In this paper, we discuss a new adaptive approach for iterative solution of sparse linear systems arising from partial differential equations (PDEs) with self-adjoint operators. The idea is to use the a posteriori estimated local distribution of the algebraic error in order to steer and guide the solve process in such way that the algebraic error is reduced more efficiently in the consecutive iterations. We first explain the motivation behind the proposed procedure and show that it can be equivalently formulated as constructing a special combination of preconditioner and initial guess for the original system. We present several numerical experiments in order to identify when the adaptive procedure can be of practical use.

在本文中，我们讨论了一种新的自适应方法，该方法适用于带有自伴随算子的偏微分方程（PDE）引起的稀疏线性系统的迭代。想法是使用代数误差的后验估计局部分布，以便以这样的方式操纵和引导求解过程，使得在连续迭代中更有效地减小代数误差。我们首先解释提出的程序背后的动机，并表明它可以等效地公式化为构造原始系统的预处理器和初始猜测的特殊组合。为了确定何时可以实际使用自适应程序，我们提出了一些数值实验。