Adaptive FEM for linear elliptic PDEs: Optimal complexity

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.