  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.
