## Convergence of Adaptive Finite Element Methods

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##### Дата

2005-12-05##### Автор

Mekchay, Khamron

##### Advisor

Nochetto, Ricardo H.

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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
$C^1$ graphs in $R^d ~(d\ge2)$. We first derive a posteriori error
estimates that account for both the energy error in $H^1$ and the
geometric error in $W^1_\infty$ 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.

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