORDS error analysis, nonlinear space-time fractional diffusion model, two-grid finite element algorithm 1 INTRODUCTION Space-time fractional partial differential equations (FPDEs) are a kind of important mathematical models. Because of the complexity of structure for this class of fractional models, people need to develop efficient numerical methods to achieve the numerical solutions. One can see some numerical studies such as difference algorithms for the advection-diffusion equation with space-time fractional