Inexact Additive Schwarz Solvers for hp-FEM Discretizations in Three Dimensions

Beuchler, Sven

doi:10.1007/978-3-642-30316-6_4

Cited by 1 publication

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“…The result κ(Â −1 in,1 A) (log k log χ log k) 3 has been proved in [9]. The assertion for d > 1 follows straightforward by using (26) and (17).…”

Section: The Preconditioner For the Velocity Partmentioning

confidence: 89%

“…We use an overlapping DD-preconditioner with inexact subproblem solvers which has been developed for scalar elliptic problems and a scalar hp-FEM basis, see [9] for more details. Let which are ordered first.…”

Section: The Preconditioner For the Velocity Partmentioning

confidence: 99%

“…For elliptic problems, preconditioned conjugate gradient (PCG) methods with additive Schwarz preconditioners (ASM) as domain decomposition (DD) are a powerful tool for the development of fast and efficient solvers for the h-version as well as for the p-version of the FEM, see [1, 5, 7, 8, 10, 17, 21, 27-29, 31, 33, 34]. The extension to linear elasticity problems is straightforward, [9]. Based on Korn's inequality, [22], an optimal preconditioner is obtained by using a preconditioner for the potential equation for each component of the displacement u.…”

Section: Introductionmentioning

confidence: 99%

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Schwarz Type Solvers for -FEM Discretizations of Mixed Problems

Beuchler¹,

Purrucker²

2012

Comput. Methods Appl. Math.

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The Stokes problem and linear elasticity problems can be viewed as a mixed variational formulation. These formulations are discretized by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complementrelated to the components of the pressure modes and the discrezation by a stable finite element pair which satisfies the discrete infsup condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast hp-FEM preconditioners for elliptic problems. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressureis discretized by discontiuous finite elements.We will prove that the system of linear algebraic equationscan be solved in almost optimal complexity if the Q k − P k−1,disc element is used. This yields to quasioptimal hp-FEM solvers for the Stokes problems and linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

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“…The result κ(Â −1 in,1 A) (log k log χ log k) 3 has been proved in [9]. The assertion for d > 1 follows straightforward by using (26) and (17).…”

Section: The Preconditioner For the Velocity Partmentioning

confidence: 89%

Section: The Preconditioner For the Velocity Partmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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