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.

scholarly journals Optimized Schwarz Methods for Maxwell's Equations

2009 ◽  
Vol 31 (3) ◽  
pp. 2193-2213 ◽  
Author(s):  
V. Dolean ◽  
M.J. Gander ◽  
L. Gerardo-Giorda
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Schwarz Methods ◽  
Optimized Schwarz Methods

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.

Screen type mixed boundary value problems for anisotropic pseudo-Maxwell’s equations

2016 ◽  
Vol 23 (4) ◽  
pp. 511-518
Author(s):  
Otar Chkadua ◽  
Roland Duduchava ◽  
David Kapanadze
Keyword(s):  
Boundary Value Problems ◽  
Unique Solvability ◽  
Bilinear Form ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Potential Method ◽  
Mixed Boundary Value Problems ◽  
Well Posed

AbstractWe investigate screen type mixed boundary value problems for anisotropic pseudo-Maxwell’s equations. We show that the problems with tangent traces are well posed in tangent Sobolev spaces. The unique solvability results are proven based on the potential method and coercivity result of Costabel on the bilinear form associated with pseudo-Maxwell’s equations.

scholarly journals Stability of a Leap-Frog Discontinuous Galerkin Method for Time-Domain Maxwell's Equations in Anisotropic Materials

2017 ◽  
Vol 21 (5) ◽  
pp. 1350-1375 ◽  
Author(s):  
Adérito Araújo ◽  
Sílvia Barbeiro ◽  
Maryam Khaksar Ghalati
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Mesh Size ◽  
Element Space ◽  
First Order ◽  
Balance Accuracy ◽  
The Stability

AbstractIn this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We present a sufficient condition for the stability and error estimates, for cases of typical boundary conditions, either perfect electric, perfect magnetic or first order Silver-Müller. The bounds of the stability region point out the influence of not only the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. In the model we consider heterogeneous anisotropic permittivity tensors which arise naturally in many applications of interest. Numerical results supporting the analysis are provided.

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