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arXiv · 1809.06234

Magnus-type integrator for the finite element discretization of semilinear parabolic non-autonomous SPDEs driven by additive noise

Jean Daniel Mukam·Antoine Tambue
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Abstract

In this paper, we investigate a numerical approximation of a general second order semilinear parabolic non-autonomous stochastic partial differential equation (SPDE) driven by additive noise. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature while the non-autonomous case is not yet well understood. We discretize the non-autonomous SPDE in space by the finite element method and in time by the Magnus-type integrator. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $L^2$ norm. Appropriate assumptions on the drift term and the noise allow to achieve optimal convergence order in time greater than $1/2$, without any logarithmic reduction of convergence order in time. In particular, for trace class noise, we achieve optimal convergence orders $\mathcal{O}\left(h^{2-ε}+Δt\right)$, where $ε$ is a positive number small enough. Numerical simulations are provided to illustrate our theoretical results.

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Jean Daniel Mukam, Antoine Tambue. 2018-09-13. Magnus-type integrator for the finite element discretization of semilinear parabolic non-autonomous SPDEs driven by additive noise. https://arxiv.org/abs/1809.06234

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