Analysis of the Cell Vertex Finite Volume Method for the Cauchy-Riemann Equations

This paper initiates a study of finite volume methods for linear first-order elliptic systems by performing a stability and convergence analysis of the cell vertex approximation of the Cauchy-Riemann equations. The approach is based on reformulating the scheme as a Petrov-Galerkin finite element method with continuous bilinear trial functions and piecewise constant test functions. Optimal error bounds are derived in a mesh-dependent norm, and the counting problem which may occur due to geometry and boundary conditions is considered.