  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.
