In this work, we develop algebraic solvers for linear systems arising from the discretization of second-order elliptic problems by saddle-point mixed finite element methods of arbitrary polynomial degree $p \ge 0$. We present a multigrid and a two-level domain decomposition approach in two or three space dimensions, which are steered by their respective a posteriori estimators of the algebraic error. First, we extend the results of [A. Mira\c{c}i, J. Pape\v{z}, and M. Vohral\'ik, SIAM J. Sci. Comput. 43 (2021), S117--S145] to the mixed finite element setting. Extending the multigrid procedure itself is rather natural. To obtain analogous theoretical results, however, a multilevel stable decomposition of the velocity space is needed. In two space dimensions, we can treat the velocity space as the curl of a stream-function space, for which the previous results apply. In three space dimensions, we design a novel stable decomposition by combining a one-level high-order local stable decomposition of [Chaumont-Frelet and Vohral\'ik, SIAM J. Numer. Anal. 61 (2023), 1783--1818] and a multilevel lowest-order stable decomposition of [Hiptmair, Wu, and Zheng, Numer. Math. Theory Methods Appl. 5 (2012), 297--332]. This allows us to prove that our multigrid solver contracts the algebraic error at each iteration and, simultaneously, that the associated a posteriori estimator is efficient. A $p$-robust contraction is shown in two space dimensions. Next, we use this multilevel methodology to define a two-level domain decomposition method where the subdomains consist of overlapping patches of coarse-level elements sharing a common coarse-level vertex. We again establish a ($p$-robust) contraction of the solver and efficiency of the a posteriori estimator. Numerical results presented both for the multigrid approach and the domain decomposition method confirm the theoretical findings.

翻译：本文针对任意多项式次数$p \ge 0$的鞍点混合有限元方法离散二阶椭圆问题所产生的线性系统，发展了代数求解器。我们在二维或三维空间中提出了一种多重网格方法和一种两层区域分解方法，这两种方法均由其相应的代数误差后验估计子引导。首先，我们将[A. Mira\c{c}i, J. Pape\v{z}, 和 M. Vohral\'ik, SIAM J. Sci. Comput. 43 (2021), S117--S145]的结果推广到混合有限元框架。扩展多重网格过程本身是相当自然的。然而，为了获得类似的理论结果，需要速度空间的多层稳定分解。在二维空间中，我们可以将速度空间视为流函数空间的旋度，从而应用先前的结果。在三维空间中，我们通过结合[Chaumont-Frelet and Vohral\'ik, SIAM J. Numer. Anal. 61 (2023), 1783--1818]的单层高阶局部稳定分解与[Hiptmair, Wu, and Zheng, Numer. Math. Theory Methods Appl. 5 (2012), 297--332]的多层最低阶稳定分解，设计了一种新颖的稳定分解。这使得我们能够证明我们的多重网格求解器在每次迭代中压缩代数误差，同时证明相关的后验估计子是高效的。在二维空间中证明了$p$-鲁棒的压缩性。接着，我们利用这种多层方法定义了一种两层区域分解方法，其中子区域由共享一个公共粗网格顶点的、重叠的粗层单元块构成。我们再次确立了求解器的（$p$-鲁棒）压缩性以及后验估计子的高效性。针对多重网格方法和区域分解方法所呈现的数值结果均验证了理论结论。