Abstract

A fundamentally new—nonsaturable—method is constructed for the numerical solution of elliptic boundary value problems for the Laplace equation in \({{C}^{\infty }}\)-smooth axisymmetric domains of fairly arbitrary shape. A distinctive feature of the method is that it has a zero leading error term. As a result, the method is automatically adjusted to any redundant (extraordinary) smoothness of the solutions to be found. The method enriches practice with a new computational tool capable of inheriting, in discretized form, both differential and spectral characteristics of the operator of the problem under study. This underlies the construction of a numerical solution of guaranteed quality (accuracy) if the elliptic problem under study has a sufficiently smooth (e.g., \({{C}^{\infty }}\)-smooth) solution. The result obtained is of fundamental importance, since, in the case of \({{C}^{\infty }}\)-smooth solutions, the solution is constructed with an absolutely sharp exponential error estimate. The sharpness of the estimate is caused by the fact that the Aleksandrov \(m\)-width of the compact set of \({{C}^{\infty }}\)-smooth functions, which contains the exact solution of the problem, is asymptotically represented in the form of an exponential function decaying to zero (with growing integer parameter \(m).\)

中文翻译：

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光滑轴对称域中拉普拉斯方程数值解的超收敛算法

摘要

针对完全任意形状的\（{{C} ^ {\ infty}} \）-光滑轴对称域中的Laplace方程，它为椭圆边界值问题的数值解构造了一种根本上不饱和的新方法。该方法的一个显着特征是其前导误差项为零。结果，该方法会自动调整为要找到的解决方案的任何冗余（非凡）平滑度。该方法以一种新的计算工具丰富了实践，该工具能够以离散形式继承所研究问题的算子的差分和频谱特性。如果所研究的椭圆问题具有足够的光滑度（例如，\（{{C} ^ {\ infty}} \）-平滑）解决方案。获得的结果具有根本的重要性，因为在\（{{C} ^ {\ infty}} \）-光滑解的情况下，该解是用绝对尖锐的指数误差估计构造的。估算的尖锐性是由于以下事实引起的：Aleksandrov \（m \） -紧缩函数\（{{C} ^ {\ infty}} \）-）光滑函数的宽度，其中包含问题，以渐减为零的指数函数形式（随着整数参数\（m）。\）渐近表示。