Abstract
This paper provides a refined a posteriori error control for the obstacle problem with an affine obstacle which allows for a proof of optimal complexity of an adaptive algorithm. This is the first adaptive mesh-refining finite element method known to be of optimal complexity for some variational inequality. The result holds for first-order conforming finite element methods in any spacial dimension based on shape-regular triangulation into simplices for an affine obstacle. The key contribution is the discrete reliability of the a posteriori error estimator from [Numer. Math. 107 (2007), 455–471] in an edge-oriented modification which circumvents the difficulties caused by the non-existence of a positive second-order approximation [Math. Comp. 71 (2002), 1405–1419].
Funding source: DFG Research Center MATHEON
Funding source: WCU program through KOSEF
Award Identifier / Grant number: R31-2008-000-10049-0
Funding source: NSFC
Award Identifier / Grant number: 11271035, 91430213, 11421101
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