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Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Yalchin Efendiev, Juan Galvis, Raytcho Lazarov, Joerg Willems (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract...

Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Yalchin Efendiev, Juan Galvis, Raytcho Lazarov, Joerg Willems (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations....

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

Maria Lymbery, Svetozar Margenov (2012)

Open Mathematics

While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study...

Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

Paola F. Antonietti, Blanca Ayuso (2007)

ESAIM: Mathematical Modelling and Numerical Analysis


We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain...

Some remarks on averaging in the BDDC method

Čertíková, Marta, Burda, Pavel, Novotný, Jaroslav, Šístek, Jakub (2010)

Programs and Algorithms of Numerical Mathematics

In this paper, we introduce a general framework for derivation of the averaging operator, from which the standard choices are recovered by simplifications. Then, an alternative approach derived by another simplification is proposed and tested on a 2D example.

Spectral discretization of Darcy equations coupled with Navier-Stokes equations by vorticity-velocity-pressure formulation

Yassine Mabrouki, Jamil Satouri (2022)

Applications of Mathematics

We consider a model coupling the Darcy equations in a porous medium with the Navier-Stokes equations in the cracks, for which the coupling is provided by the pressure's continuity on the interface. We discretize the coupled problem by the spectral element method combined with a nonoverlapping domain decomposition method. We prove the existence of solution for the discrete problem and establish an error estimation. We conclude with some numerical tests confirming the results of our analysis.

Spectral methods for singular perturbation problems

Wilhelm Heinrichs (1994)

Applications of Mathematics

We study spectral discretizations for singular perturbation problems. A special technique of stabilization for the spectral method is proposed. Boundary layer problems are accurately solved by a domain decomposition method. An effective iterative method for the solution of spectral systems is proposed. Suitable components for a multigrid method are presented.

Stabilization of a non standard FETI-DP mortar method for the Stokes problem

E. Chacón Vera, T. Chacón Rebollo (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove...

The combination technique for a two-dimensional convection-diffusion problem with exponential layers

Sebastian Franz, Fang Liu, Hans-Görg Roos, Martin Stynes, Aihui Zhou (2009)

Applications of Mathematics

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on N × N , N × N and N × N meshes. It is shown that the combination FEM yields (up to a factor ln N ) the same order of accuracy in the associated...

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