M. Page, D. Praetorius:

"Convergence of adaptive FEM for some elliptic obstacle problem";

in: "ASC Report 05/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

In this work, we treat the convergence of adaptive lowest-order FEM

for some elliptic obstacle problem with affine obstacle. For error estimation,

we use a residual error estimator from [Braess, Carstensen, Hoppe 2007].

We extend recent ideas from [Cascon, Kreuzer, Nochetto, Siebert 2008] for

the unrestricted variational problem to overcome the lack of Galerkin

orthogonality. The main result states that an appropriately weighted sum

of energy error, edge residuals, and data oscillations satisfies a contraction

property within each step of the adaptive feedback loop. This result is

superior to a prior result from [Braess, Carstensen, Hoppe 2007] in two

ways: First, it is unnecessary to control the decay of the data oscillations

explicitly. Second, our analysis avoids the use of a discrete local efficiency

so that the local mesh-refinement is fairly arbitrary.

http://www.asc.tuwien.ac.at/preprint/2010/asc05x2010.pdf

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