Condition Numbers and Local Errors in the Boundary Element Method

Dijkstra, W.; Kakuba, Godwin; Mattheij, R.M.M.

doi:10.1142/9781848165809_0010

Cited by 6 publications

(5 citation statements)

References 29 publications

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“…At the same time, similarly to BEM, 39 MMP also suffers from ill‐conditioning. However, its impact is still more limited than BEM due to the low number of degrees of freedom required for MMP, given its exponential convergence: the dense MMP blocks in the coupling matrices are therefore small.…”

Section: Discussionmentioning

confidence: 99%

Coupling finite elements and auxiliary sources for electromagnetic wave propagation

Casati

Hiptmair

Smajić

2020

Int J Numerical Modelling

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We propose four approaches to solve time-harmonic Maxwell's equations in R 3 through the finite element method (FEM) in a bounded region encompassing parameter inhomogeneities, coupled with the multiple multipole program (MMP) in the unbounded complement. MMP belongs to the class of methods of auxiliary sources and of Trefftz methods, as it employs point sources that spawn exact solutions of the homogeneous equations. Each of these sources is anchored at a point that, if singular, is placed outside the respective domain of approximation. In the MMP domain, we assume that material parameters are piecewise constant, which induces a partition into one unbounded subdomain and other bounded, but possibly very large, subdomains, each requiring its own MMP trial space. Hence, in addition to the transmission conditions between the FEM and MMP domains, one also has to impose transmission conditions connecting the MMP subdomains. Coupling approaches arise from seeking stationary points of Lagrangian functionals that both enforce the variational form of the equations in the FEM domain and match the different trial functions across subdomain interfaces. We discuss the following approaches: 1 Least-squares-based coupling using techniques from PDE-constrained optimization. 2 Discontinuous Galerkin coupling between the meshed FEM domain and the single-entity MMP subdomains. 3 Multi-field variational formulation in the spirit of mortar FEMs. 4 Coupling through the Dirichlet-to-Neumann operator. We compare these approaches in a series of numerical experiments with different geometries and material parameters, including examples that exhibit triple-point singularities and infinite layered media.

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Section: Discussionmentioning

confidence: 99%

Coupling finite elements and auxiliary sources for electromagnetic wave propagation

Casati

Hiptmair

Smajić

2020

Int J Numerical Modelling

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“…where {e 1 , e 2 } is the canonical basis of R 2 . These densities are a basis of the space of equilibrium distributions for the Stokes problem (see [7] and [6]). They are the Stokes equivalent of the electrostatic equilibrium distribution that leads to the definition of logarithmic capacity.…”

Section: Alsomentioning

confidence: 99%

Variational views of stokeslets and stresslets

Sayas

Selgás

2014

SeMA

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In this paper we present a self-contained variational theory of the layer potentials for the Stokes problem on Lipschitz boundaries. We use these weak definitions to show how to prove the main theorems about the associated Calderón projector. Finally, we relate these variational definitions to the integral forms. Instead of working these relations from scratch, we show some formulas parametrizing the Stokes layer potentials in terms of those for the Lamé and Laplace operators. While all the results in this paper are well known for smooth domains, and most might be known for non-smooth domains, the approach is novel a gives a solid structure to the theory of Stokes layer potentials. *

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“…In general, glass can be treated as an isotropic viscoelastic Maxwell material [3,8,9,61], that is the strain rate tensor can be split up into an elastic and a viscous part: 15) where the elastic and viscous strain rate tensors,Ė e andĖ v respectively, are given by [9] …”

Section: Mechanicsmentioning

confidence: 99%

“…A commonly used boundary condition to describe fluid flow at an impenetrable wall [15,21,31,53] is Navier's slip condition: can be obtained by using the Tresca model [18]. Introduce a dimensionless friction coefficient,…”

Section: Mechanicsmentioning

confidence: 99%