Mixed Finite Element Methods for the Nonlinear Stochastic Elastodynamics Equations With Multiplicative Noise

ABSTRACT This paper presents a numerical framework for solving nonlinear stochastic elastodynamics equations with a particular emphasis on the coupled displacement–stress formulation. Spatial discretization is achieved using the Arnold–Winther mixed finite element method, which enables a stable and accurate approximation of both displacement and stress fields. For temporal discretization, we employ the Euler–Maruyama scheme, from which a fully discrete numerical approximation for the stochastic system is then yielded. Within this setting, we establish rigorous stability results and derive error estimates. Numerical experiments are provided to validate the theoretical findings for both displacement and stress and demonstrate the effectiveness and robustness of the proposed scheme.