Duality estimates for the numerical solution of integral equations

Costabel, Martin; Stephan, Ernst P.

doi:10.1007/bf01396766

Cited by 37 publications

(15 citation statements)

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“…By noting that y/* = yi on r0 and using the triangle inequality, we obtain Following the line of reasoning in [4] in using Nitsche's trick, we can prove the following: and with Lx bounded from Hp to Hp for any s, t £ E. There being no need to consider a nonuniform mesh, we used a uniform mesh and investigated the global errors in this example. We chose piecewise constants as test and trial functions in the Galerkin approximation for (5.3), and used A"/, = K\ 2 given by (4.11) to average the values of «/, (see Theorem 3.1 and Remark 2).…”

Section: The Condition T < 0 Is Necessary Because In General Y/ £ H°(mentioning

confidence: 90%

“…With piecewise constant functions used as trial and test functions, it was proved that the local L2-error converges with order 0(h) in the case of smooth closed curves [11] and with order 0(hx¡2) in the case of smooth open curves [16]. However, it is well known that the highest orders of global convergence achieved (in negative norms) are 0(h3) for the closed smooth case [5] and 0(h) for the open smooth case [4,13]. The purpose of this article is to construct, from the Galerkin solution, a better approximate solution which inherits the highest possible orders of global convergence to give best local convergence in the L2-norm (e.g., in the example mentioned above, order 0(hy) for the closed case and 0(h) for the open case can be achieved locally in the L2-norm).…”

Section: Introductionmentioning

confidence: 99%

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The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates

Tran¹

1995

Math. Comp.

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Abstract. Superconvergence in the L2-norm for the Galerkin approximation of the integral equation Lu = f is studied, where I is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let Uf, be the Galerkin approximation to u . By using the ^-operator, an operator that averages the values of uh , we will construct a better approximation than uh itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm's equation on a slit the same order of convergence can be recovered if the mesh is suitably graded.

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Section: The Condition T < 0 Is Necessary Because In General Y/ £ H°(mentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates

Tran¹

1995

Math. Comp.

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“…Grundlegende Eigenschaften der Randintegraloperatoren wurden bewiesen [25,21 ], und die asymptotische Konvergenz der BE-N/iherungsl6sungen und BE/ FE-Kopplungsverfahren wurde begrfindet [123, 124,125] sowie [24]. Die Galerkin-Approximation ffir stark singul/ire sowie hypersinguliire Randintegralgleichunge wurde mathematisch untersucht [22,23,113,114].…”

Section: Oberblick Zur Randelementmethodeunclassified

Integrationsverfahren zur Galerkin-Randeiementmethode für dreidimensionale Elastizitätsprobleme

Andrä

1999

Forsch Ing-Wes

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Zusammenfassung Ffir die dreidimensionalen Elastizit~itsgleichungen wird eine L6sungsmethode fiber ein System yon singul/iren Integralgleichungen behandelt. Diese Vorgehensweise stellt eine Alternative zur Methode der Finiten Elemente (FE) dar und hat ffir dreidimensiohale Probleme den Hauptvorteil, daft kein FE-Volumennetz, sondern nur ein Oberfl~ichennetz generiert werden muff. Die Integralgleichungen werden mit der GalerkinRandelementmethode, die gegenfiber der weitverbreiteten Kollokationsmethode mehrere wesentliche Vorteile besitzt, diskretisiert. Gegenw/irtig werden im Ingenieurbereich fast keine Galerkin-Randelementmethoden eingesetzt, da diese Verfahren auf singul/ire Doppelintegrale f0hren, deren Berechnungsaufwand mit herk6mmli-chen Verfahren v6Uig inakzeptabel war. Schwerpunkt der Ver6ffentlichung ist eine neue Methode zur Berechnung der singul~iren Doppelintegrale. Das wichtigste Ergebnis ist dabei, daft auch ffir gekrfimmte Gebietsberandungen die singul~iren Integrationen auf regul~ire Integrationen mit relativ einfachen Integranden zurfickgefiihrt werden k6n-nen, wobei zur numerischen Berechnung der Integrale nur Standard-Gaufl-Quadraturformeln notwendig sind. Diese Resultate vereinfachen die Implementierung der GalerkinRandelementmethode wesentlich und machen somit diese Methode ffir praktische Anwendungen im Ingenieurbereich zug/inglich. Weiterhin wird auf die Behandlung der auftretenden reguliiren Doppelintegrale eingegangen. Numerische Testrechnungen ffir Modellprobleme der linearen Elastostatik werden diskutiert, wobei Quadratur-und Diskretisierungsfehler ausgewertet werden.Integration procedures for 6olerkin-type boundary element methods in three-dimensional elosticity Abstract The mixed boundary value problem in threedimensional linear elasticity is solved via a system of singular boundary integral equations. This procedure is an alternative to the finite element method and has the main advantage that expensive volume mesh generation is omitted and only a surface mesh is sufficient. The integral equations are discretized by the Galerkin-type boundary element method, which has essential advantages compared to the widely used collocation method. At present the Galerkin method is almost never used in engineering, because this method leads to an unacceptably high effort for the computation of singular double integrals if traditional integration methods are used. The main result of this paper is a new method for the computation of such singular double integrals. The integration procedure leads to simple regular integrand functions also in the case of curved boundary elements. This result simplifies the implementation of the Galerkin-type boundary element method and makes this method applicable in mechanical engineering. Furthermore, the integration of regular double integrals is explained. Numerical tests for model problems in linear elasticity are discussed. Quadrature and discretization errors are analyzed.

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“…This limits the orders of convergence independently of the choice of the test functions and of the number of singular functions used at the corners. Some higher orders of convergence can be obtained in Sobolev spaces with negative indices, see [5].…”

Section: Introductionmentioning

confidence: 99%

Optimal order collocation for the mixed boundary value problem on polygons

Laubin¹

2000

Math. Comp.

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Abstract. In usual boundary elements methods, the mixed Dirichlet-Neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed. We build collocation methods based on a mixed single and double layer potential. This indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of Sobolev type. The boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not 1. This condition is analogous to the one met in the use of the single layer potential, and is not a problem in practical computations. The collocation methods use smoothest splines and known singular functions generated by the corners. If splines of order 2m − 1 are used, we get quasi-optimal estimates in H m -norm. The order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function.

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Duality estimates for the numerical solution of integral equations

Cited by 37 publications

References 14 publications

The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates

The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates

Integrationsverfahren zur Galerkin-Randeiementmethode für dreidimensionale Elastizitätsprobleme

Optimal order collocation for the mixed boundary value problem on polygons

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