SIAM Journal on Scientific Computing, Ahead of Print.
We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree $p \ge 1$. Our solver employs zero pre- and one postsmoothing by the overlapping Schwarz (block-Jacobi) method and features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and levelwise and patchwise error reductions. We show the following two results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree $p$; and the estimator represents a two-sided $p$-robust bound on the algebraic error. The $p$-robustness results are obtained by carefully applying the results of [J. Schöberl et al., IMA J. Numer. Anal., 28 (2008), pp. 1--24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of [J. Xu, L. Chen, and R. H. Nochetto, Optimal multilevel methods for $H({grad})$, $H({curl})$, and $H({div})$ systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 599--659]. We consider quasi-uniform or graded bisection simplicial meshes and prove at most linear dependence on the number of mesh levels for minimal $H^1$-regularity and complete independence for $H^2$-regularity. We also present a simple and effective way for the solver to adaptively choose the number of postsmoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm $p$-robustness and show the benefits of the adaptive number of smoothing steps.

中文翻译：

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具有最佳步长和自适应步长数的Posteroriori Steered $ p $ -Robust多重网格

《 SIAM科学计算杂志》，预印本。
我们开发了由代数误差的后验估计器控制的多网格求解器。我们采用二阶椭圆扩散问题的上下文，该问题通过使任意多项式度数$ p \ ge 1 $的有限元一致来离散化。我们的求解器通过重叠的Schwarz（block-Jacobi）方法采用零个预平滑和一个后平滑，并通过行搜索在每个级别的平滑校正中对步长进行了最佳选择。在当前误差以及逐级误差和逐级误差减少方面，这导致下一步的代数误差的简单毕达哥拉斯公式。我们显示以下两个结果及其等价关系：求解器将代数误差与多项式度$ p $无关地收缩；估计器代表代数误差的两边$ p $ -robust界。$ p $ -robustness结果是通过仔细应用[J. Schöberl等人，IMA J. Numer。Anal。，28（2008），pp。1--24]结合一个网格的分段仿射多项式的多级稳定分解。Xu，L. Chen和RH Nochetto，在多尺度上针对梯度网格和非结构网格上的$ H（{grad}）$，$ H（{curl}）$和$ H（{div}）$系统的最优多级方法，非线性和自适应近似，施普林格，柏林，2009年，第599--659页]。我们考虑准均匀或渐变的二等分单纯形网格，并证明对于最小的$ H ^ 1 $-规则性和至于完全的独立性（$ H ^ 2 $-规则性），至多线性依赖于网格级别的数量。我们还为求解器提供了一种简单有效的方法，可以自适应地选择每个单独级别上所需的后期平滑步骤的数量，产生了改进的错误减少率。数值测试证实了$ p $的稳健性，并显示了自适应数量的平滑步骤的好处。