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23
Construction and Arithmetics of HMatrices
, 2003
"... In previous papers hierarchical matrices were introduced which are datasparse and allow an approximate matrix arithmetic of nearly optimal complexity. In this paper we analyse the complexity (storage, addition, multiplication and inversion) of the hierarchical matrix arithmetics. Two criteria, the ..."
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Cited by 56 (10 self)
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In previous papers hierarchical matrices were introduced which are datasparse and allow an approximate matrix arithmetic of nearly optimal complexity. In this paper we analyse the complexity (storage, addition, multiplication and inversion) of the hierarchical matrix arithmetics. Two criteria, the sparsity and idempotency, are sufficient to give the desired bounds. For standard finite element and boundary element applications we present a construction of the hierarchical matrix format for which we can give explicit bounds for the sparsity and idempotency.
Simple A Posteriori Error Estimators for the hVersion of the Boundary Element Method
, 2007
"... The hh/2strategy is one very basic and wellknown technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. Let φ denote the exact solution. One then considers ηH: = ‖φh − φh/2‖ to estimate the error ‖φ − φh‖, where φh is a Galerkin solution with ..."
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Cited by 17 (10 self)
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The hh/2strategy is one very basic and wellknown technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. Let φ denote the exact solution. One then considers ηH: = ‖φh − φh/2‖ to estimate the error ‖φ − φh‖, where φh is a Galerkin solution with respect to a mesh Th and φh/2 is a Galerkin solution for a mesh Th/2 obtained from uniform refinement of Th. We stress that ηH is always efficient – even with known efficiency constant Ceff = 1, i.e. ηH ≤ ‖φ − φh‖. Reliability of ηH follows immediately from the assumption ‖φ − φh/2 ‖ ≤ σ ‖φ − φh ‖ with some saturation constant σ ∈ (0, 1). Under this assumption, there holds ‖φ − φh ‖ ≤ 1√ 1 − σ2 ηH. However, for boundary element methods, the energy norm ‖ · ‖ is nonlocal and thus the error estimator ηH does not provide information for a local meshrefinement. Recent localization techniques from [1]
Praetorius: Averaging Techniques for the A Posteriori BEM Error Control for a Hypersingular Integral Equation in Two Dimensions
 SIAM J. Sci. Comput
"... Abstract. Averaging techniques or gradient recovery techniques are frequently employed tools for the a posteriori finite element error analysis. Their very recent mathematical justification for partial differential equations allows unstructured meshes and nonsmooth exact solutions. This paper establ ..."
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Cited by 14 (3 self)
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Abstract. Averaging techniques or gradient recovery techniques are frequently employed tools for the a posteriori finite element error analysis. Their very recent mathematical justification for partial differential equations allows unstructured meshes and nonsmooth exact solutions. This paper establishes an averaging technique for the hypersingular integral equation on a onedimensional boundary and presents numerical examples that show averaging techniques can be employed for an effective meshrefining algorithm. For the discussed test examples, the provided estimator estimates the (in general unknown) error very accurately in the sense that the quotient error/estimator stays bounded with a value close to 1.
Classical FEMBEM coupling methods: nonlinearities, wellposedness, and adaptivity
"... Efficiency and optimality of some weightedresidual error estimator for adaptive 2D boundary element methods ..."
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Cited by 10 (7 self)
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Efficiency and optimality of some weightedresidual error estimator for adaptive 2D boundary element methods
Inversetype estimates on hpfinite element spaces and applications
 Math. Comp
, 2008
"... Abstract. This work is concerned with the development of inversetype inequalities for piecewise polynomial functions and, in particular, functions belonging to hpfinite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite el ..."
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Cited by 9 (0 self)
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Abstract. This work is concerned with the development of inversetype inequalities for piecewise polynomial functions and, in particular, functions belonging to hpfinite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions. The inequalities are explicit both in the local polynomial degree and the local meshsize. The assumptions on the hpfinite element spaces are very weak, allowing anisotropic (shapeirregular) elements and varying polynomial degree across elements. Finally, the new inversetype inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of hpboundary element method discretisations of integral equations, with elementwise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials. 1.
Inverse estimates for elliptic boundary integral operators and their application to the adaptive coupling of FEM and BEM
, 2012
"... ar ..."
BEM with linear complexity for the classical boundary integral operators
 Institute of Mathematics, University of Zurich
, 2003
"... Abstract. Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panelclustering. These representations have t ..."
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Cited by 2 (1 self)
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Abstract. Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panelclustering. These representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations. Loworder discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered. We will present fully discrete Galerkin boundary element methods where the storage amount and the CPU time grow only linearly with respect to the number of unknowns. 1.
Optimal PanelClustering in the Presence of Anisotropic Mesh Refinement
, 2006
"... In this paper we propose and analyse a new enhanced version of the panelclustering algorithm for discrete boundary integral equations on polyhedral surfaces in 3D, which is designed to perform efficiently even when the meshes contain the highly stretched elements needed for efficient discretisation ..."
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Cited by 1 (0 self)
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In this paper we propose and analyse a new enhanced version of the panelclustering algorithm for discrete boundary integral equations on polyhedral surfaces in 3D, which is designed to perform efficiently even when the meshes contain the highly stretched elements needed for efficient discretisation when the solution contains edge singularities. The key features of our algorithm are: (i) the employment of partial analytic integration in the direction of stretching, yielding a new kernel function on a one dimensional manifold where the influence of the high aspect ratios in the stretched elements is removed and (ii) the introduction of a generalised admissibility condition with respect to the partially integrated kernel which ensures that certain stretched clusters which are inadmissible in the classical sense now become admissible. In the context of a model problem, we prove that our algorithm yields an accurate (up to discretisation error) matrixvector multiplication which requires O(N log κ N) operations, where N is the number of degrees of freedom and κ is small and independent of the aspect ratio of the elements. We also show that the classical admissibility condition leads to a suboptimal clustering algorithm for these problems. A numerical experiment shows that the theoretical estimates can be realised in practice. The generalised admissibility condition can be viewed as a simple addition to the classical method which may be useful in general when stretched meshes are present.
Mathematik Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators
"... Abstract We discuss several adaptive meshrefinement strategies based on (h − h/2)error estimation. This class of adaptive methods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, ..."
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Abstract We discuss several adaptive meshrefinement strategies based on (h − h/2)error estimation. This class of adaptive methods 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.
Timedependent Simulations of Multidimensional Quantum Waveguides Using
"... ISBN 9783902627032 c © Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors. ASC TUWIEN CONVERGENCE OF SOME ADAPTIVE FEMBEM COUPLING ..."
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ISBN 9783902627032 c © Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors. ASC TUWIEN CONVERGENCE OF SOME ADAPTIVE FEMBEM COUPLING