A Kind Of Quadratic Superconvergence Finite Volume Element Method For One Dimensional Elliptic And Parabolic Equations

Finite volume element methods(FVEMs),which were called box methods[25]in early times discretize the integral form of conservation law of differential equation by choosing linear or bilinear finite element space as the trial space.The method,which are also called generalized difference method(GDMs) have been widely used in numerical partial differential equations and fluid dynamacis because they keep the conservation law of mass or energy.Essentially,finite volume element is method based on interpolations.By approximation theory,we know the numerical derivatives have only k order accuracy for interpolating polynomials of order k in general.But this fact does not exclude the possibility that the approximation of derivatives may be of higher order accuracy at some special points,which are called optimal stress points.For quadratic interpolation associated with the points-1,0,1 on the element[-1,1],third order accuracy is achieved at points -1/3^1/2,1/3^1/2.We call these points the optimal stress points.In this paper,we present a kind of quadratic finite volume element method based on optimal stress points for two-point boundary value problems and obtain the superconvergence of derivatives at optimal stress points and superconvergence at nodes.In chapter one,quadratic superconvergence finite volume element method is presented for two-point boundary value problems. In section one,computational scheme is derived and in the section two,we give H¹ and L² norm error estimates and prove second order with H¹ norm and third order accuracy with L² norm.In the section three,the derivative superconvergence at optimal stress points is showed theoretically and prove the superconvergence at nodes for the simply case.Finally,in the section four,two numerical examples show that the method is very effective by comparison.In chapter two,superconvergent finite volume element method based on quadratic interpolation is generalized to one dimensional parabolic problems.In section one,the scheme is derived.In section two,L² norm is analyzed and finally,numerical examples are given to show the efficiency of the method.