$hp$-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

TL;DR

This work develops a geometric multigrid solver for locally refined FEM discretizations of symmetric elliptic diffusion problems that is robust with respect to both mesh size $h$ and polynomial degree $p$. The authors design an iterative solver with a built-in algebraic error estimator $\zeta_L$, prove an $hp$-robust contraction $||u_L^{\star}-\Phi(v_L)|| \le q_{ctr}||u_L^{\star}-v_L||$ with $q_{ctr}<1$ independent of $L$ and $p$, and establish a two-sided bound $\zeta_L(v_L) \le ||u_L^{\star}-v_L|| \le C_{rel}\zeta_L(v_L)$. They integrate this solver into an AFEM framework with nested iterations and an adaptive stopping criterion, and prove optimal computational complexity for fixed $p$ under standard adaptivity assumptions. Numerical experiments on 2D problems, including L-shaped domains and jumping coefficients, confirm hp-robust contraction, optimal estimator decay $-p/2$, and favorable performance relative to direct solvers. The results yield a practically efficient, hp-robust solver suite for adaptive FEM on locally refined meshes, with provable reliability of the algebraic error estimator guiding adaptive SOLVE steps.

Abstract

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.

Figures (7)

• Figure 1: Schematic of 2D NVB refinement pattern: For each triangle $T \in \mathcal{T}$, there is one fixed refinement edge$E_T$ indicated by the extra pink line. The pink dots indicate edges that are marked for refinement. If an element is marked for refinement, at least its refinement edge is marked for refinement (top). Iterated bisection refines a marked element into 2, 3, or 4 children (bottom).
• Figure 2: Illustration of degrees of freedom ($p=2$) for the space $\mathbb{V}_{L,z}^{p}$ associated to the patch $\mathcal{T}_{L,z}$.
• Figure 3: Contraction of the algebraic solver. History plot of the contraction factors $q_{\rm ctr}$ from \ref{['equation:contraction']} for various polynomial degrees $p$ with parameter $\mu = 10^{-5}$ for the presented polynomial hierarchy from \ref{['equation:space_nestedness']} in the adaptive algorithm from Algorithm \ref{['algorithm:afem']} stopping once the final mesh consists of $10^6$ degrees of freedom (left) and the comparison with polynomial hierarchy motivated by Mir_Pap_Voh_lam_21 with localized smoothing for a fixed number of levels $L=10$ (right).
• Figure 4: Optimality of AFEM on L-shape. The convergence history plot of the discretization error estimator $\eta_{L}(u_L)$ with respect to the total computational cost (left) and the cumulative computational time (right).
• Figure 5: Optimality of the local multigrid solver. History plot of the cumulative computational time and the relative computational time per degree of freedom for the polynomial degrees $p=1$ and $p=4$. We compare the overall time with the direct solve (square) to the overall time of the AFEM algorithm with the multigrid solver (diamond). In particular, the displayed times include setup, marking, and mesh refinement.

Theorems & Definitions (27)

• Remark 1: Construction of the new iterate
• Remark 2: Computational effort and speed of convergence
• Remark 3: Nested iterations
• Theorem 4
• Corollary 5
• Remark 6
• Theorem 7
• Remark 8
• Remark 9
• proof : Proof of Theorem \ref{['theorem:optimal-cost']}