Volume 54, Number 6, November-December 2020
|Page(s)||2199 - 2227|
|Published online||03 November 2020|
Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, PO Box 278, H-1444 Budapest, Hungary
2 Department of Mathematical Sciences, Chalmers University of Technology & University of Gothenburg, S-412 96 Gothenburg, Sweden
* Corresponding author: firstname.lastname@example.org
Accepted: 17 February 2020
The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.
Mathematics Subject Classification: 60H15 / 65M12 / 0H35 / 65C30 / 65M60 / 60H07
Key words: Stochastic partial differential equations / stochastic wave equations / stochastic hyperbolic equations / weak convergence / finite element methods / Galerkin methods / rational approximations of semigroups / Crank–Nicolson method / Malliavin calculus
© EDP Sciences, SMAI 2020
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