Spectral Element Methods For Stochastic Differential Equations Driven By Additive Noise

Local and global categories can be used as the classification of numerical methods for solving differential equations.For example,the finite difference and finite element methods are local methods,which are suitable to solve problems in complex geome-tries;while spectral methods are global methods with high accuracy.Spectral element methods combine the idea of the finite element methods,not only enhance the domain adaptability of the method,but also keep high precision,which well inherit the merits of both the global and local methods.We use spectral element methods to solving stochastic differential equations driv-en by additive noise numerically in this paper.For the case stochastic differential equa-tions driven by white noise,we first approximate the white noise process by piecewise constant random process,and provide its regularity estimate.Next,apply spectral ele-ment methods to solve the corresponding stochastic differential equations numerically,the approximation errors are derived in this paper.Furthermore,the errors results are listed and they indicate the effectiveness of the proposed methods.For another case stochastic differential equations driven by colored noise,first of all,we use an abstract formulation to simulate colored noise.And then,the relevant stochastic differential e-quations are capable of being approximated by Galerkin spectral element schemes,the corresponding error estimates are obtained.Finally,we represent the error results coin-cided with the theoretical analysis,which prove the accuracy of the proposed schemes.