Journal of Scientific Computing

, Volume 74, Issue 2, pp 937–978 | Cite as

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

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Abstract

In this paper, we consider the numerical approximation of a general second order semilinear stochastic spartial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise with finite trace and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square \(L^2\) norm. Numerical experiments to sustain theoretical results are provided.

Keywords

Exponential Rosenbrock–Euler method Stochastic partial differential equations Multiplicative & additive noise Strong convergence Finite element method Errors estimate Stochastic reactive dominated transport equations 

Mathematics Subject Classification

65C30 74S05 74S60 

Notes

Acknowledgements

This work was supported by the German Academic Exchange Service (DAAD) (DAAD-Project 57142917) and the Robert Bosch Stiftung through the AIMS ARETE Chair programme (Grant No 11.5.8040.0033.0). Part of this work was done when Antoine Tambue visited TU Chemnitz. The visit was supported by TWAS-DFG Cooperation Visits Programme.

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Technische Universität ChemnitzChemnitzGermany
  2. 2.The African Institute for Mathematical Sciences (AIMS) of South Africa and Stellenbosch UniversityMuizenbergSouth Africa
  3. 3.Center for Research in Computational and Applied Mechanics (CERECAM), Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

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