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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.

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Notes

  1. Full implicit or semi-implicit methods.

  2. The proposition indeed is general and provides some estimates for any semigroup and its generator.

  3. Think about for example a multiple of Laplace operator \(A=\alpha \varDelta \), when \(\alpha \rightarrow 0\).

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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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Authors and Affiliations

  1. Technische Universität Chemnitz, 09126, Chemnitz, Germany

    Jean Daniel Mukam

  2. The African Institute for Mathematical Sciences (AIMS) of South Africa and Stellenbosch University, Muizenberg, South Africa

    Antoine Tambue

  3. Center for Research in Computational and Applied Mechanics (CERECAM), Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa

    Antoine Tambue

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  1. Jean Daniel Mukam
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  2. Antoine Tambue
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Correspondence to Antoine Tambue.

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Mukam, J.D., Tambue, A. Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise. J Sci Comput 74, 937–978 (2018). https://doi.org/10.1007/s10915-017-0475-y

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