Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
Artikel i vetenskaplig tidskrift, 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.

Malliavin calculus

Galerkin methods

finite element methods

rational approximations of semigroups

stochastic wave equations

stochastic hyperbolic equations

weak convergence

Crank--Nicolson method

Stochastic partial differential equations

Författare

Mihaly Kovacs

Pázmány Péter Katolikus Egyetem

Annika Lang

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Andreas Petersson

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Mathematical Modelling and Numerical Analysis

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

Vol. 54 6 2199-2227

Approximation och simulering av Lévy-drivna SPDE

Vetenskapsrådet (VR), 2015-01-01 -- 2018-12-31.

Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Matematisk analys

Fundament

Grundläggande vetenskaper

DOI

10.1051/m2an/2020012

Mer information

Senast uppdaterat

2020-12-29