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

Ruisheng Qi, Xiaojie Wang

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 $τ$. Finally, numerical results are presented to verify the previous theoretical findings.

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

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 . 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 , 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 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 , 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 and . Finally, numerical results are presented to verify the previous theoretical findings.
Paper Structure (11 sections, 23 theorems, 291 equations, 3 figures)

This paper contains 11 sections, 23 theorems, 291 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The considered SPDEs with uniform-in-time regularity
  3. The fully discrete schemes
  4. Error estimates in Banach spaces for the finite element fully discretizations of the deterministic linear parabolic equation
  5. Uniform-in-time moment bounds of the fully discrete schemes
  6. Strong convergence analysis of the full discretization scheme
  7. Uniform-in-time weak convergence analysis
  8. Malliavin regularity estimates of fully discrete finite element approximations
  9. Uniform-in-time weak convergence rate
  10. Numerical experiments
  11. Conclusion

Key Result

Lemma 2.3

Let Assumption assum:nonlinearity be fulfilled. Then, there exist constants $L_f\in \mathbb{R}$ and $R_f, c_0, c_1$, $c_2, c_3, c_4, c_5>0$ such that, for all $x,y\in \mathbb{R}$

Figures (3)

  • Figure 1: Long-time behaviors for the scheme in the trace noise case ($\mathrm{Tr}(Q )<\infty$)
  • Figure 2: Long-time strong convergence rates in time (Left: $Q=I$; Right: $Q=A^{-0.5005}$)
  • Figure 3: Long-time weak convergence rates in time (Left: $Q=I$; Right: $Q=A^{-0.5005}$)

Theorems & Definitions (23)

  • Lemma 2.3
  • Lemma 2.6
  • Theorem 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Lemma 5.1
  • Lemma 5.2
  • Theorem 5.3
  • Proposition 5.4
  • ...and 13 more