204 research outputs found
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDE) driven
by additive space-time noise. We introduce a new modified scheme using a linear
functional of the noise with a semi--implicit Euler--Maruyama method in time
and in space we analyse a finite element method (although extension to finite
differences or finite volumes would be possible). We prove convergence in the
root mean square norm for a diffusion reaction equation and diffusion
advection reaction equation. We present numerical results for a linear reaction
diffusion equation in two dimensions as well as a nonlinear example of
two-dimensional stochastic advection diffusion reaction equation. We see from
both the analysis and numerics that the proposed scheme has better convergence
properties than the standard semi--implicit Euler--Maruyama method
A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations
In this paper the numerical solution of non-autonomous semilinear stochastic
evolution equations driven by an additive Wiener noise is investigated. We
introduce a novel fully discrete numerical approximation that combines a
standard Galerkin finite element method with a randomized Runge-Kutta scheme.
Convergence of the method to the mild solution is proven with respect to the
-norm, . We obtain the same temporal order of
convergence as for Milstein-Galerkin finite element methods but without
imposing any differentiability condition on the nonlinearity. The results are
extended to also incorporate a spectral approximation of the driving Wiener
process. An application to a stochastic partial differential equation is
discussed and illustrated through a numerical experiment.Comment: 31 pages, 1 figur
Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise
We consider Galerkin finite element methods for semilinear stochastic partial
differential equations (SPDEs) with multiplicative noise and Lipschitz
continuous nonlinearities. We analyze the strong error of convergence for
spatially semidiscrete approximations as well as a spatio-temporal
discretization which is based on a linear implicit Euler-Maruyama method. In
both cases we obtain optimal error estimates.
The proofs are based on sharp integral versions of well-known error estimates
for the corresponding deterministic linear homogeneous equation together with
optimal regularity results for the mild solution of the SPDE. The results hold
for different Galerkin methods such as the standard finite element method or
spectral Galerkin approximations.Comment: 30 page
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