SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1618-1638, January 2021.
Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate approximate mass matrix which is diagonal, making it computationally efficient for explicit simulations. In this article it is shown that for the standard space of polynomials used with triangular elements, denoted $\mathcal{T}(p)$, where $p$ is the degree of the space, there is no diagonal approximate mass matrix that permits accurate solutions where accuracy is defined as giving an exact projection of functions in ${\cal T} (p - 1)$. In light of this, a lower-triangular pseudo-mass matrix method is introduced that requires only local operations, and the method's accuracy is demonstrated for the space $\mathcal{T}(3)$. The pseudo-mass matrix and accompanying high-order basis allow for computationally efficient time-stepping techniques without sacrificing the accuracy of the spatial approximation for unstructured triangular meshes.

中文翻译：

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hp三角有限元方法显式时间推进的高阶低三角伪质量矩阵

SIAM 数值分析杂志，第 59 卷，第 3 期，第 1618-1638 页，2021 年 1 月。
连续有限元的显式时间推进需要全局质量矩阵的求逆。对于四边形和六面体的谱元模拟，有一个精确的近似质量矩阵，它是对角线，这使得它对于显式模拟具有计算效率。在这篇文章中表明，对于三角元素使用的多项式的标准空间，记为 $\mathcal{T}(p)$，其中 $p$ 是空间的度数，没有对角近似质量矩阵允许准确的解决方案，其中准确度定义为在 ${\cal T} (p - 1)$ 中给出函数的精确投影。有鉴于此，引入了只需要局部运算的下三角伪质量矩阵方法，并在空间$\mathcal{T}(3)$证明了该方法的准确性。