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。我们提出并分析了一种具有任意多项式次数的自适应有限元算法，该算法可引导自适应网格细化及所生成线性系统的非精确迭代求解。具体而言，迭代求解器采用Zarantonello迭代作为外循环以对称化系统，并采用一致压缩代数求解器作为内循环（例如最优预条件共轭梯度法或最优几何多重网格算法）。我们证明了所提出的非精确自适应迭代对称化有限元方法（AISFEM）具有完全线性收敛性，并且当自适应参数足够小时，该方法在总计算代价（即总计算时间）意义上达到最优收敛率。数值实验验证了理论结果。