A finite element method for elliptic problems with observational boundary data

TL;DR

This paper introduces a finite element method for elliptic problems with noisy observational boundary data, demonstrating convergence and probabilistic error bounds, supported by numerical experiments.

Contribution

It develops a finite element approach using Lagrangian multipliers for elliptic problems with noisy boundary data, providing convergence analysis and probabilistic error estimates.

Findings

Convergence of finite element error in expectation

Exponential decay of error violation probability under sub-Gaussian noise

Numerical examples validate theoretical results

Abstract

In this paper we propose a finite element method for solving elliptic equations with the observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier. We show the convergence of the random finite element error in expectation and, when the noise is sub-Gaussian, in the Orlicz 2- norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.