Unified Analysis of Finite Volume Methods for Second Order Elliptic Problems

We establish a general framework for analyzing the class of finite volume methods which employ continuous or totally discontinuous trial functions and piecewise constant test functions. Under the framework, optimal order convergence in the H1 and L2 norms can be obtained in a natural and systematic way for classical finite volume methods and new finite volume methods such as discontinuous finite volume methods applied to second order elliptic problems. Key words, finite lement methods, finite volume methods, discontinuous Galerkin methods, finite volume element AMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50 DOI. 10.1137/050643994