An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise

TL;DR

This paper develops an efficient numerical method for solving stochastic evolution PDEs with complex random diffusion coefficients and multiplicative noise, including theoretical analysis and numerical validation.

Contribution

It introduces a novel sampling technique for the random coefficient and a combined semi-implicit Euler-Maruyama and finite element discretization with convergence analysis.

Findings

Proved well-posedness of the stochastic evolution equations.

Established strong convergence rates for the numerical scheme.

Numerical experiments confirm theoretical convergence results.

Abstract

In this paper, we investigate the stochastic evolution equations (SEEs) driven by $\log$-Whittle-Mat$\acute{{\mathrm{e}}}$rn (W-M) random diffusion coefficient field and $Q$-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.