Quadrature Methods for Strongly Elliptic Equations of Negative Order on Smooth Closed Curves

In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms. Mathematics Subject Classifications (2000) 65R20 · 65N38

show abstract

“…Notice that this special choice for logarithmic integral equations had already been observed in [4,19].…”

Section: Superconvergent Methodsmentioning

confidence: 89%

“…Quadrature methods for equations of logarithmic type had been previously studied by an equivalent formulation with trigonometric polynomials in [19] or as Dirac delta approximations in [4]. For equations of the second kind, the approach is usually based on the classical analysis of Nyström methods (see, for instance, [2,14]).…”

Section: Introductionmentioning

confidence: 99%

Dirac delta methods for Helmholtz transmission problems

2006

View full text Add to dashboard Cite

show abstract

“…The proof of this result can be found in [16] Lemma 2.1. The uniform boundedness of P h , that is, Hypothesis (H1), is a direct consequence of (31).…”

Section: Trigonometric Collocationmentioning

confidence: 99%

“…Taking into account (16) and κ ≥ 0, we obtain κ β = 0 on R and v = 0 in ω. Going to the boundary we deduce that v(x) = ı α− β is constant on R. Since κ does not vanish identically, the real β is 0, α is constant and y = α x .…”

Section: The Convex Casementioning

confidence: 99%

Theoretical and numerical study of a free boundary problem by boundary integral methods

2001

View full text Add to dashboard Cite

Abstract. In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.Résumé. Dans cet article on considère un problèmeà frontière libre intervenant en formageélectro-magnétique. Après l'avoir ramenéà un système intégro-différentiel où l'inconnue est la représentation paramétrique de la frontière, on enétudie les propriétés mathématiques essentielles. On s'intéresse ensuiteà l'approximation numérique par des méthodes de type Galerkin ou de collocation en utilisant pour l'approximation des polynômes trigonométriques ou des fonctions splines.Mathematics Subject Classification. 35R35, 41A15, 42A12, 45G05, 65R20.

show abstract

“…As far as we know the localization has not been applied before to the case of numerical solution of parabolic boundary integral equations. Here we adapt the approach given by Saranen and Schroderus in [23] for a quadrature method. This approach makes an effective use of the stability and yields the final result more directly than that of [4].…”

Section: Introductionmentioning

confidence: 99%

The spline collocation method for parabolic boundary integral equations on smooth curves

Costabel

Saranen

2003

Numerische Mathematik

Self Cite

View full text Add to dashboard Cite

We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces.

show abstract

Quadrature Methods for Strongly Elliptic Equations of Negative Order on Smooth Closed Curves

Cited by 29 publications

References 31 publications

Dirac delta methods for Helmholtz transmission problems

Dirac delta methods for Helmholtz transmission problems

Theoretical and numerical study of a free boundary problem by boundary integral methods

The spline collocation method for parabolic boundary integral equations on smooth curves

Contact Info

Product

Resources

About