Quadratic And Cubic Finite Volume Element Method Based On Optimal Stress Points For Semilinear Parabolic Equations

Finite volume element method(FVEM) is a discrete method for solving partial differen-tial equations. Firstly, a mesh partition and a dual mesh partition is placed on computation domain, where the dual elements are called control volumes. Secondly, the integral form of conservation law of differential equation is derived by integrating the equation in every control volume. Finally, trial function space is chosen as linear or higher-order finite element subspace to discretize the integral form of conservation law to obtain the scheme. FVEM, which has the simplicity of finite difference and the flexibility and accuracy of finite element, is the bridge between finite difference method and finite element method. The method has wide applications in the computation of science and engineering, because it keeps the conservation law of mass or energy.Essentially, finite volume element is numerical method based on interpolations. 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 first order derivative superconvergent points or optimal stress points in mechanics. If we choose optimal stress points as control volume nodes, then we can obtain superconvergent finite volume element method. This topic has some research results. In this paper, we discuss superconvergent FVEM for semilinear parabolic equations.Three chapters are included. Chapter one is an introduction of FVEM. In chapter two, a kind of quadratic FVEM based on optimal stress points is presented for semilinear parabolic equations with mixed initial and boundary conditions. Based on the Crank-Nicolson discrete idea, we discretize the parabolic term with second order accuracy. For nonlinear right hand-side source term, by using a linear extrapolation of two nearest time levels, we obtain a class of linearized quadratic FVEM. L2 norm error estimate is given. Two numerical examples illustrate the correctness of the theoretical analysis and the effectiveness of the scheme. In chapter three, cubic superconvergent FVEM is presented for semilinear parabolic equations. Unlike quadratic element method, it uses another linearization method to nonlinear right hand-side source term to obtain a two level scheme, and so it is not necessary to make special treatment for the first level. This chapter uses energy analysis method to prove the convergence. A numerical example is given to show the efficiency of the method.