粘弹性波传播方程的半离散弱有限元方法
四川大学, 锦江学院数学教学部, 四川彭山 620860
| 摘 要： | 本文主要基于Maxwell模型的准静态粘弹性方程，针对Maxwell模型的粘弹性波传播方程，提出了一类任意阶的弱有限元方法。先在模型问题中引入速度变量，建立了速度-应力模型的弱变分形式；再采用弱有限元方法对空间变量进行离散，得到速度-应力模型的半离散格式。在弱有限元方法的离散中，利用分片k次（k≥1）和k+1次分别逼近单元内部的应力和速度，为了逼近速度在单元边界上的迹，采用了分片k次多项式逼近。本文利用Gronwall不等式证明了半离散格式的解的稳定性；通过对方法进行分析，最后证明该方法关于空间网格尺度的最优性的误差估计。本文的研究结果表明，通过引入速度变量和采用弱有限元方法，可以有效求解Maxwell模型的粘弹性波传播方程。 |
| 关 键 词： | 粘弹性波传播; 弱有限元方法; 误差估计 |
| DOI： | 10.57237/j.wjms.2024.02.002 |
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收稿日期: 2024-09-12
接受日期: 2024-11-19
在线出版日期: 2024-12-23
Semi-discrete Weak Galerkin Finite Element Method of Viscoelastic Wave Propagation
Department of Mathematics of Jinjiang College, Sichuan University, Pengshan 620860, China
| Abstract: | This paper primarily proposes a weak Galerkin finite element of arbitrary order for the wave propagation equation of the Maxwell viscoelasticity model. The quasi-static viscoelastic equations of the Maxwell model serve as the foundation for this approach. By introducing the velocity variable in the model problem, the weak variational form of the velocity-stress model is established. The spatial variables are then discretized using the weak Galerkin finite element method, resulting in a semi-discrete format for the velocity-stress model. In the discretization process of the weak Galerkin finite element method, piecewise polynomials of degree k (k≥1) and k+1 are used to approximate the stress and velocity within each element, and piecewise polynomials of degree k are employed to approximate the trace of the velocity on the element boundaries. By analyzing and proving the method, this paper demonstrates the stability of the semi-discrete format's solution using Gronwall's inequality, and finally establishes the optimal error estimation with respect to the spatial mesh size for this method. The research findings of this paper indicate that by introducing the velocity variable and utilizing the weak Galerkin finite element method, the wave propagation equation of the viscoelasticity based on the Maxwell model can be effectively solved. |
| Keywords: | Viscoelastic Wave Propagation; Weak Galerkin Finite Element; Error Estimate |
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