On the Convergence of Collocation Methods for Boundary Integral Equations on Polygons

Costabel, Martin; Stephan, Ernst P.

doi:10.2307/2008322

Cited by 14 publications

(5 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This natural connection is possibly the reason for some early attempts to include CAD representations in a BEM framework. For example, spline collocation methods were used to develop convergence estimates for the BEM in two [5,6,7,8,9] and three dimensions [10,11,12]. A BEM formulation based on cubic splines was proposed by [13] and [14] to solve groundwater problems and the Laplace's equation, respectively.…”

Section: Introductionmentioning

confidence: 99%

Fast isogeometric boundary element method based on independent field approximation

Marussig

Zechner

Beer

et al. 2015

Computer Methods in Applied Mechanics and Engineering

134

View full text Add to dashboard Cite

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.

show abstract

Section: Introductionmentioning

confidence: 99%

Fast isogeometric boundary element method based on independent field approximation

Marussig

Zechner

Beer

et al. 2015

Computer Methods in Applied Mechanics and Engineering

134

View full text Add to dashboard Cite

show abstract

“…In this paper we examine the numerical approximation of equation (1.2) by Galerkin methods based on piecewise polynomials. Galerkin and collocation methods with finite elements for boundary integral equations for two-dimensional boundary value problems in polygonal domains have been studied by many authors, where high convergence rates were obtained either by augmenting the space of trial functions by special singular elements or by using appropriate mesh refinement in the vicinity of the corners; see, for example, [2][3][4]8,9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

The double‐layer potential operator over polyhedral domains II: Spline Galerkin methods

Elschner¹

1992

Math Methods in App Sciences

View full text Add to dashboard Cite

We examine the numerical approximation of the integral equation (A -K)u =f, where K is the double layer (harmonic) potential operator on a closed polyhedral surface in R3 and I, lAl 2 1, is a complex constant. The solution is approximated by Galerkin's method, which is based on piecewise polynomials of arbitrary degree on graded triangulations. By utilizing spline spaces which are modified in that the trial functions vanish on some of the triangles closest to the vertices and edges, we investigate the stability of this method in Lz. Furthermore, the use of suitably graded meshes leads to the same quasioptimal error estimates as in the case of a smooth surface.

show abstract

“…However, if Γ possesses corner points, the situation becomes more involved. One of the simplest cases to treat is a polygonal boundary or a boundary with polygonally shaped corners and there are a number of works investigating approximation methods for the equation (1) on such curves [3,5,6,16,19,21]. For a comprehensive survey, we refer the reader to [2,20].…”

Section: Introductionmentioning

confidence: 99%

Spline Galerkin Methods for the Double Layer Potential Equations on Contours with Corners

Didenko

2017

Recent Trends in Operator Theory and Partial Differential Equations

View full text Add to dashboard Cite

Spline Galerkin methods for the double layer potential equation on contours with corners are studied. The stability of the method depends on the invertibility of some operators R τ associated with the corner points τ . The operators R τ do not depend on the shape of the contour but only on the opening angles of the corner points τ . The invertibility of these operators is studied numerically via the stability of the method on model curves, all corner points of which have the same opening angle. The case of the splines of order 0, 1 and 2 is considered. It is shown that no opening angle located in the interval [0.1π, 1.9π] can cause the instability of the method. This result is in strong contrast with the Nyström method, which has four instability angles in the interval mentioned. Numerical experiments show a good convergence of the methods even if the right-hand side of the equation has discontinuities located at the corner points of the contour.

show abstract

On the Convergence of Collocation Methods for Boundary Integral Equations on Polygons

Cited by 14 publications

References 6 publications

Fast isogeometric boundary element method based on independent field approximation

Fast isogeometric boundary element method based on independent field approximation

The double‐layer potential operator over polyhedral domains II: Spline Galerkin methods

Spline Galerkin Methods for the Double Layer Potential Equations on Contours with Corners

Contact Info

Product

Resources

About