Conforming VEM for general second-order elliptic problems with rough data on polygonal meshes and its application to a Poisson inverse source problem
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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(Ω), 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(Ω). This paper has three main contributions. The first contribution is the design of a conforming companion operator J from the conforming virtual element space to the Sobolev space V:=H^1_0(Ω), 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 V^*. The 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.
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