Strong convergence of the backward Euler approximation for the finite element discretization of semilinear parabolic SPDEs with non-global Lipschitz drift driven by additive noise

TL;DR

This paper proves strong convergence of a backward Euler finite element scheme for semilinear parabolic SPDEs with non-globally Lipschitz drift driven by additive noise, extending previous results to more general nonlinearities and non-self-adjoint operators.

Contribution

It introduces a novel convergence analysis for fully discrete schemes applied to broader classes of SPDEs with polynomial nonlinearities and non-self-adjoint operators, including numerical validation.

Findings

Achieves convergence order of O(h^2 + Δt^{1-ε}) for trace class noise.

Extends analysis to nonlinear drifts of polynomial type with negative leading term.

Provides numerical experiments confirming theoretical convergence rates.

Abstract

This paper deals with the backward Euler method applied to semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. The SPDE is discretized in space by the finite element method and in time by the backward Euler. We consider a larger class of nonlinear drift functions, which are of Nemytskii type and polynomial of any odd degree with negative leading term, instead of only dealing with the special case of stochastic Allen-Chan equation as in the up to date literature.Moreover our linear operator is of second order and not necessarily self-adjoint, therefore makes estimates more challenging than in the case of self-adjoint operator. We prove the strong convergence of our fully discrete schemes toward the mild solution and results indicate how the convergence rates depend on the regularities of the initial data and the noise. In particular, for trace class noise, we achieve convergence order $\mathcal{O} (h^{2}+\Delta t^{1-\epsilon})$, where $\epsilon>0$ is positive number, small enough. We also provide numerical experiments to illustrate our theoretical results.