215 research outputs found
Adaptive BEM with optimal convergence rates for the Helmholtz equation
We analyze an adaptive boundary element method for the weakly-singular and
hypersingular integral equations for the 2D and 3D Helmholtz problem. The
proposed adaptive algorithm is steered by a residual error estimator and does
not rely on any a priori information that the underlying meshes are
sufficiently fine. We prove convergence of the error estimator with optimal
algebraic rates, independently of the (coarse) initial mesh. As a technical
contribution, we prove certain local inverse-type estimates for the boundary
integral operators associated with the Helmholtz equation
Inverse estimates for elliptic boundary integral operators and their application to the adaptive coupling of FEM and BEM
We prove inverse-type estimates for the four classical boundary integral
operators associated with the Laplace operator. These estimates are used to
show convergence of an h-adaptive algorithm for the coupling of a finite
element method with a boundary element method which is driven by a weighted
residual error estimator
ZZ-type aposteriori error estimators for adaptive boundary element methods on a curve
In the context of the adaptive finite element method (FEM), ZZ-error
estimators named after Zienkiewicz and Zhu are mathematically well-established
and widely used in practice. In this work, we propose and analyze ZZ-type error
estimators for the adaptive boundary element method (BEM). We consider
weakly-singular and hyper-singular integral equations and prove, in particular,
convergence of the related adaptive mesh-refining algorithms
Adaptive boundary element methods with convergence rates
This paper presents adaptive boundary element methods for positive, negative,
as well as zero order operator equations, together with proofs that they
converge at certain rates. The convergence rates are quasi-optimal in a certain
sense under mild assumptions that are analogous to what is typically assumed in
the theory of adaptive finite element methods. In particular, no
saturation-type assumption is used. 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.Comment: 48 pages. A journal version. The previous version (v3) is a bit
lengthie
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FE/BE coupling for an acoustic fluid-structure interaction problem. Residual a posteriori error estimates
This is the author's accepted manuscript. The final published article is available from the link below. Copyright © 2011 John Wiley & Sons, Ltd.In this paper, we developed an a posteriori error analysis of a coupling of finite elements and boundary elements for a fluid–structure interaction problem in two and three dimensions. This problem is governed by the acoustic and the elastodynamic equations in time-harmonic vibration. Our methods combined integral equations for the exterior fluid and FEMs for the elastic structure. It is well-known that because of the reduction of the boundary value problem to boundary integral equations, the solution is not unique in general. However, because of superposition of various potentials, we consider a boundary integral equation that is uniquely solvable and avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures were considered; one is based on the nonsymmetric formulation and the other is based on a symmetric formulation. For both formulations, we derived reliable residual a posteriori error estimates. From the estimators we computed local error indicators that allowed us to develop an adaptive mesh refinement strategy. For the two-dimensional case we performed an adaptive algorithm on triangles, and for the three-dimensional case we used hanging nodes on hexahedrons. Numerical experiments underline our theoretical results.DFG German Research Foundatio
Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators
We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general
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