A probabilistic finite element method based on random meshes: Error estimators and Bayesian inverse problems

TL;DR

This paper introduces a probabilistic finite element method using random meshes for solving elliptic PDEs, providing new error estimators and improving Bayesian inverse problem solutions through uncertainty quantification.

Contribution

It presents a novel probabilistic FEM framework based on random meshes, with new error estimators and applications to Bayesian inverse problems.

Findings

Numerical experiments validate the error estimators.

RM-FEM improves solution quality in Bayesian inverse problems.

Probabilistic approach enhances uncertainty quantification.

Abstract

We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a probability measure on standard piecewise linear FEM. We present a posteriori error estimators based uniquely on probabilistic information. A series of numerical experiments illustrates the potential of the RM-FEM for error estimation and validates our analysis. We furthermore demonstrate how employing the RM-FEM enhances the quality of the solution of Bayesian inverse problems, thus allowing a better quantification of numerical errors in pipelines of computations.