Error Estimates for the Finite Volume Element Method for Parabolic Equations in Convex Polygonal Domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data which yield optimal order error estimates in L2 and H. The convergence rate in the L∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method.