Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

Abstract

We consider a general nonsymmetric second-order linear elliptic PDE 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 mesh-refinement 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, e.g., an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method (AISFEM) 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.

References in corpus (1)

• MooAFEM: An object oriented Matlab code for higher-order adaptive FEM for (nonlinear) elliptic PDEs

Cited by in corpus (4)

• $hp$-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs
• Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEs
• On full linear convergence and optimal complexity of adaptive FEM with inexact solver
• Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs