Optimal Order Space-Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations with Multiplicative Noise
Xiaobing Feng, Yukun Li, Liet Vo
TL;DR
The authors address nonlinear stochastic elastic wave equations with multiplicative noise by introducing an optimal-time-stepping scheme and a standard finite element spatial discretization. The nonstandard time discretization achieves exacting temporal accuracy, with $O(\tau)$ in the energy norm and $O(\tau^{3/2})$ in $L^2$ for the semi-discrete case, and the fully discrete method attains $O(\tau + h)$ in the energy norm and $O(\tau^{3/2} + h^2)$ in $L^2$. Central to the analysis are high-moment stability results and refined trapezoidal quadrature error controls that handle nonlinear drift and multiplicative noise; these are complemented by rigorous space-time error estimates and numerical experiments validating the theory. The results yield robust, optimal-order convergence for stochastic elastic wave simulations, with potential impact on accurate probabilistic predictions in engineering and physics applications under stochastic forcing.
Abstract
This paper develops and analyzes an optimal-order semi-discrete scheme and its fully discrete finite element approximation for nonlinear stochastic elastic wave equations with multiplicative noise. A non-standard time-stepping scheme is introduced for time discretization, it is showed that the scheme converges with rates $O(τ)$ and $O(τ^{\frac32})$ respectively in the energy- and $L^2$-norm, which are optimal with respect to the time regularity of the PDE solution. For spatial discretization, the standard finite element method is employed. It is proven that the fully discrete method converges with optimal rates $O(τ+ h)$ and $O(τ^{\frac{3}{2}} + h^2)$ respectively in the energy- and $L^2$-norm. The cruxes of the analysis are to establish some high-moment stability results and utilize a refined error estimate for the trapezoidal quadrature rule to control the nonlinearities from the drift term and the multiplicative noise. Numerical experiments are also provided to validate the theoretical results.