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.

翻译：我们考虑在Lax-Milgram引理框架下的一般非对称二阶线性椭圆型偏微分方程。我们提出并分析一种具有任意多项式次数的自适应有限元算法，该算法引导自适应网格细化以及所产生线性系统的非精确迭代求解。更精确地说，该迭代求解器采用外层循环——所谓的Zarantonello迭代来对称化系统，以及内层循环——一种一致收缩的代数求解器，例如最优预条件的共轭梯度法或最优几何多重网格算法。我们证明，所提出的非精确自适应迭代对称化有限元方法（AISFEM）能够实现完全线性收敛，并且对于足够小的自适应参数，能够相对于整体计算成本（即总计算时间）达到最优收敛率。数值实验验证了该理论。