Numerical Analysis Of Volume Element Methods For Two Kinds Of Partial Differential Equations

In this paper, an expanded mixed covolume method on triangular grids for soboiev equation and a finite volume element method along characteristics for the 2-dimensional mathematic model of sediment transport equation are considered. By a thorough numerical analysis, the error estimates are obstained.In Chapter one. we consider the expanded mixed covolume method for the invitial-boundary value problem of sobolev equationThe method has the advantages both of expanded mixed element method and the mixed covolume element method. The method does not need the inverse, so it can be used to solve degenerate problems. Meanwhile, it satisfies the local conservation law with the discrete solution. By making the numerical analysis, we obtain the optimal error estimates of L~2 norm about the unknown function and its gradient as well as L~2 and H(div) norm about its flux. In addition, using of the regularized Green function, we obtain the quasi-optimal order estimates of L~∞norm about the unknown function and its derivative. The numerical example results show that the error estimates are effective.In Chapter two. we study a characteristic finite volume element method for the mathematic model consisting of following equations: Water continuity equationWater dynamic equationsSilt transport equationThe equation of bottom topography changeThe governing equations are uniformly convection-domained. so that it is appropriate to use the characteristic finite volume element method for discreting. Because the method not only has the advantages of the characteristic finite difference method but of the characteristic finite element. It has been widely used in comoutational fluid mechanics and heat transfer problems. It possesses the important and crucial property of inheriting the physical conservation laws of the original problem locally. But the first equation of the model has the strongly hyperbolic nature, we only obstain the quasi-optimal L~2 norm error estimates between the exact solution and the discrete solution.