Long-time error analysis of finite element fully discrete schemes for SPDEs with non-globally Lipschitz coefficients

TL;DR

This paper introduces new fully discrete finite element schemes for long-time approximation of SPDEs with non-globally Lipschitz coefficients, providing uniform-in-time error bounds and stability analysis.

Contribution

The paper develops a novel family of linearly implicit schemes that preserve uniform moment bounds without step size restrictions, and derives new long-time error estimates in $L^r$ spaces for SPDEs.

Findings

Uniform-in-time moment bounds established for the schemes.

Convergence rates obtained for both strong and weak errors.

Numerical results confirm theoretical error estimates.

Abstract

The present paper proposes new fully discrete schemes for long-time approximations of stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients in a bounded domain $D \subset \R^d, d =1,2,3 $. A novel family of linearly implicit time-stepping schemes is introduced, based on a standard Galerkin finite element spatial semi-discretization. A distinguishing feature of the schemes is that the proposed finite element fully discrete approximations preserve uniform-in-time moment bounds in a Banach space $L^{r}(D), r >2$, without requiring any restriction on the time-space discretization stepsize ratio. %established... To show it, some non-standard arguments are developed. First, we derive long-time error estimates in the Banach space $L^r(D)$ for finite element fully discrete approximations of the deterministic linear parabolic equation with non-smooth initial value, which is, to our knowledge, new for the literature on numerical PDEs and of independent interest. These error estimates together with the contractive property of the semi-group in $L^{r}(D), r > 2$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy help us establish the desired uniform-in-time moment bounds. Then both strong and weak error bounds of the proposed schemes are carefully analyzed in a setting of low regularity, with uniform-in-time convergence rates obtained for cases of both space-time white and trace-class noises. The analysis is highly nontrivial, due to the finite element discretization, the low regularity and the presence of the super-linearly growing nonlinearity. %in the long-time scenario... the discretization parameters $h$ and $\tau$. Finally, numerical results are presented to verify the previous theoretical findings.