A robust multilevel approach for minimizing H(div)‐dominated functionals in an H1‐conforming finite element space

Abstract

AbstractThe standard multigrid algorithm is widely known to yield optimal convergence whenever all high‐frequency error components correspond to large relative eigenvalues. This property guarantees that smoothers like Gauss–Seidel and Jacobi will significantly dampen all the high‐frequency error components, and thus, produce a smooth error. This has been established for matrices generated from standard discretizations of most elliptic equations. In this paper, we address a system of equations that is generated from a perturbation of the non‐elliptic operator I‐grad div by a negative ε Δ. For εnear to one, this operator is elliptic, but as εapproaches zero, the operator becomes non‐elliptic as it is dominated by its non‐elliptic part. Previous research on the non‐elliptic part has revealed that discretizing I‐grad div with the proper finite element space allows one to define a robust geometric multigrid algorithm. The robustness of the multigrid algorithm depends on a relaxation operator that yields a smooth error. We use this research to assist in developing a robust discretization and solution method for the perturbed problem. To this end, we introduce a new finite element space for tensor product meshes that is used in the discretization, and a relaxation operator that succeeds in dampening all high‐frequency error components. The success of the corresponding multigrid algorithm is first demonstrated by numerical results that quantitatively imply convergence for any εis bounded by the convergence for εequal to zero. Then we prove that convergence of this multigrid algorithm for the case of ε equal to zero is independent of mesh size. Copyright © 2004 John Wiley & Sons, Ltd.