Superconvergence of quadratic finite volume method on triangular meshes

This paper is concerned with the superconvergence properties of the quadratic finite volume method (FVM) on triangular meshes for elliptic equations. We proved the 3rd order superconvergence rate of the gradient approximation ‖uh−uI‖1=O(h3) and the 4th order superconvergence rate of the function value approximation ‖uh−uI‖0=O(h4) for the quadratic FVM on triangular meshes. Here uh is the FVM solution and uI is the piecewise quadratic Lagrange interpolation of the exact solution. It should be pointed out that the superconvergence phenomena of FVM strongly depends on the construction of dual mesh. Specially for quadratic FVMs, the scheme presented in this paper is the unique scheme which holds superconvergence.