2022/12/01 by Maximilian Brunner, Pascal Heid, Michael Innerberger +3 · 1 voice · 9 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Finite element method #Mathematical analysis #Mathematics #Numerical methods in engineering #Structural engineering #math.NA
paper · pdf · doi:10.1093/imanum/drad039
published in IMA Journal of Numerical Analysis 44(3), 1560-1596 (Oxford University Press)
arxiv published 2022/12/01 · openalex publication_date 2023/06/19 · openalex created_date 2023/06/22 · arxiv updated 2023/11/21 · openalex updated_date 2026/06/17
Abstract We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.