Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:130

Article

Mukam, Jean Daniel¹  Tambue, Antoine^2,3,4,5 

[1] Tech Univ Chemnitz, Fak Math, D-09126 Chemnitz, Germany

[2] Western Norway Univ Appl Sci, Dept Comp Sci Elect Engn & Math Sci, Inndalsveien 28, N-5063 Bergen, Norway

[3] Univ Cape Town, Ctr Res Computat & Appl Mech CERECAM, ZA-7701 Rondebosch, South Africa

[4] Univ Cape Town, Dept Math & Appl Math, ZA-7701 Rondebosch, South Africa

[5] African Inst Math Sci AIMS South Africa, 6-8 Melrose Rd, ZA-7945 Muizenberg, South Africa

关键词: Rosenbrock-type scheme;    Stochastic partial differential equations;    Multiplicative & additive noise;    Strong convergence;    Finite element method;   

DOI  :  10.1016/j.spa.2020.02.008

来源: Elsevier

【 摘 要 】

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial 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, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free