95,489 research outputs found
Convergence of Adaptive Finite Element Methods
Adaptive finite element methods (FEMs) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Nochetto, Ricardo Horacio. University of Maryland; Estados UnidosFil: Siebert, Kunibert G.. Universität Heidelberg
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
Convergence and Optimality of Adaptive Mixed Finite Element Methods
The convergence and optimality of adaptive mixed finite element methods for
the Poisson equation are established in this paper. The main difficulty for
mixed finite element methods is the lack of minimization principle and thus the
failure of orthogonality. A quasi-orthogonality property is proved using the
fact that the error is orthogonal to the divergence free subspace, while the
part of the error that is not divergence free can be bounded by the data
oscillation using a discrete stability result. This discrete stability result
is also used to get a localized discrete upper bound which is crucial for the
proof of the optimality of the adaptive approximation
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
Convergence of Adaptive Finite Element Methods
We develop adaptive finite element methods (AFEMs) for elliptic
problems, and prove their convergence, based on ideas introduced
by D\"{o}rfler \cite{Dw96}, and Morin, Nochetto, and Siebert
\cite{MNS00, MNS02}. We first study an AFEM for general second
order linear elliptic PDEs, thereby extending the results of Morin
et al \cite{MNS00,MNS02} that are valid for the Laplace operator.
The proof of convergence relies on quasi-orthogonality, which
accounts for the bilinear form not being a scalar product,
together with novel error and oscillation reduction estimates,
which now do not decouple. We show that AFEM is a contraction for
the sum of energy error plus oscillation. Numerical experiments,
including oscillatory coefficients and {both coercive and
non-coercive} convection-diffusion PDEs, illustrate the theory and
yield optimal meshes. The role of oscillation control is now more
crucial than in \cite{MNS00,MNS02} and is discussed and documented
in the experiments.
We next introduce an AFEM for the Laplace-Beltrami operator on
graphs in . We first derive a posteriori error
estimates that account for both the energy error in and the
geometric error in due to approximation of the
surface by a polyhedral one. We devise a marking strategy to
reduce the energy and geometric errors as well as the geometric
oscillation. We prove that AFEM is a contraction on a suitably
scaled sum of these three quantities as soon as the geometric
oscillation has been reduced beyond a threshold. The resulting
AFEM converges without knowing such threshold or any constants,
and starting from any coarse initial triangulation. Several
numerical experiments illustrate the theory.
Finally, we introduce and analyze an AFEM for the Laplace-Beltrami
operator on parametric surfaces, thereby extending the results for
graphs. Note that, due to the nature of parametric surfaces, the
geometric oscillation is now measured in terms of the differences
of tangential gradients rather than differences of normals as for
graphs. Numerical experiments with closed surfaces are provided to
illustrate the theory
Convergence and Optimality of Adaptive Mixed Methods on Surfaces
In a 1988 article, Dziuk introduced a nodal finite element method for the
Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear
triangulation, initiating a line of research into surface finite element
methods (SFEM). Demlow and Dziuk built on the original results, introducing an
adaptive method for problems on 2-surfaces, and Demlow later extended the a
priori theory to 3-surfaces and higher order elements. In a separate line of
research, the Finite Element Exterior Calculus (FEEC) framework has been
developed over the last decade by Arnold, Falk and Winther and others as a way
to exploit the observation that mixed variational problems can be posed on a
Hilbert complex, and Galerkin-type mixed methods can be obtained by solving
finite dimensional subproblems. In 2011, Holst and Stern merged these two lines
of research by developing a framework for variational crimes in abstract
Hilbert complexes, allowing for application of the FEEC framework to problems
that violate the subcomplex assumption of Arnold, Falk and Winther. When
applied to Euclidean hypersurfaces, this new framework recovers the original a
priori results and extends the theory to problems posed on surfaces of
arbitrary dimensions. In yet another seemingly distinct line of research,
Holst, Mihalik and Szypowski developed a convergence theory for a specific
class of adaptive problems in the FEEC framework. Here, we bring these ideas
together, showing convergence and optimality of an adaptive finite element
method for the mixed formulation of the Hodge Laplacian on hypersurfaces.Comment: 22 pages, no figures. arXiv admin note: substantial text overlap with
arXiv:1306.188
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