Strong convergence rates of Galerkin finite element methods for SWEs with cubic polynomial nonlinearity

TL;DR

This paper analyzes the strong convergence rates of Galerkin finite element methods for stochastic wave equations with cubic nonlinearities, introducing a novel exponential integrator and proving energy-based trace formulas.

Contribution

It introduces a new exponential time integrator combined with Galerkin finite elements for SWEs with cubic nonlinearities, overcoming the lack of global monotonicity.

Findings

Proves the scheme satisfies an energy-based trace formula.

Establishes strong convergence rates in space and time.

Numerical results confirm theoretical convergence rates.

Abstract

In the present work, strong approximation errors are analyzed for both the spatial semi-discretization and the spatio-temporal fully discretization of stochastic wave equations (SWEs) with cubic polynomial nonlinearities and additive noises. The fully discretization is achieved by the standard Galerkin ffnite element method in space and a novel exponential time integrator combined with the averaged vector ffeld approach. The newly proposed scheme is proved to exactly satisfy a trace formula based on an energy functional. Recovering the convergence rates of the scheme, however, meets essential difffculties, due to the lack of the global monotonicity condition. To overcome this issue, we derive the exponential integrability property of the considered numerical approximations, by the energy functional. Armed with these properties, we obtain the strong convergence rates of the approximations in both spatial and temporal direction. Finally, numerical results are presented to verify the previously theoretical findings.