scholarly journals Hybrid Discontinuous Galerkin Discretisation and Domain Decomposition Preconditioners for the Stokes Problem

2019 ◽  
Vol 19 (4) ◽  
pp. 703-722 ◽  
Author(s):  
Gabriel R. Barrenechea ◽  
Michał Bosy ◽  
Victorita Dolean ◽  
Frédéric Nataf ◽  
Pierre-Henri Tournier
Keyword(s):  
Domain Decomposition ◽  
Discontinuous Galerkin ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Degrees Of Freedom ◽  
Domain Decomposition Method ◽  
Stokes Problem ◽  
Stability And Convergence ◽  
Interface Conditions ◽  
Optimal Domain

AbstractSolving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis of the discretisation method is presented, and the results are corroborated by numerical experiments. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

A NEW INTERFACE CEMENT EQUILIBRATED MORTAR (NICEM) METHOD WITH ROBIN INTERFACE CONDITIONS: THE P1 FINITE ELEMENT CASE

2013 ◽  
Vol 23 (12) ◽  
pp. 2253-2292 ◽  
Author(s):  
CAROLINE JAPHET ◽  
YVON MADAY ◽  
FREDERIC NATAF
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Domain Decomposition ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Numerical Results ◽  
New Method ◽  
Interface Conditions ◽  
Well Posed

We design and analyze a new non-conforming domain decomposition method, named the NICEM method, based on Schwarz-type approaches that allows for the use of Robin interface conditions on non-conforming grids. The method is proven to be well posed. The error analysis is performed in 2D and in 3D for P1 elements. Numerical results in 2D illustrate the new method.

scholarly journals Coupling Parareal with Non-Overlapping Domain Decomposition Method

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Rim GUETAT
Keyword(s):  
Domain Decomposition ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Time Dependent ◽  
Interface Problem ◽  
Time Interval ◽  
Overlapping Domain Decomposition ◽  
Time Dependent Problems

In this paper, we present a new parallel algorithm for time dependent problems based on coupling parareal with non-overlapping domain decomposition method in order to increase parallelism in time and in space. For this we focus on the iterative methods of parallization in space to solve the interface problem like Neumann-Neumann method. In the new algorithm, the coarse temporel propagator is defined on the global domain and the Neumann-Neumann method is chosen as a fine propagator with a few iterations. We present the rigorous convergence analysis of the new coupled algorithm on bounded time interval. Numerical experiments illustrate the performance of this new algorithm and confirm our analysis. RÉSUMÉ. Dans ce papier, nous présentons un nouvel algorithme parallèle pour les problèmes dé-pendant du temps basé sur le couplage du pararéel avec les méthodes de décomposition de domaine sans recouvrement afin d'augmenter le parallélisme dans le temps et l'espace. Nous nous concen-trons sur les méthodes itératives de parallélisation en espace pour résoudre le problème d'interface par la méthode de Neumann-Neumann. Dans ce nouvel algorithme, le propagateur grossier est dé-finie sur le domaine global et la méthode de Neumann-Neumann est choisi pour le propagateur fin avec quelques itérations. Nous présentons l'analyse rigoureuse de convergence du nouvel algorithme couplé sur un intervalle de temps borné. Des expèriences numériques illustrent les performances de ce nouvel algorithme et confirment notre analyse. Dans ce papier, nous présentons un nouvel algorithme parallèle pour les problèmes dépendantdu temps basé sur le couplage du pararéel avec les méthodes de décomposition de domainesans recouvrement afin d’augmenter le parallélisme dans le temps et l’espace. Nous nous concentronssur les méthodes itératives de parallélisation en espace pour résoudre le problème d’interfacepar la méthode de Neumann-Neumann. Dans ce nouvel algorithme, le propagateur grossier est définiesur le domaine global et la méthode de Neumann-Neumann est choisi pour le propagateur finavec quelques itérations. Nous présentons l’analyse rigoureuse de convergence du nouvel algorithmecouplé sur un intervalle de temps borné. Des expèriences numériques illustrent les performances dece nouvel algorithme et confirment notre analyse.

A Non-Overlapping Domain Decomposition Method for the Stokes Problem

2002 ◽  
Author(s):  
Attila Zsaki ◽  
Marius Paraschivoiu
Keyword(s):  
Domain Decomposition ◽  
Lagrange Multipliers ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Stokes Problem ◽  
Uzawa Algorithm ◽  
Driven Cavity ◽  
Numerical Tests ◽  
Overlapping Domain Decomposition ◽  
The Cost

A domain decomposition method for the Stokes problem using Lagrange multipliers is described. The dual system associated with the Lagrange multipliers is solved based on an iterative procedure using the two-level finite element tearing and interconnecting (FETI) method. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer iterations and an evaluation of the cost of the inner iterations are reported. Comparison with the well-known Uzawa algorithm shows a reduction in the floating point operations count of the inner iterations while achieving the same number of outer iterations.

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