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Convergence analysis of the adaptive stochastic collocation finite element method

Alex Bespalov, Andrey Savinov

TL;DR

This work develops a rigorous convergence analysis for an adaptive single-level SC-FEM applied to parametric elliptic PDEs with affine and nonaffine dependence. It extends a posteriori error estimators to general parametric inputs and proves that the adaptive algorithm drives the total error estimate $\mu_\ell+\tau_\ell$ to zero, leveraging a summability property of Taylor coefficients for semidiscrete solutions and, in the nonaffine case, analyticity in the parameter domain. The analysis covers both parametric and spatial refinements, establishing convergence of the parametric indicators $\tau_\ell$ and the spatial indicators $\mu_{\ell\mathbf{z}}$ under standard marking strategies, and provides a robust framework for adaptive enrichment driven by hierarchical surpluses. Numerical experiments on affine (cookie) and nonaffine (Fourier-mode) test cases corroborate the theory, showing reliable error decay and highlighting practical differences between Leja and Clenshaw–Curtis collocation nodes.

Abstract

This paper is focused on the convergence analysis of an adaptive stochastic collocation algorithm for the stationary diffusion equation with parametric coefficient. The algorithm employs sparse grid collocation in the parameter domain alongside finite element approximations in the spatial domain, and adaptivity is driven by recently proposed parametric and spatial a posteriori error indicators. We prove that for a general diffusion coefficient with finite-dimensional parametrization, the algorithm drives the underlying error estimates to zero. Thus, our analysis covers problems with affine and nonaffine parametric coefficient dependence.

Convergence analysis of the adaptive stochastic collocation finite element method

TL;DR

This work develops a rigorous convergence analysis for an adaptive single-level SC-FEM applied to parametric elliptic PDEs with affine and nonaffine dependence. It extends a posteriori error estimators to general parametric inputs and proves that the adaptive algorithm drives the total error estimate to zero, leveraging a summability property of Taylor coefficients for semidiscrete solutions and, in the nonaffine case, analyticity in the parameter domain. The analysis covers both parametric and spatial refinements, establishing convergence of the parametric indicators and the spatial indicators under standard marking strategies, and provides a robust framework for adaptive enrichment driven by hierarchical surpluses. Numerical experiments on affine (cookie) and nonaffine (Fourier-mode) test cases corroborate the theory, showing reliable error decay and highlighting practical differences between Leja and Clenshaw–Curtis collocation nodes.

Abstract

This paper is focused on the convergence analysis of an adaptive stochastic collocation algorithm for the stationary diffusion equation with parametric coefficient. The algorithm employs sparse grid collocation in the parameter domain alongside finite element approximations in the spatial domain, and adaptivity is driven by recently proposed parametric and spatial a posteriori error indicators. We prove that for a general diffusion coefficient with finite-dimensional parametrization, the algorithm drives the underlying error estimates to zero. Thus, our analysis covers problems with affine and nonaffine parametric coefficient dependence.
Paper Structure (13 sections, 10 theorems, 86 equations, 6 figures, 2 algorithms)

This paper contains 13 sections, 10 theorems, 86 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Parametric model problem
  3. Stochastic collocation finite element method
  4. Sparse grid interpolation
  5. The summability property of Taylor coefficients
  6. Error estimates, error indicators and adaptive algorithm
  7. Convergence of parametric error estimates
  8. Convergence of spatial error estimates
  9. Convergence of the adaptive algorithm
  10. Numerical results
  11. Test case I: affine coefficient data (cookie problem)
  12. Test case II: nonaffine coefficient data
  13. Concluding remarks

Key Result

Lemma 1

Suppose that the diffusion coefficient has affine representation, i.e., If the expansion coefficients $a_m \in L^{\infty}(D)$, $m = 0,1,\ldots,M$, satisfy the uniform ellipticity assumption, i.e., then inequalities eq:amin:amax hold and the Taylor coefficients $w_{\mathbf{i}}(x)$ given by eq:Taylor:coeff satisfy the summability property eq:summability:property.

Figures (6)

  • Figure 1: An example of iteration subsequences illustrating Scenario 3 in the proof of Theorem \ref{['theorem:main']}.
  • Figure 2: Test case I: spatial domain and subdomains (left) and the refined mesh after 18 iterations of Algorithm \ref{['algorithm']} using Leja collocation points (right).
  • Figure 3: Test case I: evolution of the weighted sums of error indicators for Leja (left) and CC (right) points. The axes limits are identical in the left and right plots.
  • Figure 4: Test case I: evolution of the error estimates for Leja (left) and CC (right) points. The axes limits are identical in the left and right plots.
  • Figure 5: Test case II: evolution of the weighted sums of error indicators for Leja (left) and CC (right) points. The axes limits are identical in the left and right plots.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 8
  • Lemma 9: FeischlS21
  • Lemma 10: bprr18
  • ...and 10 more