Second Order Discontinuous Finite Volume Element Method For Elliptic Equation

As a discretization technique for solving partial differential equations,finite volume method is an important and popular methods in the numerical simulation of many practical problems in physics and engineering.In recent years,some devel-opments of discontinuous finite volume method have achieved.However,the main limitation of discontinuous finite volume method is the low approximation order.Litter work has been done using the high order discontinuous finite volume method.In this paper,a quadratic discontinuous finite volume method for second order ellip-tic equations is developed by mixing quadratic discontinuous finite element method and linear discontinuous finite volume method and using the hierarchical decompo-sition to realize the hybridization.We analyze the continuous and coerciveness of the bilinear form,then get error estimate in a mesh dependent norm.Our anal-ysis shows that the new method does not require continuity of the approximation functions across the interelement boundary conditions,which makes it convinient to construct the space.Then we develop a discontinuous finite volume formulation for the parabolic problem based on the method for elliptic equation.And the existence and uniqueness of the solution is proved.