粘弹性方程的时间间断Galerkin时空有限元方法

A Time Discontinuous Galerkin Space-Time Finite Element Method for the Viscoelastic Equation

摘要: 粘弹性方程是研究地震波、断裂力学问题的主要数学物理方程之一。为时空统一高精度求解位移$u$ 和速度$u_t$ 的近似解，针对方程特性，引入中间变量$\sigma =u_t$ 将原问题转换为方程组系统，并建立时间间断时空有限元格式去获得位移和速度的近似解。本文采用有限元分析，结合基于Radau积分节点的Lagrange插值，推导出位移关于$L^{\infty }\left(\left[0,T\right];H^1\left(\Omega \right)\right)$ –模和速度关于$L^{\infty }\left(\left[0,T\right];L^2\left(\Omega \right)\right)$ –模最优误差估计。最后，通过一个数值例子证明了方法的可行性和误差分析结果的合理性。

Abstract: The viscoelastic equation is one of the main mathematical and physical equations for studying seismic waves and fracture mechanics problems. To uniformly and precisely solve for the approximate solutions of displacement $u$ and velocity $u_t$ in space-time, by analyzing the equation’s properties, intermediate variables $\sigma =u_t$ are introduced to reformulate the original problem as a system of equations. Then, a temporally discontinuous space-time finite element framework is established to obtain the approximate solutions of both displacement and velocity fields. By using finite element analysis and combining with Lagrange interpolation based on Radau integration nodes, the optimal error estimation of displacement with respect to the $L^{\infty }\left(\left[0,T\right];H^1\left(\Omega \right)\right)$ -norm and velocity with respect to the $L^{\infty }\left(\left[0,T\right];L^2\left(\Omega \right)\right)$ -norm is derived. Finally, a numerical example was provided to demonstrate the feasibility of the method and the rationality of the theoretical results of the error analysis.