The Conditioning of Boundary Element Equations on Locally Refined Meshes and Preconditioning by Diagonal Scaling

Ainsworth, Mark; McLean, William; Tran, Thanh

doi:10.1137/s0036142997330809

Cited by 83 publications

(90 citation statements)

References 10 publications

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“…Recent work on the conditioning of finite element matrices has focused on upper bounds for the Euclidean condition number in the case of locally refined meshes; see, e.g., [1,3]. The objective of the present paper is to give upper and lower bounds on κ p (A) for p ∈ [1, +∞] when A is the stiffness matrix associated with the finite element approximation of a linear, abstract model problem posed in Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

Evaluation of the condition number in linear systems arising in finite element approximations

Ern¹,

Guermond²

2006

ESAIM: M2AN

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Abstract. This paper derives upper and lower bounds for the p -condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsize h are obtained. The theoretical results are applied to finite element approximations of elliptic PDE's in variational and in mixed form, and to first-order PDE's approximated using the Galerkin-Least Squares technique or by means of a non-standard Galerkin technique in L 1 (Ω). Numerical simulations are presented to illustrate the theoretical results.Mathematics Subject Classification. 65F35, 65N30.

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Section: Introductionmentioning

confidence: 99%

Evaluation of the condition number in linear systems arising in finite element approximations

Ern¹,

Guermond²

2006

ESAIM: M2AN

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“…Our analysis will cover faults that can be represented by a random matrix belonging to S ε . This means that the analysis will cover each of the cases (i) when the fault is detectable and mitigated using the laissez-faire strategy, and (ii) when the fault is silent but is relatively small in the sense that it may be modelled using (2). However, our analysis will not cover the important case involving faults which result in entries in the matrices or right hand sides in the problem being corrupted.…”

Section: Probabilistic Model Of Faultsmentioning

confidence: 99%

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

Ainsworth

Glusa

2017

SIAM J. Sci. Comput.

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Abstract. The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant.The purpose of this work is to provide an analysis of the behaviour of the multigrid algorithm in the presence of faults. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations and it is of considerable importance to anticipate its behaviour on an exascale machine. The analysis of resilience of algorithms is in its infancy and the current work is perhaps the first to provide a mathematical model for faults and analyse the behaviour of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method. IntroductionPresident Obama's executive order in the summer of 2015 establishing the National Strategic Computing Initiative 1 committed the US to the development of a capable exascale computing system. Given that the performance of the current number one machine Tianhe-2 is roughly one thirtieth of that of an exascale system, it is easy to underestimate the challenge posed by this task. One way to envisage the scale of the undertaking is that the combined processing power of the entire TOP 500 list is less than half of one exaflop (10 18 floating point operations per second).It is widely accepted that an exascale machine should respect a 20MW power envelope. Tianhe-2 already consumes 18MW of power, and if it were possible to simply upscale to exascale using the current technology, would require around 2010 Mathematics Subject Classification. 65F10, 65N22, 65N55, 68M15.

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“…Inequality (3.4) was given in [32], Lemma 3.2, for the case when the norm in H s is defined by the method of complex interpolation, and was proved in [4] in the case of real interpolation.…”

Section: Lemma 34 Let K H Be a Triangle (Respectively A Parallelogmentioning

confidence: 99%

Thehp-version of the boundary element method with quasi-uniform meshes in three dimensions

Bespalov¹,

Heuer²

2008

ESAIM: M2AN

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Abstract.We prove an a priori error estimate for the hp-version of the boundary element method with hypersingular operators on piecewise plane open or closed surfaces. The underlying meshes are supposed to be quasi-uniform. The solutions of problems on polyhedral or piecewise plane open surfaces exhibit typical singularities which limit the convergence rate of the boundary element method. On closed surfaces, and for sufficiently smooth given data, the solution is H 1 -regular whereas, on open surfaces, edge singularities are strong enough to prevent the solution from being in H 1 . In this paper we cover both cases and, in particular, prove an a priori error estimate for the h-version with quasiuniform meshes. For open surfaces we prove a convergence like O(h

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The Conditioning of Boundary Element Equations on Locally Refined Meshes and Preconditioning by Diagonal Scaling

Cited by 83 publications

References 10 publications

Evaluation of the condition number in linear systems arising in finite element approximations

Evaluation of the condition number in linear systems arising in finite element approximations

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

Thehp-version of the boundary element method with quasi-uniform meshes in three dimensions

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