scholarly journals On the convergence of collocation methods for boundary integral equations on polygons

1987 ◽  
Vol 49 (180) ◽  
pp. 461-461 ◽  
Author(s):  
Martin Costabel ◽  
Ernst P. Stephan
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Collocation Methods

Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1447-1480 ◽  
Author(s):  
Matthias Taus ◽  
Gregory J. Rodin ◽  
Thomas J. R. Hughes
Keyword(s):  
Integral Equations ◽  
Isogeometric Analysis ◽  
Computer Aided Design ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
High Order ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Integration Schemes ◽  
Linear Algebraic Equations

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

The Collocation Method and the Splitting Extrapolation for the First Kind of Boundary Integral Equations on Polygonal Regions

2012 ◽  
Vol 4 (5) ◽  
pp. 603-616
Author(s):  
Li Wang
Keyword(s):  
Asymptotic Expansion ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Collocation Methods ◽  
A Posteriori ◽  
Splitting Extrapolation ◽  
Periodic Transformation ◽  
Posteriori Estimation

AbstractIn this paper, the collocation methods are used to solve the boundary integral equations of the first kind on the polygon. By means of Sidi’s periodic transformation and domain decomposition, the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers (i = 1,...,d), which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations. Numerical experiments are carried out to show that the methods are very efficient.

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