Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.

中文翻译：

---

完全操作员预调节和求解线性系统的准确性


除非有特殊条件，否则尝试使用标准数值方法求解条件不良的线性方程组会导致无法控制的高数值误差，并且迭代求解器的收敛速度通常会很慢。在许多情况下，这种系统是由具有大量离散变量的算子方程的离散化产生的，并且病态是通过预处理来解决的。本文的一个关键观察是有时被忽视的事实，即虽然传统的预处理有效地加速了迭代方法的收敛，但它通常不会提高解的准确性。尽管如此，有时还是可以克服这个障碍：如果在离散化之前转换方程，则可以显著提高精度，这个过程我们称之为完全算子预整 （FOP）。我们强调这一原理已经用于各个领域，包括第二类积分方程和 Olver-Townsend 谱法。我们制定了一个充分的条件，在该条件下，FOP 可以获得高精度。我们用一个四阶微分方程来说明这一点，该方程使用有限元进行离散化。