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

TL;DR

This paper proves that adaptive finite element methods can achieve optimal convergence rates for compactly perturbed elliptic problems, including the Helmholtz equation, without requiring initially fine meshes.

Contribution

It demonstrates that adaptivity stabilizes and overcomes preasymptotic limitations, ensuring optimal convergence without a-priori mesh assumptions.

Findings

Adaptive mesh-refinement achieves optimal algebraic rates.

Adaptivity overcomes preasymptotic behavior in elliptic problems.

Results apply to Helmholtz equation discretization.

Abstract

We prove that for compactly perturbed elliptic problems, where the corresponding bilinear form satisfies a Garding inequality, adaptive mesh-refinement is capable 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.