标题： 非线性随机弹性波方程具有乘性噪声的最优阶时空离散化方法 显示英文标题

标题： Optimal Order Space-Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations with Multiplicative Noise

摘要： 本文研究了非线性随机弹性波方程（带有乘性噪声）的一个最优阶半离散格式及其全离散有限元逼近。 为了时间离散化，引入了一个非常规的时间步进方案，证明了该方案分别在能量模和$L^2$-模下的收敛率分别为$O(\tau)$和$O(\tau^{\frac32})$，这些收敛率与偏微分方程解的时间正则性相一致。 对于空间离散化，采用了标准的有限元方法。 证明了全离散方法分别在能量模和$L^2$模下的收敛率分别为$O(\tau + h)$和$O(\tau^{\frac{3}{2}} + h^2)$。 分析的关键在于建立一些高矩稳定性结果，并利用梯形求积规则的精细误差估计来控制漂移项和乘性噪声中的非线性。 还提供了数值实验以验证理论结果。 显示英文摘要

摘要： 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 $O(\tau)$ and $O(\tau^{\frac32})$ 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 $O(\tau + h)$ and $O(\tau^{\frac{3}{2}} + h^2)$ 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.