Several Numerical Schemes Of Stochastic Partial Differential Equations

This paper is a general survey about numerical solution methods to stochastic partial differential equations. We introduces several types of effective numerical method, like Galerkin finite element method and collocation method. We compare and analyze these methods.The model problem has the form L(a)u=f in D Where L is an elliptic operator in a domain D(?)Rd, which depends on some stochastic coefficients a(x,ω), with x∈D,ω∈ΩandΩindicating the set of possible outcomes. Similarly,the forcing term f=f(x,ω) can be assumed to be stochastic as well.About the above previous SPDE, this paper describes spectral Galerkin approximate, Monte Carlo Galerkin approximate, stochastic collocation approximate, their standard approximate (finite element,finite volume,spectral method,collocation method and so on)in space and polynomial approximate. Then, we show an orthogonal polynomial collocation method, which has some features:it will lead to uncoupled deterministic problem; it can effectively deal with auxiliary density; easily handle unbounded, and analyze convergence.There exist positive rn, n=1,…,N, and C,independent of h and p,such that Yield collocation method has the same accuracy with Galerkin method, gets to exponential convergent and analyzes probability exponential convergent under comprehensive circumstances. But it only has effect on a small number of random variables; it leads to curse of dimensionality when it deals with large dimensionality of the probability space. If it happens, we should convert it to sparse tensor product space to study.Discussing the spatial discretization is done by piecewise linear finite elements on globally quasi-uniform meshes. This paper compares the asymptotical numerical complexity for the Monte Carlo Galerkin finite element method with the Stochastic Galerkin finite element method by and (?)υ>0, the constant C>0 is independent fromυ,M and h, moreover estimatesκ×h-SGFEM approximations of u by and then get the estimate which is a superconvergence result with respect toκThen it has two conclusions:Galerkin finite element method controls the errors in a probability sense. Compute work is related to the number of random variables describing the problem and only increases in a polynomial way; it can control bound of errors of Galerkin finite element method.