Additive Schwarz methods for thep-version finite element method

1993 ◽  
Vol 66 (1) ◽  
pp. 493-515 ◽  
Author(s):  
Luca F. Pavarino
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Schwarz Methods ◽  
Additive Schwarz ◽  
Additive Schwarz Methods ◽  
Element Method

Additive Average Schwarz Methods for Discretization of Elliptic Problems with Highly Discontinuous Coefficients

2010 ◽  
Vol 10 (2) ◽  
pp. 164-176 ◽  
Author(s):  
M. Dryja ◽  
M. Sarkis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Thin Layer ◽  
Discontinuous Galerkin Method ◽  
Discontinuous Coefficients ◽  
Schwarz Methods ◽  
Piecewise Constant ◽  
Additive Schwarz ◽  
Order Elliptic Problem ◽  
Additive Schwarz Methods

AbstractA second order elliptic problem with highly discontinuous coefficients has been considered. The problem is discretized by two methods: 1) continuous finite element method (FEM) and 2) composite discretization given by a continuous FEM inside the substructures and a discontinuous Galerkin method (DG) across the boundaries of these substructures. The main goal of this paper is to design and analyze parallel algorithms for the resulting discretizations. These algorithms are additive Schwarz methods (ASMs) with special coarse spaces spanned by functions that are almost piecewise constant with respect to the substructures for the first discretization and by piecewise constant functions for the second discretization. It has been established that the condition number of the preconditioned systems does not depend on the jumps of the coefficients across the substructure boundaries and outside of a thin layer along the substructure boundaries. The algorithms are very well suited for parallel computations.

scholarly journals A BDDC Preconditioner for the RotatedQ1FEM for Elliptic Problems with Discontinuous Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yaqin Jiang
Keyword(s):  
Finite Element Method ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Mesh Size ◽  
Discontinuous Coefficients ◽  
Schwarz Methods ◽  
Additive Schwarz ◽  
Variational Form ◽  
Second Order Elliptic Equations ◽  
Additive Schwarz Methods

We propose a BDDC preconditioner for the rotatedQ1finite element method for second order elliptic equations with piecewise but discontinuous coefficients. In the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasioptimal convergence behavior; that is, the condition number of the preconditioned problem is independent of the jumps of the coefficients and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.

Additive Schwarz Methods for Elliptic Mortar Finite Element Problems

2003 ◽  
Vol 95 (3) ◽  
pp. 427-457 ◽  
Author(s):  
Petter E. Bj�rstad ◽  
Maksymilian Dryja ◽  
Talal Rahman
Keyword(s):  
Finite Element ◽  
Schwarz Methods ◽  
Additive Schwarz ◽  
Additive Schwarz Methods
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