46,964 research outputs found
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
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
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
Global boundary conditions for a Dirac operator on the solid torus
We study a Dirac operator subject to Atiayh-Patodi-Singer like boundary
conditions on the solid torus and show that the corresponding boundary value
problem is elliptic, in the sense that the Dirac operator has a compact
parametrix
A posteriori error estimates for the Electric Field Integral Equation on polyhedra
We present a residual-based a posteriori error estimate for the Electric
Field Integral Equation (EFIE) on a bounded polyhedron. The EFIE is a
variational equation formulated in a negative order Sobolev space on the
surface of the polyhedron. We express the estimate in terms of
square-integrable and thus computable quantities and derive global lower and
upper bounds (up to oscillation terms).Comment: Submitted to Mathematics of Computatio
Local convergence of the FEM for the integral fractional Laplacian
We provide for first order discretizations of the integral fractional
Laplacian sharp local error estimates on proper subdomains in both the local
-norm and the localized energy norm. Our estimates have the form of a
local best approximation error plus a global error measured in a weaker norm
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