Goal-oriented adaptive finite element methods with optimal computational complexity

TL;DR

This paper develops a goal-oriented adaptive finite element method that achieves optimal convergence rates and computational complexity for solving linear elliptic PDEs with goal functionals, integrating mesh refinement and iterative solvers.

Contribution

It introduces a novel adaptive algorithm that guarantees optimal complexity and convergence rates, considering both mesh refinement and iterative solver efficiency.

Findings

Proves linear convergence of the adaptive method.

Establishes optimal algebraic convergence rates.

Demonstrates optimal computational complexity.

Abstract

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.