LBB stability of a mixed Galerkin finite element pair for fluid flow simulations
LBB stability of a mixed Galerkin finite element pair for fluid flow simulations
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DOI：
10.1016/j.jcp.2008.09.014
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发表时间：
2009-01
期刊：
影响因子：
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通讯作者：
C. Cotter;D. Ham;C. Pain;S. Reich
中科院分区：
文献类型：
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作者：
C. Cotter;D. Ham;C. Pain;S. Reich
本文介绍一种新的混合有限元，用于求解二维和三维波动方程和不可压缩流动方程。我们称之为P1 D-P2的单元使用不连续的分段线性函数表示速度，使用连续的分段二次函数表示压力。引入混合制剂的目的是产生一个新的灵活的元素选择三角形和四面体网格，满足LBB稳定性条件，因此没有虚假的零能量模式。这种特殊元素选择的优点是速度的质量矩阵是块对角矩阵，因此可以进行简单的反演;它还允许将压力的阶数增加到二次，同时保持LBB稳定性，这在具有科里奥利力的地球物理应用中具有优势。本文给出了一维半离散波动方程的简正模分析，证明了单元对是稳定的;通过二维波动方程的数值积分，证明了单元对是稳定的;通过对二维和三维离散拉普拉斯算子在不同网格上的分析，证明了单元对不存在任何伪模。我们提供的元素对，确认该元素是稳定的，因为数值解的收敛速度是二次的收敛性测试。
We introduce a new mixed finite element for solving the 2- and 3-dimensional wave equations and equations of incompressible flow. The element, which we refer to as P1D–P2, uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure. The aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the LBB stability condition and hence has no spurious zero-energy modes. The advantage of this particular element choice is that the mass matrix for velocity is block diagonal so it can be trivially inverted; it also allows the order of the pressure to be increased to quadratic whilst maintaining LBB stability which has benefits in geophysical applications with Coriolis forces. We give a normal mode analysis of the semi-discrete wave equation in one dimension which shows that the element pair is stable, and demonstrate that the element is stable with numerical integrations of the wave equation in two dimensions, an analysis of the resultant discrete Laplace operator in two and three dimensions on various meshes which shows that the element pair does not have any spurious modes. We provide convergence tests for the element pair which confirm that the element is stable since the convergence rate of the numerical solution is quadratic.