The Finite Volume Element Method For Solving Convection-dominated Reaction-diffusion Problems

Convection-dominated reaction-diffusion equations used to describe many physical phenomena are one of important mathematical models, and because the equations themselves have very strong hyperbolic characteristics, the usual numerical methods for solving such problems often generate excessive numerical diffusion and non-physical numerical oscillation. This thesis uses the finite volume element method that has the advantages of finite difference method and finite element method to study the convection-dominated reaction-diffusion problems, then it analyses theoretically the stability and error estimate of the finite volume element scheme constructed and demonstrates its effectiveness to solve the strong convection-dominated reaction-diffusion problems.Firstly, this dissertation transforms the general convection-dominated reaction-diffusion equation into the equation of conservative main part, then it uses a kind of bilinear finite volume element method to solve this equation, that is, the trial function space is the finite element space that consists of piecewise bilinear interpolation functions and the test function space is the finite element space that consists of piecewise constant functions. By analyzing the stability and error estimate of the finite volume element scheme constructed, the paper concludes its uniform stability that has nothing to do with the Peclet number of the equation and mesh step length as well as O(h2) order error estimates in L∞-norm.Secondly, this dissertation also uses a kind of biquadratic finite volume element method to solve convection-dominated reaction-diffusion problem, that is, the trial function space is the finite element space that consists of piecewise biquadratic interpolation functions and the test function space is the finite element space that consists of piecewise constant functions. By analyzing the stability and error estimate of the finite volume element scheme constructed, the paper also concludes its uniform stability in L∞-norm that has nothing to do with the Peclet number of the equation and mesh step length as well as O(h3) order error estimates in discrete energy norm under certain conditions. Finally, this paper conducts numerical experiments to calculate and compares the two kinds of finite volume element methods in the text with the central difference method, then it uses Matlab as a tool to give the numerical results and images under different equation Peclet numbers to verify the theoretical analysis results.Through theoretical analysis and numerical experiments, the discretization scheme that constructed by the finite volume element method in the dissertation has very good stability and approximation accuracy, and it is an effective numerical method to solve the convection-dominated reaction-diffusion problems.