2020/03/24 by Gregor Gantner, Alexander Haberl, Dirk Praetorius +1 · 31 citations
Computer Science · Engineering · Mathematics · #Adaptive mesh refinement #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Computer science #Convergence (economics) #Electromagnetic Simulation and Numerical Methods #Finite element method #Iterative method #Key (lock) #Linear system #Mathematical analysis #Mathematical optimization #Mathematics #Monotone polygon #Nonlinear system #Rate of convergence #cs.NA #math.NA #msc:41A25 #msc:65N22 #msc:65N30 #msc:65N50 #msc:65Y20
paper · pdf · doi:10.1090/mcom/3654
published in Mathematics of Computation 90(331), 2011-2040 (American Mathematical Society)
arxiv created 2020/03/24 · openalex publication_date 2021/04/21 · arxiv updated 2021/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.