Superconvergent Quadratic Finite Volume Element Methods For Systems Of One Dimensional Elliptic And Parabolic Equations

Finite volume element is a discrete method for solving partial differential equations based on interpolations, which has been widely used in the computations of science and engineering. It is well known for rth Langrange interpolation, the numerical derivatives have only rth order accuracy 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 in mechanics. The finite volume element method based on optimal stress points has the super convergence of gradient or displacement, which is very important for computation. At present, a lot of the main research is limited to a single equation. In this paper, we present quadratic superconvergent finite volume element method for systems of one dimensional second-order elliptic and parabolic equations.Three chapters are included. Chapter one is an introduction of finite volume element method. In chapter two, quadratic superconvergent finite volume element method is presented for systems of second-order elliptic ordinary differential equations with mixed boundary conditions. In section one, the quadratic superconvergent scheme is derived, and in section two, we prove H1and L2norm error estimates, which have second and third order accuracy respectively. In this section we also give the third gradient superconvergence estimates at optimal stress points. In the last section, a numerical example is given to show the correctness of theoretical analysis and the efficiency of the scheme by implementing Fortran codes.In chapter three, the superconvergent finite volume element method based on quadratic interpolation is generalized to systems of one dimensional parabolic equations with mixed boundary conditions. In section one, the Crank-Nicolson full discrete scheme is presented. In section two, L2norm error estimate is analyzed by using symmetrization technique. It is proved that the scheme has second order accuracy in the direction of time and third order accuracy in the direction of space. Finally, a numerical example is given to show the efficiency of the method.