A convergent adaptive finite element stochastic Galerkin method based on multilevel expansions of random fields

TL;DR

The paper addresses adaptive stochastic Galerkin finite element approximation for parametric elliptic PDEs with coefficient expansions $a(y)=\theta_0+\sum_{\mu} y_{\mu}\theta_{\mu}$. It introduces independent spatial meshes for each polynomial coefficient and leverages multilevel, wavelet-like expansions of random fields to obtain sparse polynomial representations and localized spatial features; error reduction is guaranteed via a saturation property proven using finite element frames (BPX) and operator compression in the stochastic variables. The resulting algorithm achieves convergence with near-optimal computational complexity, demonstrated by numerical experiments even for random fields with low regularity. The approach is robust to the choice of parameter expansions (including KL-type and multilevel wavelets) and yields a framework for scalable, adaptive stochastic Galerkin discretizations that can handle interactions across many spatial meshes within a single solver cycle.

Abstract

The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.

Figures (4)

• Figure 1: Plots of Legendre coefficients $u_\nu \in H^1_0(0,1)$ (normalized to equal $\lVert u_\nu\rVert_{L\infty}$) in the expansion \ref{['eq:legendreexpansion']} with $a$ given by the hierarchical hat function expansion \ref{['eq:lc1d']} with $\alpha=1$, and nodes of adaptively generated piecewise linear approximations on dyadic subintervals.
• Figure 2: Triangulation of approximations of different components $u_\nu$ in \ref{['eq:legendreexpansion']}, analogous to Figure \ref{['fig:unu']}, generated by the adaptive method for different $\nu$ for $d=2$. Top: $\nu = 0$, bottom: $\nu = e_\mu$ for two different $\mu \in \mathcal{M}$.
• Figure 3: Computed residual bounds for $d=1$ as a function of total number of degrees of freedom $\dim \mathcal{V}({\mathbb{T}}) \eqsim N(\mathbb{T})$ of the current approximation of $u$ (solid gray lines), degrees of stochastic freedom $\#F$ (solid black lines), and elapsed computation time in seconds (dash-dotted line).
• Figure 4: Computed residual bounds for $d=2$ as a function of total number of degrees of freedom $\dim \mathcal{V}({\mathbb{T}}) \eqsim N(\mathbb{T})$ of the current approximation of $u$ (solid gray lines), number of product Legendre polynomials $\#F$ (solid black lines), and elapsed computation time in seconds (dash-dotted line).

Theorems & Definitions (31)

• Remark 2.1
• Lemma 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Proposition 3.1
• Proposition 3.2
• Remark 3.3
• Lemma 3.4
• proof