Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces
Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces
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DOI：
10.1137/060660588
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发表时间：
2007-10
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通讯作者：
R. Hiptmair;Jinchao Xu
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文献类型：
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作者：
R. Hiptmair;Jinchao Xu
在本文中，我们开发并分析了一种预处理线性方程组的通用方法，该方法是由 H(curl, )- 和 H(div, )- 椭圆变分问题的一致有限元离散化产生的。预处理器完全依赖于离散泊松问题的求解器。我们利用辅助空间预处理的抽象理论证明了预处理器的网格独立有效性。主要工具是函数空间 H(curl, ) 和 H(div, ) 中所谓正则分解结果的离散模拟。我们的 H(curl, ) 预处理器类似于 [R. Beck，通过对简单三角剖分上的边缘元素进行组件分割的代数多重网格，技术。代表。 SC 99-40，ZIB，德国柏林，1999 年]。
In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, )- and H(div, )-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning. The main tools are discrete analogues of so-called regular decomposition results in the function spaces H(curl, ) and H(div, ). Our preconditioner for H(curl, ) is similar to an algorithm proposed in [R. Beck, Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany, 1999].