Optimal Order Space-Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations with Multiplicative Noise
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| Format: | Preprint |
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2025
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| _version_ | 1866909568441778176 |
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| author | Feng, Xiaobing Li, Yukun Vo, Liet |
| author_facet | Feng, Xiaobing Li, Yukun Vo, Liet |
| contents | This paper develops and analyzes an optimal-order semi-discrete scheme and its fully discrete finite element approximation for nonlinear stochastic elastic wave equations with multiplicative noise. A non-standard time-stepping scheme is introduced for time discretization, it is showed that the scheme converges with rates $O(τ)$ and $O(τ^{\frac32})$ respectively in the energy- and $L^2$-norm, which are optimal with respect to the time regularity of the PDE solution. For spatial discretization, the standard finite element method is employed. It is proven that the fully discrete method converges with optimal rates $O(τ+ h)$ and $O(τ^{\frac{3}{2}} + h^2)$ respectively in the energy- and $L^2$-norm. The cruxes of the analysis are to establish some high-moment stability results and utilize a refined error estimate for the trapezoidal quadrature rule to control the nonlinearities from the drift term and the multiplicative noise. Numerical experiments are also provided to validate the theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Order Space-Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations with Multiplicative Noise Feng, Xiaobing Li, Yukun Vo, Liet Numerical Analysis 65N12, %Stability and convergence of numerical methods 65N15, %Error bounds 65N30, 65N12, 65N15 This paper develops and analyzes an optimal-order semi-discrete scheme and its fully discrete finite element approximation for nonlinear stochastic elastic wave equations with multiplicative noise. A non-standard time-stepping scheme is introduced for time discretization, it is showed that the scheme converges with rates $O(τ)$ and $O(τ^{\frac32})$ respectively in the energy- and $L^2$-norm, which are optimal with respect to the time regularity of the PDE solution. For spatial discretization, the standard finite element method is employed. It is proven that the fully discrete method converges with optimal rates $O(τ+ h)$ and $O(τ^{\frac{3}{2}} + h^2)$ respectively in the energy- and $L^2$-norm. The cruxes of the analysis are to establish some high-moment stability results and utilize a refined error estimate for the trapezoidal quadrature rule to control the nonlinearities from the drift term and the multiplicative noise. Numerical experiments are also provided to validate the theoretical results. |
| title | Optimal Order Space-Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations with Multiplicative Noise |
| topic | Numerical Analysis 65N12, %Stability and convergence of numerical methods 65N15, %Error bounds 65N30, 65N12, 65N15 |
| url | https://arxiv.org/abs/2504.04531 |