Applications of Multigrid Algorithms to Finite Difference Schemes for Elliptic Equations with Variable Coefficients

Abstract

This paper is devoted to a study of multigrid algorithms applied to finite difference schemes. If the elliptic equation has variable coefficients, the analysis of multigrid algorithms in the existent literature only gave a convergence rate depending on the number of levels. In this paper, for multigrid algorithms applied to finite difference schemes for elliptic equations with variable coefficients, we establish a convergence rate independent of the number of levels. Our convergence analysis does not require any additional regularity of the solution and is valid for commonly used smoothing operators including the standard Gauss-Seidel method. Under guidance of the general theory, we give details of implementation of the inherited multigrid V(1,1) algorithm. Furthermore, we will provide numerical examples to illustrate the general theory and demonstrate that the inherited multigrid algorithm is efficient for numerical solutions of elliptic equations with variable coefficients. In particular, we will consider elliptic equations on an L-shaped domain whose solutions do not have full regularity and show that the multigrid V(1,1) algorithm performs well in such situations.

Uniform convergence of V-cycle multigrid finite element method for one-dimensional time-dependent fractional problem

“…The elegant theoretical framework and uniform convergence of the V-cycle MGM for second order elliptic equation is well established in [7,9,55]. The convergence rate independent of the number of levels is presented by multigrid finite difference method for elliptic equations with variable coefficients [38]. In the case of multilevel matrix algebras, for special prolongation operators [35], the convergence rate of the V-cycle MGM is derived in [3,4,8] for the elliptic PDEs.…”