Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
Journal article, 2020

We consider the numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise. For the spatial approximation we consider a standard finite element method and for the temporal approximation, a rational approximation of the exponential function. We first show strong convergence of this approximation in both positive and negative order norms. 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.

Galerkin methods

stochastic hyperbolic equations

Malliavin calculus

finite element methods

weak convergence

rational approximations of semigroups

Stochastic partial differential equations

crank-nicolson method

stochastic wave equations

Author

Mihaly Kovacs

Pázmány Péter Catholic University

Annika Lang

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Andreas Petersson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Mathematical Modelling and Numerical Analysis

0764-583X (ISSN) 1290-3841 (eISSN)

Vol. 54 6 2199-2227

Approximation and simulation of Lévy-driven SPDE

Swedish Research Council (VR) (2014-3995), 2015-01-01 -- 2018-12-31.

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

Roots

Basic sciences

DOI

10.1051/m2an/2020012

More information

Latest update

10/7/2021