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Travis Austin, Thomas Manteuffel and Steve McCormick, "A robust multilevel approach for minimizing H(div)-dominated functionals in an H1-conforming finite element space", Numerical Linear Algebra , vol. 11, pp. 115-140, 2004


The 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 of 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 - ∇ ∇ ] by a negative ε Δ. For ε near to one, this operator is elliptic, but as ε approaches 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 - ∇ ∇ ] 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 ε is bounded by the convergence for ε equal to zero. Then we prove that convergence of this multigrid algorithm for the case of ε equal to zero is independent of mesh size.

BibTeX Entry

author = {Travis Austin and Thomas Manteuffel and Steve McCormick},
title = {A robust multilevel approach for minimizing H(div)-dominated functionals in an H1-conforming finite element space},
year = {2004},
urlpdf = {http://math.lanl.gov/~austint/Papers/austin-2004-robust.pdf},
journal = { Numerical Linear Algebra },
volume = {11},
pages = {115-140}