Research On Several Numerical Methods For Elliptic Equation

Elliptic partial differential equations are a kind of important partial differential equations.They are widely used in fluid mechanics,elasticity, electromagnetics, geometry and calculus of variations But because of the precise solution is difficult to obtain,so finding numerical solutions is especially important. In this paper, the numerical solution of elliptic partial differential problem is analyzed and studied by using the finite difference method and Krylov subspace method (BICGSTAB, CG, GMRES and LSQR) as well as the spectral method(Barycentric interpolation collocation method and Chebyshev spectrum method). The main research work is as follows:(1) Based on the analysis of the background and significance of partial differential equations and the research status at home and abroad, the paper focuses on the research status and research significance of elliptic equation boundary problem.(2) On the basis of the finite difference method, the discrete linear equations, based on the Krylov subspace,using BICGSTAB algorithm,CG algorithm,GMRES algorithm and LSQR algorithm of the numerical simulation and the results were compared and analyzed.(3) Based on the spectral method, the barycentric interpolation collocation method and the Chebyshev spectral method are studied and applied to the solution of a class of elliptic partial differential equations with variable coefficients. Numerical simulations are also given.(4) The precision and error of several solving methods are analyzed, the results show that the Krylov subspace method is easy to calculate, but the precision is not high; spectral method in equidistant node number is large, the accuracy is poor in accuracy on Chebyshev points is very high, but we can't determine the value of any point. Therefore, the pretreatment and interpolation methods are improved and the effect is better.