Non-negative mixed finite element formulations for a tensorial diffusion equation
Non-negative mixed finite element formulations for a tensorial diffusion equation
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DOI：
10.1016/j.jcp.2009.05.039
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发表时间：
2008-10
期刊：
影响因子：
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通讯作者：
K. Nakshatrala;A. Valocchi
中科院分区：
文献类型：
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作者：
K. Nakshatrala;A. Valocchi
我们考虑张量扩散方程，并解决离散的最大-最小原则的混合有限元制剂。特别是，我们解决非负解（这是一个特殊情况下的最大值-最小值原则）的混合有限元制剂。众所周知，经典的有限元公式（如单场Galerkin公式，Raviart-Thomas，变分多尺度和Galerkin/最小二乘混合公式）在任意网格和强各向异性扩散系数上不产生非负解（即，它们不满足离散最大-最小原理）。本文基于约束优化技术，给出了张量扩散方程的两个非负混合有限元列式。这些提出的混合公式在任意网格上为低阶（即，线性、双线性和三线性）有限元。第一个公式是基于Raviart-Thomas空间，第二个非负公式是基于变分多尺度公式。对于前者的配方，我们评论的效果，增加了非负约束的局部质量平衡性质的Raviart-Thomas配方。我们进行数值收敛性分析所提出的基于优化的非负混合配方。我们还研究了有效集策略解决由此产生的约束优化问题的性能。所提出的配方的整体性能说明了三个典型的测试问题。
We consider the tensorial diffusion equation, and address the discrete maximum–minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum–minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart–Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum–minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. In this paper, we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart–Thomas spaces, and the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart–Thomas formulation. We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.