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

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.