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The Weak Galerkin Finite Element Method for the Symmetric Hyperbolic Systems
arXiv - CS - Numerical Analysis Pub Date : 2020-11-23 , DOI: arxiv-2011.11196
Tie Zhang, Shangyou Zhang

In this paper, we present and analyze a weak Galerkin finite element (WG) method for solving the symmetric hyperbolic systems. This method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other. By introducing special weak derivative, we construct a stable weak Galerkin scheme and derive the optimal $L_2$-error estimate of $O(h^{k+\frac{1}{2}})$-order for the discrete solution when the $k$-order polynomials are used for $k\geq 0$. As application, we discuss this WG method for solving the singularly perturbed convection-diffusion-reaction equation and derive an $\varepsilon$-uniform error estimate of order $k+1/2$. Numerical examples are provided to show the effectiveness of the proposed WG method.

中文翻译:

对称双曲系统的弱Galerkin有限元方法

在本文中,我们提出并分析了一种用于求解对称双曲系统的弱Galerkin有限元(WG)方法。通过允许在元素及其边界上彼此独立使用不连续有限元素,此方法具有很高的灵活性。通过引入特殊的弱导数,我们构造了一个稳定的弱Galerkin方案,并针对离散解在以下情况下获得了$ O(h ^ {k + \ frac {1} {2}})$阶的最优$ L_2 $误差估计。 $ k $阶多项式用于$ k \ geq 0 $。作为应用,我们讨论了用于求解奇摄动对流-扩散-反应方程的这种WG方法,并得出阶数为\ k + 1/2 $的$ \ varepsilon $-均匀误差估计。数值例子表明了所提出的WG方法的有效性。
更新日期:2020-11-25
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