Additive schwarz methods for the H-version boundary element method

Tran, Thanh; Stephan, Ernst P.

doi:10.1080/00036819608840418

Cited by 59 publications

(40 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The well-known equivalence of (20) and (26) was discussed explicitly in [25]. In fact, if u X is a solution of (20), then from the definition of P j and g j we deduce…”

Section: Lemma 32 For Any Two Zonal Functions ψ(T) and φ(T) Of The mentioning

confidence: 96%

See 1 more Smart Citation

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

2009

View full text Add to dashboard Cite

Abstract. We present an overlapping domain decomposition technique for solving elliptic partial differential equations on the sphere. The approximate solution is constructed using shifts of a strictly positive definite kernel on the sphere. The condition number of the Schwarz operator depends on the way we decompose the scattered set into smaller subsets. The method is illustrated by numerical experiments on relatively large scattered point sets taken from MAGSAT satellite data.

show abstract

“…The well-known equivalence of (20) and (26) was discussed explicitly in [25]. In fact, if u X is a solution of (20), then from the definition of P j and g j we deduce…”

Section: Lemma 32 For Any Two Zonal Functions ψ(T) and φ(T) Of The mentioning

confidence: 96%

“…The Schwarz operator P is then defined by (24) and (25). Functions in V j have supports in Γ j , where…”

Section: Subspace Decompositionmentioning

confidence: 99%

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

2009

View full text Add to dashboard Cite

show abstract

“…In [25, 7] stability proofs are given for closed curves, in [18, 1] for surfaces and adaptive refinement. These results were extended to the p and hp version of BEM in [9, 24, 23] and to screen problems in [26].…”

Section: Multilevel Preconditioningmentioning

confidence: 99%

“…This has been established for elliptic variational problems in H 1 (Ω) [17,28,27], and in H(curl, Ω) [10, 14,29]. The proofs boil down to establishing stability of a splitting V = i V i of the approximation space V in the sense that the multilevel normis equivalent to the problem induced energy norm · A independent of the level of refinement.Such stability results are available for the BEM spacesIn [25, 7] stability proofs are given for closed curves, in [18, 1] for surfaces and adaptive refinement. These results were extended to the p and hp version of BEM in [9, 24, 23] and to screen problems in [26].…”

mentioning

confidence: 99%

Schnelle Löser für Partielle Differentialgleichungen

Quarteroni¹,

Manzoni²,

Rozza³

2011

Oberwolfach Rep.

View full text Add to dashboard Cite

Abstract. The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd -May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds. (2000): 65N55, 65M55, 65M60, 65N30, 65N50, 65F10, 65F50, 65H10, 65K10. Mathematics Subject Classification Introduction by the OrganisersThe conference was organized by Randolph E. Bank, UCSD, La Jolla, Wolfgang Hackbusch, MPI Leipzig, and Gabriel Wittum, University of Frankfurt. This was the fifth one in a series of conferences on fast solvers held at Oberwolfach since 1999. The idea of these workshops is to bring together experts from the different thriving areas of solvers and offer a platform for scientific exchange and progress.The field of solvers for the algebraic systems arising from the discretization of partial differential equations has developed to a major area of numerical mathematics and scientific computing. Solvers are the essential part of simulation codes for problems from science and technology, in many cases determining the complexity of the whole simulation. By virtue of that, the choice of the solver can decide on the realiability of a simulation and if it can be done at all. Thus, solvers are a substantial mathematical component of most simulation tools and a major contribution of mathematics to quite a lot of applied disciplines. This has increased 2 Oberwolfach Report 28/2011 the interest in mathematics of colleagues from the applied sciencies over the last decade substantially.Major areas of solvers represented at the workshop are: Multigrid methods, H-matrices, domain decomposition methods, and conjugate gradient methods and their scalable parallelization on huge numbers of cores. Often these methods are combined, e.g. conjugate gradient like methods are used as accelerator for multigrid. Besides that, several talks were given on other aspects of solving partial differential equations, such as discretization schemes and the algebraic properties of the resulting stiffness matrices, overall solution strategies, and application areas where solving plays a crucial rle. The question of the right solver for critical application problems is still open, but new approaches have been developed in recent years. New light is shed on the solver question by the recent change of paradigm in computer architecture. The modern multicore processors with additional strong GPU and MIC accelerators pose a new and serious challenge for the development of fast solvers. A total of . . . presentations gave a nice overview over the current research, open problems and new developments. Intense discussions provided the opportunity to go into details of novel algorithms and approaches. In multigrid methods, a lot of research is going in the direction of developing robust methods for special applications. T...

show abstract

“…In [18,56] stability proofs are given for closed curves, in [2,44] for surfaces and adaptive refinement. These results were extended to the p and hp version of BEM in [25,27,28,53,54,57] and to screen problems in [24,58].…”

Section: Introductionmentioning

confidence: 99%

Stable multilevel splittings of boundary edge element spaces

Hiptmair

Mao

2012

Bit Numer Math

View full text Add to dashboard Cite

We establish the stability of nodal multilevel decompositions of lowestorder conforming boundary element subspaces of the trace space Hon boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by uniform refinement and the stability bounds are uniform in the number of refinement levels.The main tool is the general theory of P. Oswald (Interface preconditioners and multilevel extension operators, in Proc. 11th Intern. Conf. on Domain Decomposition Methods, London, 1998, pp. 96-103) that teaches, when stability of decompositions of boundary element spaces with respect to trace norms can be inferred from corresponding stability results for finite element spaces. H (curl, Ω)-stable discrete extension operators are instrumental in this.Stable multilevel decompositions immediately spawn subspace correction preconditioners whose performance will not degrade on very fine surface meshes. Thus, the results of this article demonstrate how to construct optimal iterative solvers for the linear systems of equations arising from the Galerkin edge element discretization of boundary integral equations for eddy current problems.

show abstract

Additive schwarz methods for the H-version boundary element method

Cited by 59 publications

References 7 publications

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

Schnelle Löser für Partielle Differentialgleichungen

Stable multilevel splittings of boundary edge element spaces

Contact Info

Product

Resources

About