Approximation of SPDE covariance operators by finite elements: a semigroup approach
Journal article, 2023

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 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 wave equations

covariance operators

finite element method

integral equations

stochastic advection-diffusion equations

stochastic partial differential equations

Author

Mihaly Kovacs

Chalmers, Mathematical Sciences

Budapest University of Technology and Economics

Pázmány Péter Catholic University

University of Gothenburg

Annika Lang

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Andreas Petersson

University of Oslo

IMA Journal of Numerical Analysis

0272-4979 (ISSN) 1464-3642 (eISSN)

Vol. 43 3 1324-1357

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Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

DOI

10.1093/imanum/drac020

More information

Latest update

7/5/2023 1