Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
Preprint, 2019

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.

stochastic wave equations

finite element methods

rational approximations of semigroups

Stochastic partial differential equations

Crank--Nicolson method

weak convergence

Galerkin methods

Malliavin calculus

stochastic hyperbolic equations

Author

Mihaly Kovacs

Pázmány Péter Catholic University

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Annika Lang

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Andreas Petersson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Approximation and simulation of Lévy-driven SPDE

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

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

Roots

Basic sciences

More information

Created

12/20/2019