2021/06/28 by Yukun Li, Shuonan Wu, Li, Yukun +3
Earth and Planetary Sciences · Economics, Econometrics and Finance · Engineering · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Meteorological Phenomena and Simulations #Numerical Analysis (math.NA) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2106.14998
openalex publication_date 2021/06/28 · openalex created_date 2021/07/05 · openalex updated_date 2026/07/28
Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite element method for a class of nonlinear stochastic wave equations, where the diffusion term is globally Lipschitz continuous while the drift term is only assumed to satisfy weaker conditions as in [11]. The novelties of this paper are threefold. First, the error estimates cannot not be directly obtained if the numerical scheme in primal form is used. The numerical scheme in mixed form is introduced and several Hölder continuity results of the strong solution are proved, which are used to establish the error estimates in both L2 norm and energy norms. Second, two types of discretization of the nonlinear term are proposed to establish the L2 stability and energy stability results of the discrete solutions. These two types of discretization and proper test functions are designed to overcome the challenges arising from the stochastic scaling in time issues and the nonlinear interaction. These stability results play key roles in proving the probability of the set on which the error estimates hold approaches one. Third, higher order moment stability results of the discrete solutions are proved based on an energy argument and the underlying energy decaying property of the method. Numerical experiments are also presented to show the stability results of the discrete solutions and the convergence rates in various norms.