A Finite Volume Method to Solve the Ill-Posed Elliptic Problems

Abstract

In this paper, we propose a finite volume element method of primal-dual type to solve the ill-posed elliptic problem, that is, the elliptic problem with lacking or overlapping boundary value condition. We first establish the primal-dual finite volume element scheme by introducing the Lagrange multiplier λ and prove the well-posedness of the discrete scheme. Then, the error estimations of the finite volume solution are derived under some proper norms including the H1-norm. Numerical experiments are provided to verify the effectiveness of the proposed finite volume element method at last.