Efficient Numerical Schemes For Two Kinds Of Evolution Equations

In modern society,many physical,chemical,biological and other practical prob-lems are eventually transformed into numerical solutions of partial differential equa-tions.Therefore,it is very important to study the numerical solutions of partial differential equations.This paper is devoted to a two-grid finite volume element algorithm based on Crank-Nicolson scheme for nonlinear parabolic equations.For nonlinear parabolic equations,the Crank-Nicolson scheme is adopted to discretion in time and the two-grid finite volume element method is adopted to discretion in space.It is proved that the two-grid method can obtain asymptotically optimal error estimates in spaces and second order accuracy in time.Compared with the finite volume element method,the two grid finite volume element method can significantly improve the computing efficiency while keeping the accuracy.Meanwhile,numerical results are provided to verify the theoretical results.Secondly,the Ginzburg-Landau equation is one of the most important nonlinear equations,which plays an important role in the exploration of many physics phenom-ena.First,combined with the backward Euler method and the two-grid finite volume element method,the fully discrete scheme is obtained.The fully discrete scheme is divided into real part and imaginary part,because the coefficient of the equation is complex.Combined with Taylor expansion,a fully discrete scheme of two-grid fi-nite volume element is obtained.Meanwhile,the optimal theoretical order of error is analyzed in L~2-norm.