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

Abstract

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.

arXiv comment31 pages, 8 figures; added derivation of trace error formula; to appear in IMA Journal of Numerical Analysis

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Cited by in corpus (1)

• Hilbert--Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations