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}
}
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
TLDR
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
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
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
No . 20 / 2012 Convergence of adaptive 3 D BEM for weakly singular integral equations based on isotropic mesh-refinement
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
...
...