Improved Schwarz methods for an elliptic problem in a nonconvex domain with discontinuous coefficients

2009 ◽  
Vol 25 (2) ◽  
pp. 440-459 ◽  
Author(s):  
Chokri Chniti
Keyword(s):  
Elliptic Problem ◽  
Discontinuous Coefficients ◽  
Schwarz Methods ◽  
Nonconvex Domain

On the error estimates of a hybridizable discontinuous Galerkin method for second-order elliptic problem with discontinuous coefficients

2019 ◽  
Vol 40 (2) ◽  
pp. 1577-1600
Author(s):  
Gang Chen ◽  
Jintao Cui
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Elliptic Problem ◽  
Posteriori Error ◽  
Discontinuous Coefficients ◽  
Second Order ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Hybridizable Discontinuous Galerkin ◽  
Order Elliptic Problem

Abstract Hybridizable discontinuous Galerkin (HDG) methods retain the main advantages of standard discontinuous Galerkin (DG) methods, including their flexibility in meshing, ease of design and implementation, ease of use within an $hp$-adaptive strategy and preservation of local conservation of physical quantities. Moreover, HDG methods can significantly reduce the number of degrees of freedom, resulting in a substantial reduction of computational cost. In this paper, we study an HDG method for the second-order elliptic problem with discontinuous coefficients. The numerical scheme is proposed on general polygonal and polyhedral meshes with specially designed stabilization parameters. Robust a priori and a posteriori error estimates are derived without a full elliptic regularity assumption. The proposed a posteriori error estimators are proved to be efficient and reliable without a quasi-monotonicity assumption on the diffusion coefficient.

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.

Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions

1996 ◽  
Vol 72 (3) ◽  
pp. 313-348 ◽  
Author(s):  
Maksymilian Dryja ◽  
Marcus V. Sarkis ◽  
Olof B. Widlund
Keyword(s):  
Elliptic Problems ◽  
Discontinuous Coefficients ◽  
Three Dimensions ◽  
Schwarz Methods

A FETI-DP Method for Crouzeix-Raviart Finite Element Discretizations

2012 ◽  
Vol 12 (1) ◽  
pp. 73-91 ◽  
Author(s):  
Leszek Marcinkowski ◽  
Talal Rahman
Keyword(s):  
Finite Element ◽  
Condition Number ◽  
Elliptic Problem ◽  
Discontinuous Coefficients ◽  
Second Order ◽  
System Of Equations ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Finite Element Discretizations ◽  
Element Discretizations

AbstractThis paper is concerned with the construction and analysis of a parallel preconditioner for a FETI-DP system of equations arising from the nonconforming Crouzeix-Raviart finite element discretization of a model elliptic problem of second order with discontinuous coefficients. We show that the condition number of the preconditioned problem is independent of the coefficient jumps, and the preconditioner is quasi optimal.

scholarly journals Asymptotic analysis of optimized Schwarz methods for maxwell’s equations with discontinuous coefficients

2018 ◽  
Vol 52 (6) ◽  
pp. 2457-2477
Author(s):  
Victorita Dolean ◽  
Martin J. Gander ◽  
Erwin Veneros
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Single Layer ◽  
Mesh Size ◽  
Discontinuous Coefficients ◽  
Potential Method ◽  
Transmission Conditions ◽  
Schwarz Methods ◽  
Layer Potential ◽  
Optimized Schwarz Methods

Discretized time harmonic Maxwell’s equations are hard to solve by iterative methods, and the best currently available methods are based on domain decomposition and optimized transmission conditions. Optimized Schwarz methods were the first ones to use such transmission conditions, and this approach turned out to be so fundamentally important that it has been rediscovered over the last years under the name sweeping preconditioners, source transfer, single layer potential method and the method of polarized traces. We show here how one can optimize transmission conditions to take benefit from the jumps in the coefficients of the problem, when they are aligned with the subdomain interface, and obtain methods which converge for two subdomains in certain situations independently of the mesh size, which would not be possible without jumps in the coefficients.

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.

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