Conforming VEM for general second-order elliptic problems with rough data on polygonal meshes and its application to a Poisson inverse source problem

TL;DR

This paper develops a conforming virtual element method with a novel companion operator for second-order elliptic problems with rough data, extending analysis to inverse source problems and demonstrating effectiveness through numerical tests.

Contribution

It introduces a new conforming companion operator and modified virtual element scheme for rough data, extending error analysis to inverse problems.

Findings

Error estimates without data oscillations

Effective approximation of inverse source problems

Numerical validation on polygonal meshes

Abstract

This paper focuses on the analysis of conforming virtual element methods for general second-order linear elliptic problems with rough source terms and applies it to a Poisson inverse source problem with rough measurements. For the forward problem, when the source term belongs to $H^{-1}(\Omega)$, the right-hand side for the discrete approximation defined through polynomial projections is not meaningful even for standard conforming virtual element method. The modified discrete scheme in this paper introduces a novel companion operator in the context of conforming virtual element method and allows data in $H^{-1}(\Omega)$. This paper has {\it three} main contributions. The {\it first} contribution is the design of a conforming companion operator $J$ from the {\it conforming virtual element space} to the Sobolev space $V:=H^1_0(\Omega)$, a modified virtual element scheme, and the \textit{a priori} error estimate for the Poisson problem in the best-approximation form without data oscillations. The {\it second} contribution is the extension of the \textit{a priori} analysis to general second-order elliptic problems with source term in $V^*$. The {\it third} contribution is an application of the companion operator in a Poisson inverse source problem when the measurements belong to $V^*$. The Tikhonov's regularization technique regularizes the ill-posed inverse problem, and the conforming virtual element method approximates the regularized problem given a finite measurement data. The inverse problem is also discretised using the conforming virtual element method and error estimates are established. Numerical tests on different polygonal meshes for general second-order problems, and for a Poisson inverse source problem with finite measurement data verify the theoretical results.