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：本文在Lax-Milgram引理框架下，研究一般非对称二阶线性椭圆型偏微分方程。我们构建并分析了一种具有任意多项式次数的自适应有限元算法，该算法能够引导自适应网格细化及求解线性方程组时的不精确迭代过程。具体而言，该迭代求解器采用外循环（即Zarantonello迭代）来对称化系统，并以内循环形式融入一致收缩代数求解器（例如最优预条件共轭梯度法或最优几何多重网格算法）。我们证明，所提出的不精确自适应迭代对称化有限元方法（AISFEM）可实现完全线性收敛，且当自适应参数足够小时，该方法能就整体计算成本（即总计算时间）达到最优收敛速率。数值实验验证了理论分析结果。