On Multigrid Methods for Generalized Finite Element Methods
This paper reports investigations on how multigrid methods can be applied for the solution of some generalized finite element methods based on the partition of unity technique. One feature of the generalized finite element method is that the underlying al
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Abstract. This paper reports investigations on how multigrid methods can be applied for the solution of some generalized finite element methods based on the partition of unity technique. One feature of the generalized finite element method is that the underlying algebraic system is often singular due to the overlapping from the partition of unity. While standard iterative methods such as the conjugate gradient method, Jacobi, Gauss-Seidel methods, multigrid methods and domain decomposition methods are still convergent for this type of singular systems, we observe that a standard multigrid method does not converge uniformly with respect to mesh parameters. Using a simple model problem, we will carefully investigate why these method do not work. We will then propose a multigrid method that does converge uniformly as in the standard finite element method.
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Introduction
The Generalized Finite Element Method (GFEM), based on a partition of unity, represents a wide class of discretization methods for partial differential equations (see Babuska and Melenk [4,5)). These methods can be used in various situations, such as discretizations on nonmatching grids (see Huang and Xu [13)), improving the accuracy of the finite element discretization for equations with rough data and coefficients (see Babuska and Osborn [2,3], Babuska, Caloz and Osborn [1)). The GFEM framework is also a base for the construction of other discretization techniques, such as: meshless h, p and hp methods (see Oden and Duarte [9], Matache, Babuska and Schwab [15)), particle methods (see Liu, Jun, Adee and Belytschko [7,14], Griebel and Schweitzer [10)). These methods have a wide range of applications including homogenization [12,15]' structural mechanics [6,8,16-18) and dynamics [14], divide and conquer methods on nonmatching grids [13) and many other research areas of computational science and engineering. In the recent years, the research efforts have been mostly focused on the construction of generalized finite element methods, and investigation of their convergence properties. Very little attention has been paid to the development of effective solution methods for the resulting algebraic systems. The existing solution schemes are mainly based on LU decomposition (see [16,18)). Apparently, direct solution techniques can be used when the problem size is small, but for practically interesting cases, a direct method will be computationally very expensive. Moreover, standard techniques for LU-decomposition M. Griebel et al. (eds.), Meshfree Methods for Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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J. Xu, L. T. Zikatanov
will not work well because the resulting algebraic system is often singular or nearly singular. The goal of this paper is on the design of optimal multigrid method for the solution of the corresponding discrete problems. To better explain some of the difficulties related to solving of the discretized systems of equations obtained from the GFEM, we will restrict our considerations to a very special case
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