English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper develops and analyzes a fully discrete finite element method for aclass of semilinear stochastic partial differential equations (SPDEs) withmultiplicative noise. The nonlinearity in the diffusion term of the SPDEs isassumed to be globally Lipschitz and the nonlinearity in the drift term is onlyassumed to satisfy a one-side Lipschitz condition. The semilinear SPDEsconsidered in this paper is a direct generalization of the SODEs considered in[13]. There are several difficulties which need to be overcome for thisgeneralization. First, obviously the spatial discretization, which does notappear in the SODE case, adds an extra layer of difficulty. It turns out aspecial discretization must be designed to guarantee certain properties for thenumerical scheme and its stiffness matrix. In this paper we use a finiteelement interpolation technique to discretize the nonlinear drift term. Second,in order to prove the strong convergence of the proposed fully discrete finiteelement method, stability estimates for higher order moments of the$H^1$-seminorm of the numerical solution must be established, which aredifficult and delicate. A judicious combination of the properties of the driftand diffusion terms and a nontrivial technique borrowed from [16] is used inthis paper to achieve the goal. Finally, stability estimates for the second andhigher order moments of the $L^2$-norm of the numerical solution is alsodifficult to obtain due to the fact that the mass matrix may not be diagonallydominant. This is done by utilizing the interpolation theory and the highermoment estimates for the $H^1$-seminorm of the numerical solution. Afterovercoming these difficulties, it is proved that the proposed fully discretefinite element method is convergent in strong norms with nearly optimal ratesof convergence.

Strong convergence of a fully discrete finite element method for a class  of semilinear stochastic partial differential equations with multiplicative  noise