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，用于迭代求解由对称二阶线性椭圆扩散问题的有限元离散所产生的离散系统。我们证明了该迭代求解器对多项式次数 $p \ge 1$ 和（局部）网格尺寸 $h$ 具有鲁棒性，能有效压缩代数误差。进一步，我们证明了该求解器内置的代数误差估计子与代数误差具有 $hp$-鲁棒等价性。文中概述了该方法在自适应有限元框架中的应用，并指出其计算成本接近最优。数值实验验证了理论结果。