论文标题
高阶有限元DE RHAM复合物的低阶预处理
Low-order preconditioning for the high-order finite element de Rham complex
论文作者
论文摘要
在本文中，我们提出了一个统一的框架，用于为高阶有限元元素DE RHAM复合物构建频谱等效的低阶离散化。该理论涵盖了$ h^1 $，$ h（{\ rm curl}）$和$ h（{\ rm div}）$中的扩散问题，并基于将低阶的低阶离散化和高阶基础相结合的高阶基础，用于使用nédélec和raviart-thomas的高阶基础，该元素使用多种元素的概念（构成多种元素）的概念（在某些地区）。这种光谱等效性与使用低阶离散化构建的代数多机方法相结合，在完整的DE RHAM复合体中，为高阶有限元问题提供了高度可扩展的无基质预处理。此外，为高阶内部惩罚不连续的盖尔金（DG）离散开发了一种新的最低阶（分段常数）预处理，为此提供了频谱等效性结果和代数多族方法的收敛证明。在所有情况下，光谱等效性结果与多项式程度和网格大小无关。对于DG方法，它们也独立于惩罚参数。这些新求解器灵活且易于使用；任何针对低阶问题的“黑框”预处理程序都可以用于为相应的高阶问题创建有效，有效的预处理。基于有限元库MFEM中的隐含性，提供了许多数值实验。这些预处理的理论特性得到了证实，该方法的灵活性和可扩展性在一系列具有挑战性的三维问题上得到了证明。
In this paper we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in $H^1$, $H({\rm curl})$, and $H({\rm div})$, and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart-Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any "black-box" preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implmentation in the finite element library MFEM. The theoretical properties of these preconditioners are corroborated, and the flexibility and scalability of the method are demonstrated on a range of challenging three-dimensional problems.
