Finite Element Approximations of a Class of Nonlinear Stochastic Wave Equation with Multiplicative Noise

06/28/2021
by   Yukun Li, et al.
0

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 L^2 norm and energy norms. Second, two types of discretization of the nonlinear term are proposed to establish the L^2 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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

01/22/2021

Mixed Finite Element Discretization for Maxwell Viscoelastic Model of Wave Propagation

This paper considers semi-discrete and fully discrete mixed finite eleme...
09/16/2020

Strong convergence of a Verlet integrator for the semi-linear stochastic wave equation

The full discretization of the semi-linear stochastic wave equation is c...
09/07/2020

Energy-preserving mixed finite element methods for the Hodge wave equation

Energy-preserving numerical methods for solving the Hodge wave equation ...
07/05/2019

A note on optimal H^1-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation

In this paper we consider a mass- and energy--conserving Crank-Nicolson ...
06/18/2020

Well-posedness, discretization and preconditioners for a class of models for mixed-dimensional problems with high dimensional gap

In this work, we illustrate the underlying mathematical structure of mix...
12/09/2019

Convective transport in nanofluids: regularity of solutions and error estimates for finite element approximations

We study the stationary version of a thermodynamically consistent varian...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.