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Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises

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  • Yang, Xu
  • Zhao, Weidong

Abstract

In this paper, we investigate the mean square error of numerical methods for SPDEs driven by Gaussian and non-Gaussian noises. The Gaussian noise considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Lévy process. As the models consider the influences of Gaussian and non-Gaussian noises simultaneously, this makes the models more realistic when the models are also influenced by some randomly abrupt factors, but more complicated. As a consequence, the numerical analysis of the problems becomes more involved. We first study the regularity for the mild solution. Next, we propose a semidiscrete finite element scheme in space and a fully discrete linear implicit Euler scheme for the SPDEs, and rigorously obtain their error estimates. Both the regularity results of the mild solution and error estimates obtained in the paper are novel.

Suggested Citation

  • Yang, Xu & Zhao, Weidong, 2018. "Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 58-75.
    • Handle: RePEc:eee:apmaco:v:332:y:2018:i:c:p:58-75
    DOI: 10.1016/j.amc.2018.03.039
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    1. Barth, Andrea & Stüwe, Tobias, 2018. "Weak convergence of Galerkin approximations of stochastic partial differential equations driven by additive Lévy noise," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 143(C), pages 215-225.
    2. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    3. Albeverio, S. & Mandrekar, V. & Rüdiger, B., 2009. "Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 835-863, March.
    4. NicolaBruti-Liberati & Eckhard Platen, 2007. "Strong approximations of stochastic differential equations with jumps," Published Paper Series 2007-7, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    5. G. N. Milstein & Eckhard Platen & H. Schurz, 1998. "Balanced Implicit Methods for Stiff Stochastic Systems," Published Paper Series 1998-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
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