A short note on plain convergence of adaptive least-squares finite element methods

TL;DR

This paper proves that adaptive least-squares finite element methods converge under weak conditions without requiring fine initial meshes or strict marking rules, even when using iterative solvers, within a broad abstract framework.

Contribution

It establishes plain convergence of adaptive least-squares FEM without restrictive assumptions, extending previous results to more general and practical settings.

Findings

Convergence holds under weak conditions on PDE operator and mesh refinement.

No need for fine initial meshes or strict marking parameters.

Convergence remains valid with iterative solvers like preconditioned conjugate gradient.

Abstract

We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.