Rate optimality of adaptive finite element methods with respect to the overall computational costs

Abstract

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.

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• Goal-oriented adaptive finite element methods with optimal computational complexity
• Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs
• A link between the steepest descent method and fixed-point iterations
• $hp$-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs
• On full linear convergence and optimal complexity of adaptive FEM with inexact solver
• Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEs
• Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs
• Plain convergence of goal-oriented adaptive FEM
• Optimal multilevel adaptive FEM for the Argyris element