An FFT-Based MAC scheme for Stokes equations with periodic boundary conditions and its application to elasticity problems

Abstract

This paper presents a fast Fourier transform (FFT) based finite difference MAC scheme to compute the Stokes equations with periodic boundary conditions. The numerical implementation of this solver is very convenient and only requires three simple steps. The remarkable superiorities of such a solver compared to classical iterative approaches are computational time-saving and memory storage saving. The application of the proposed solver to nearly incompressible linear elasticity problems is also introduced. In addition, following the similar spirit used in Dong et al. (2020) for Stokes equations with Dirichlet boundary conditions, a rigorous error analysis is provided for Stokes equations with periodic boundary conditions. The results include: a second-order convergence in the ℓ2-norm for velocity, pressure as well as the gradient of velocity; a second-order accuracy in maximum norm for both velocity and its gradient. Moreover, the properties of the discrete Green functions play an important part in maximum error estimation. Most notably, the major difficulties of the extension from Dirichlet boundary conditions to periodic boundary conditions are the discrete version of Green’s formulae, Poincaré inequality and LBB condition. A variety of two-dimensional and three-dimensional numerical examples are given to illustrate the efficiency of the proposed FFT solver and validate the theoretical results.