We investigate the solution of linear systems of equations that arise when singularly perturbed reaction–diffusion partial differential equations are solved using a standard finite difference method on layer adapted grids. It is known that there are difficulties in solving such systems by direct methods when the perturbation parameter, \(\varepsilon \), is small (MacLachlan and Madden in SIAM J Sci Comput 35(5):A2225–A2254, 2013). Therefore, iterative methods are natural choices. However, we show that, in two dimensions, the condition number of the coefficient matrix grows unboundedly when \(\varepsilon \) tends to zero, and so unpreconditioned iterative schemes, such as the conjugate gradient algorithm, perform poorly with respect to \(\varepsilon \). We provide a careful analysis of diagonal and incomplete Cholesky preconditioning methods, and show that the condition number of the preconditioned linear system is independent of the perturbation parameter. We demonstrate numerically the surprising fact that these schemes are more efficient when \(\varepsilon \) is small, than when \(\varepsilon \) is \(\mathcal {O}(1)\). Furthermore, our analysis shows that when the singularly perturbed problem features no corner layers, an incomplete Cholesky preconditioner performs extremely well when \(\varepsilon \ll 1\). We provide numerical evidence that our findings extend to three-dimensional problems.

我们研究了在层自适应网格上使用标准有限差分法求解奇摄动反应扩散部分微分方程时产生的线性方程组的解。众所周知，当扰动参数\（\ varepsilon \）较小时，通过直接方法求解此类系统存在困难（MacLachlan和Madden，SIAM J Sci Comput 35（5）：A2225–A2254，2013）。因此，迭代方法是自然的选择。但是，我们表明，在两个维度上，当\（\ varepsilon \）趋于零时，系数矩阵的条件数会无限制地增长，因此未预处理的迭代方案（例如共轭梯度算法）对于\（ \ varepsilon \）。我们对对角线和不完整的Cholesky预处理方法进行了仔细的分析，并证明了预处理线性系统的条件数与扰动参数无关。我们用数字证明了令人惊讶的事实，即当\（\ varepsilon \）较小时，这些方案比\（\ varepsilon \）为\（\ mathcal {O}（1）\）时更有效。此外，我们的分析表明，当奇摄动问题不具有角层时，当\（\ varepsilon \ ll 1 \）时，不完整的Cholesky预调节器的性能将非常好。我们提供了数值证据，表明我们的发现扩展到了三维问题。