Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity

Abstract

This paper adopts an accurate and robust algorithm for the nonlocal heat equation having a weakly singular kernel in three dimensions. The proposed method approximates the unknown solution in two steps. First, the temporal discretization is achieved through a Crank-Nicolson finite difference with the nonuniform temporal meshes, which are applied to tackle the singularity behavior for the exact solution when t=0. Second, the spatial discretization is derived by means of the Galerkin finite element. Additionally, the alternating direction implicit (ADI) approach is adopted to decrease computational burden. The proposed method in terms of the convergence and stability analysis is studied by using the discrete energy method in detail with optimal rates of convergence O(k2+hr+1), r≥1 based on some appropriate assumptions, where k and h represent step sizes in the temporal and spatial directions, respectively. Numerical results highlight the effectiveness of the proposed method and the correctness of the theoretical prediction.