An adaptive stochastic Galerkin method for random elliptic operators

Gittelson, Claude

doi:10.1090/S0025-5718-2013-02654-3

An adaptive stochastic Galerkin method for random elliptic operators

Author: Claude Jeffrey Gittelson
Journal: Math. Comp. 82 (2013), 1515-1541
MSC (2010): Primary 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20
DOI: https://doi.org/10.1090/S0025-5718-2013-02654-3
Published electronically: February 12, 2013
MathSciNet review: 3042573
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Abstract: We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence of the random solution, which combines with any discretization on the spatial domain. In addition to selecting active polynomial modes, this solver can adaptively construct a separate spatial discretization for each of their coefficients. We show convergence of the solver in this general setting, along with a computable bound for the mean square error, and an optimality property in the case of a single spatial discretization. Numerical computations demonstrate convergence of the solver and compare it to a sparse tensor product construction.

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