Journal
IMA JOURNAL OF NUMERICAL ANALYSIS
Volume 34, Issue 1, Pages 217-251Publisher
OXFORD UNIV PRESS
DOI: 10.1093/imanum/drs055
Keywords
SPDE; finite element method; spectral Galerkin method; multiplicative noise; spatially semidiscrete; Lipschitz nonlinearities; optimal error estimates; spatio-temporal discretization
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Funding
- German Research Foundation (DFG) through the Collaborative Research Centre 701
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We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyse 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.
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