Optimal Order Space‐Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations With Multiplicative Noise

ABSTRACT This paper develops and analyzes an optimal‐order semi‐discrete scheme and its fully discrete finite element approximation for a second‐order system of nonlinear stochastic equations with multiplicative noise, which governs wave propagation in elastic media under the influence of random damping and force. A nonstandard time‐stepping scheme is introduced for time discretization. It is shown that the scheme converges with rates and respectively in the energy‐ and ‐norm, which are optimal with respect to the time regularity of the PDE solution. For spatial discretization, the finite element method is constructed. It is proven that the fully discrete method converges with optimal rates and in the energy‐ and ‐norm. The crux of the analysis is 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.