On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion

2020/08/17 by Martin Eigel, Oliver P. Ernst, Eigel, Martin +5 · 1 citation
Engineering · #35R60 #60H25 #65C30 #65D05 #65D15 #65N12 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2008.07186

openalex publication_date 2020/08/17 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper "Dimension-adaptive tensor-product quadratuture" Computing (2003) by T. Gerstner and M. Griebel) is explored.

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