Adaptive FEM with optimal convergence rates for a certain class of non-symmetric and possibly non-linear problems

TL;DR

This paper proves that adaptive finite element methods can achieve optimal convergence rates for certain non-linear and non-symmetric second-order PDEs, broadening the scope beyond previous linear cases.

Contribution

It introduces a convergence analysis for adaptive FEM applied to non-linear, non-symmetric problems using residual error estimators, without relying on the interior node property.

Findings

Achieves optimal algebraic convergence rates.

Covers general linear second-order elliptic operators.

Does not require interior node property for refinement.

Abstract

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear non-symmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\r{a}rding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.