In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree $p \ge 1$ and the (local) mesh size $h$. We further prove that the built-in algebraic error estimator which comes with the solver is $hp$-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

翻译：本文提出并分析了一种几何多重网格方法，用于求解对称二阶线性椭圆扩散问题有限元离散所生成的代数系统。我们证明该迭代求解器能够对代数误差实现关于多项式次数 $p \ge 1$ 和（局部）网格尺寸 $h$ 的鲁棒压缩。进一步证明了求解器内置的代数误差估计器与代数误差之间满足 $hp$-鲁棒等价性。概述了该求解器在具有准最优计算成本的自适应有限元方法框架中的应用。数值实验验证了理论结果。