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A. Bespalov, A. Haberl, D. Praetorius:
"Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems";
in: "ASC Report 14/2016", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, 1 - 26.

We prove that for compactly perturbed elliptic problems, where the corre-sponding bilinear form satisfies a Gårding inequality, adaptive mesh-refinement is capa-ble of overcoming the preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. As an important consequence of our analysis, one does not have to deal with the a priori assumption that the underlying meshes are sufficiently fine. Hence, the overall conclusion of our results is that adaptivity has stabilizing effects and can overcome possibly pessimistic restrictions on the meshes. In particular, our analysis covers adaptive mesh-refinement for the finite element discretization of the Helmholtz equation from where our interest originated.

adaptive mesh-refinement, optimal convergence rates, a posteriori error esti-mate, Helmholtz equation.

http://www.asc.tuwien.ac.at/preprint/2016/asc14x2016.pdf