RATE OPTIMALITY OF ADAPTIVE FINITE ELEMENT METHODS WITH RESPECT TO OVERALL COMPUTATIONAL COSTS

We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the adaptive mesh-refinement as well as the inexact solution of the arising discrete systems. We prove that the proposed strategy leads to linear convergence with optimal algebraic rates. Unlike prior works, however, we focus on convergence rates with respect to the overall computational costs. In explicit terms, the proposed adaptive strategy thus guarantees quasi-optimal computational time. In particular, our analysis covers linear problems, where the linear systems are solved by an optimally preconditioned CG method as well as nonlinear problems with strongly monotone nonlinearity which are linearized by the so-called Zarantonello iteration.

Automated summary

The paper presents an adaptive finite element method for second-order elliptic PDEs, incorporating an adaptive algorithm that monitors and guides mesh refinement and approximate solutions of discrete systems. It is shown to have linear convergence with optimal algebraic rates, focusing on convergence rates with respect to overall computational costs. Unlike prior works, the proposed adaptive strategy guarantees quasi-optimal computational time, covering both linear problems solved with optimally preconditioned CG method and nonlinear problems linearized by Zarantonello iteration for strongly monotone nonlinearity.