Published

Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids

Long Chen, Michael Holst, Jinchao Xu and Yunrong Zhu

Computing and Visualization in Science, 15(5), 271--289, 2012.

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ABSTRACT:

The goal of this paper is to design optimal multilevel
solvers for the finite element approximation of second order linear
elliptic problems with piecewise constant coefficients on bisection
grids. Local multigrid and BPX preconditioners are constructed based
on local smoothing only at the newest vertices and their immediate
neighbors. The analysis of eigenvalue distributions for these local
multilevel preconditioned systems shows that there are only a fixed
number of eigenvalues which are deteriorated by the large jump. The
remaining eigenvalues are bounded uniformly with respect to the
coefficients and the meshsize. Therefore, the resulting preconditioned
conjugate gradient algorithm will converge with an asymptotic rate
independent of the coefficients and logarithmically with respect to
the meshsize. As a result, the overall computational complexity is
nearly optimal.