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We prove that for compactly perturbed elliptic problems, where thecorresponding bilinear form satisfies a Garding inequality, adaptivemesh-refinement is capable of overcoming the preasymptotic behavior andeventually leads to convergence with optimal algebraic rates. As an importantconsequence of our analysis, one does not have to deal with the a-prioriassumption that the underlying meshes are sufficiently fine. Hence, the overallconclusion of our results is that adaptivity has stabilizing effects and canovercome possibly pessimistic restrictions on the meshes. In particular, ouranalysis covers adaptive mesh-refinement for the finite element discretizationof the Helmholtz equation from where our interest originated.

Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems