An algebraic multigrid method for finite element systems on criss-cross grids
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Springer 2006
An algebraic multigrid method for finite element systems on criss-cross grids Shi Shu a , Jinchao Xu b , Ying Yang c and Haiyuan Yu a a Institute for Computational and Applied Mathematics of Xiangtan University, China b Department of Mathematics and Center for Computational Mathematics and Application of
Pennsylvania State University, USA E-mail: [email protected] c Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, China
Received 12 July 2003; accepted 27 December 2003 Communicated by A. Zhou
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
In this paper, we design and analyze an algebraic multigrid method for a condensed finite element system on criss-cross grids and then provide a convergence analysis. Criss-cross grid finite element systems represent a large class of finite element systems that can be reduced to a smaller system by first eliminating certain degrees of freedoms. The algebraic multigrid method that we construct is analogous to many other algebraic multigrid methods for more complicated problems such as unstructured grids, but, because of the specialty of our problem, we are able to provide a rigorous convergence analysis to our algebraic multigrid method. Keywords: algebraic multigrid method, finite element, criss-cross grids, convergence analysis. Mathematics subject classifications (2000): ?
1.
Introduction
The criss-cross grids (see figures 1 and 3) are an important class of grids in practical applications (see, e.g., [1, 2]). Because of the special structure of criss-cross grids, i.e., the support of the basis function corresponding to type-b (see figure 1) is contained in a square element, it is possible to eliminate certain variables locally and the size of the system can then be reduced (about half of the original size). There are also many other types of finite element that share such features. One simple example is the high order The work was supported in part by NSAF(10376031) and National Major Key Project for basic researches
and by National High-Tech ICF Committee in China.
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Figure 1. The uniform criss-cross grids.
(a)
(b)
Figure 2. (a) A third-order Lagrangian element; (b) a third order conforming macro-element.
Lagrangian elements for second order elliptic boundary value problem. Another example is the so-called macro elements which are used to construct lower order conforming finite element space for fourth order partial differential equations (such as biharmonic for plate problems). Figure 2 illustrates the cubic finite element and the third-order conforming finite element respectively, in which the degree of freedom in the interior of element can be locally eliminated. Further examples include the finite element spaces for Stokes element in which bubble functions are often introduced for the stability consideration [8]. Since every bubble function is only supported inside
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