# Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

@article{Feischl2013QuasioptimalCR,
title={Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method},
author={Michael Feischl and Michael Karkulik and Jens Markus Melenk and Dirk Praetorius},
journal={SIAM J. Numer. Anal.},
year={2013},
volume={51},
pages={1327-1348}
}
• Published 30 April 2013
• Computer Science, Mathematics
• SIAM J. Numer. Anal.
For the simple layer potential $V$ associated with the three-dimensional (3D) Laplacian, we consider the weakly singular integral equation $V\phi=f$. This equation is discretized by the lowest-order Galerkin boundary element method. We prove convergence of an $h$-adaptive algorithm that is driven by a weighted residual error estimator. Moreover, we identify the approximation class for which the adaptive algorithm converges quasi-optimally with respect to the number of elements. In particular…
Adaptive boundary element methods for optimal convergence of point errors
• Mathematics
Numerische Mathematik
• 2016
Two adaptive mesh-refining algorithms are proposed and proved to have quasi-optimal convergence behavior with respect to an a posteriori computable bound for the point error in the representation formula.
Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation
• Mathematics, Computer Science
• 2014
We analyze an adaptive boundary element method for Symm’s integral equation in 2D and 3D which incorporates the approximation of the Dirichlet data $$g$$g into the adaptive scheme. We prove
Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods
• Computer Science, Mathematics
Comput. Methods Appl. Math.
• 2013
Abstract. We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh-refinement is driven by the
Adaptive BEM with optimal convergence rates for the Helmholtz equation
• Mathematics
Computer Methods in Applied Mechanics and Engineering
• 2019
No . 20 / 2012 Convergence of adaptive 3 D BEM for weakly singular integral equations based on isotropic mesh-refinement
• Mathematics
• 2012
We consider the adaptive lowest-order boundary element method (ABEM) based on isotropic mesh-refinement for the weakly-singular integral equation for the 3D Laplacian. The proposed scheme resolves
Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic mesh‐refinement
• Mathematics, Computer Science
• 2013
It is proved that the usual adaptive mesh‐refining algorithm drives the corresponding error estimator to zero, and the sequence of discrete solutions thus tends to the exact solution within the energy norm.
Adaptive Vertex-Centered Finite Volume Methods with Convergence Rates
• Computer Science, Mathematics
SIAM J. Numer. Anal.
• 2016
This work proves that the adaptive algorithm leads to linear convergence with generically optimal algebraic rates for the error estimator and the sum of energy error plus data oscillations in the vertex-centered finite volume method.
Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations
• Mathematics
Numerische Mathematik
• 2017
An adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots leads to convergence even with optimal algebraic rates.
Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian
• Mathematics, Computer Science
Math. Comput.
• 2021
A reliable weighted residual a posteriori error estimator for the discretization of the integral fractional Laplacian based on piecewise linear functions is presented and it is proved optimal convergence rates for an $h$-adaptive algorithm driven by this error estimators.
No . 16 / 2015 Optimal convergence for adaptive IGA boundary element methods for weakly-singular intergral equations
• Mathematics, Computer Science
• 2015
An adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots leads to convergence even with optimal algebraic rates.

## References

SHOWING 1-10 OF 69 REFERENCES
Adaptive Boundary Element Methods for Some First Kind Integral Equations
• Mathematics, Computer Science
• 1996
An adaptive boundary element method for the boundary integral equations of the first kind concerning the Dirichlet problem and the Neumann problem for the Laplacian in a two-dimensional Lipschitz domain is presented and a posteriori error estimates are derived.
Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods
• Computer Science, Mathematics
Comput. Methods Appl. Math.
• 2013
Abstract. We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh-refinement is driven by the
An adaptive boundary element method for the exterior Stokes problem in three dimensions
• Mathematics, Computer Science
• 2006
An adaptive refinement strategy for the h-version of the boundary element method with weakly singular operators on surfaces with optimal lower a priori error estimates for edge singularities on uniform and graded meshes is presented.
Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic mesh‐refinement
• Mathematics, Computer Science
• 2013
It is proved that the usual adaptive mesh‐refining algorithm drives the corresponding error estimator to zero, and the sequence of discrete solutions thus tends to the exact solution within the energy norm.
Convergence of adaptive boundary element methods
• Computer Science
• 2012
In this contribution, recent mathematical results which give first positive answers for adaptive BEM are presented and discussed.
An a posteriori error estimate for a first-kind integral equation
A new a posteriori error estimate for the Galerkin boundary element method applied to an integral equation of the first kind is presented, local and sharp for quasi-uniform meshes and so improves earlier work of the authors'.
Adaptive boundary element methods with convergence rates
The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.
Two-level methods for the single layer potential in ℝ3
• Computer Science, Mathematics
Computing
• 2007
An adaptive multilevel algorithm with easily computable local error indicators which allows direction control of the local refinements is introduced and an a posteriori error estimate for the difference between the exact solution and the Galerkin solution is derived.
Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators
• Computer Science, Mathematics
Numerische Mathematik
• 2010
This paper discusses several adaptive mesh-refinement strategies based on (h − h/2)-error estimation and proves that, under the saturation assumption, these adaptive algorithms are convergent.
Convergence of Adaptive Finite Element Methods
• Computer Science
SIAM Rev.
• 2002
This work constructs a simple and efficient adaptive FEM for elliptic partial differential equations and proves that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants.