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

Abstract

This paper focuses on the analysis of conforming virtual element methods (VEMs) 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 [Formula: see text], the right-hand side for the discrete approximation defined through polynomial projections is not meaningful even for standard conforming VEM. The modified discrete scheme in this paper introduces a novel companion operator in the context of conforming VEM and allows data in [Formula: see text]. This paper has three main contributions. The first contribution is the design of a conforming companion operator [Formula: see text] from the conforming virtual element space to the Sobolev space [Formula: see text], a modified virtual element scheme, and the a priori error estimate for the Poisson problem in the best-approximation form without data oscillations. The second contribution is the extension of the a priori analysis to general second-order elliptic problems with source term in [Formula: see text]. The third contribution is an application of the companion operator in a Poisson inverse source problem when the measurements belong to [Formula: see text]. Tikhonov’s regularization technique regularizes the ill-posed inverse problem, and the conforming VEM approximates the regularized problem given a finite measurement data. The inverse problem is also discretized using the conforming VEM 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.