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Adaptive algorithm for stochastic Galerkin method

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Abstract

We introduce a new tool for obtaining efficient a posteriori estimates of errors of approximate solutions of differential equations the data of which depend linearly on random parameters. The solution method is the stochastic Galerkin method. Polynomial chaos expansion of the solution is considered and the approximation spaces are tensor products of univariate polynomials in random variables and of finite element basis functions. We derive a uniform upper bound to the strengthened Cauchy-Bunyakowski-Schwarz constant for a certain hierarchical decomposition of these spaces. Based on this, an adaptive algorithm is proposed. A simple numerical example illustrates the efficiency of the algorithm. Only the uniform distribution of random variables is considered in this paper, but the results obtained can be modified to any other type of random variables.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29, Praha 6, Czech Republic

    Ivana Pultarová

  2. Department of Mathematics, College of Polytechnics Jihlava, Tolstého 16, 586 01, Jihlava, Czech Republic

    Ivana Pultarová

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  1. Ivana Pultarová
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Correspondence to Ivana Pultarová.

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Pultarová, I. Adaptive algorithm for stochastic Galerkin method. Appl Math 60, 551–571 (2015). https://doi.org/10.1007/s10492-015-0111-9

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  • DOI: https://doi.org/10.1007/s10492-015-0111-9

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