Domain Decomposition Preconditioning for Problems with Highly Varying Coefficients

2012 ◽  
Vol 12 (4) ◽  
pp. 469-485
Author(s):  
Aleksandr Matsokin ◽  
Sergei Nepomnyaschikh
Keyword(s):  
Domain Decomposition ◽  
Numerical Experiments ◽  
Elliptic Boundary ◽  
Decomposition Methods ◽  
Second Order ◽  
Two Dimensional ◽  
Domain Decomposition Methods ◽  
Boundary Problems ◽  
Varying Coefficients ◽  
Small Holes

AbstractIn this paper we construct efficient domain decomposition methods for solving scalar second-order elliptic boundary problems in bounded two-dimensional domains with small holes, and present some results of numerical experiments, confirming the efficiency and robustness of the proposed domain decomposition methods.

scholarly journals Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018 ◽  
Vol 52 (4) ◽  
pp. 1569-1596 ◽  
Author(s):  
Xavier Antoine ◽  
Fengji Hou ◽  
Emmanuel Lorin
Keyword(s):  
Domain Decomposition ◽  
Decomposition Methods ◽  
Nonlinear Schrödinger Equations ◽  
Waveform Relaxation ◽  
Schrödinger Equations ◽  
Two Dimensional ◽  
Domain Decomposition Methods ◽  
Schwarz Waveform Relaxation ◽  
Nonlinear Schrodinger Equations ◽  
Linear And Nonlinear

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin, Numer. Math. 137 (2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

On the approximation of a virtual coarse space for domain decomposition methods in two dimensions

2018 ◽  
Vol 28 (07) ◽  
pp. 1267-1289 ◽  
Author(s):  
Juan G. Calvo
Keyword(s):  
Domain Decomposition ◽  
Numerical Experiments ◽  
Decomposition Methods ◽  
Extension Operator ◽  
Two Dimensions ◽  
Computational Time ◽  
Domain Decomposition Methods ◽  
Polygonal Elements ◽  
Schwarz Algorithm ◽  
Coarse Space

A new extension operator for a virtual coarse space is presented which can be used in domain decomposition methods for nodal elliptic problems in two dimensions. In particular, a two-level overlapping Schwarz algorithm is considered and a bound for the condition number of the preconditioned system is obtained. This bound is independent of discontinuities across the interface. The extension operator saves computational time compared to previous studies where discrete harmonic extensions are required and it is suitable for general polygonal meshes and irregular subdomains. Numerical experiments that verify the result are shown, including some with regular and irregular polygonal elements and with subdomains obtained by a mesh partitioner.

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