Approximation of SPDE covariance operators by finite elements: A semigroup approach
Preprint, 2021

The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition formula is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert--Schmidt norms with numerical simulations illustrating the results.

stochastic partial differential equations

finite element method

stochastic wave equations

covariance operators

stochastic advection-diffusion equations

integral equations

Författare

Mihaly Kovacs

Chalmers, Matematiska vetenskaper

Annika Lang

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Andreas Petersson

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

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Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Matematisk analys

Fundament

Grundläggande vetenskaper

Relaterade dataset

arXiv:2107.10109 [math.NA] [dataset]

URI: https://arxiv.org/abs/2107.10109

Mer information

Skapat

2021-08-11