Discontinuous Finite Volume Element Method For Evolution Equations

In this article, we consider the semi-discrete and the backward Euler fully-discretediscontinuous finite volume element methods for the second-order parabolic problemsand obtain the optimal order error estimates in a mesh dependent norm and in theL2-norm. On the original grid, piecewise linear polynomial function space is used as thetrial function space in this method, while on the dual grid, piecewise constant functionspace is used as the test function space. A discontinuous finite volume element method(finite volume method) for the second-order elliptic problems was recently proposed byYe [21] and the optimal order error estimate in a mesh dependent norm was obtained.Ye [22] considered the discontinuous finite volume element method for solving the Stokesproblems on both triangular and rectangular meshes. In [10], Chou and Ye establisheda general framework for analyzing the finite volume element methods which employcontinuous or totally discontinuous test functions for the elliptic problems. Under thisframework, optimal order error estimates in the H~1 and L~2 norms can be obtained ina systematic way for the classical finite volume element method and the discontinuousfinite volume element method. In addition, in this article, the semi-discrete discontinu-ous finite volume element method for the second-order hyperbolic problems is presentedby the similar way, and the optimal order error estimate in a mesh dependent norm isobtained.