Two-level overlapping Schwarz preconditioners with universal coarse spaces for $2m$th-order elliptic problems

TL;DR

The paper develops a universal two-level overlapping Schwarz preconditioner for $2m$th-order elliptic problems, with coarse spaces built on a conforming coarse mesh that are independent of the order $m$ and discretization. A quasi-interpolation operator $J_H$ and a stable decomposition framework yield a condition-number bound $\kappa(M^{-1}A) \lesssim 1 + (H/\delta)^{2m-1}$, ensuring scalability when $H/\delta$ is fixed. The coarse-space construction relies on partition-of-unity functions $\phi_i$ supported on local regions and a coarse space $V_0 = (\sum_i \phi_i \mathbb{P}_{m-1}(\omega_i)) \cap H_0^m(\Omega)$, which is universal across discretizations. Numerical experiments across BFS, Adini, C0IP, and Jin–Wu elements confirm improved convergence rates and discretization-independent scalability, highlighting the method’s practical impact for high-order elliptic problems and suggesting directions for future extensions to heterogeneities and nonlinear settings.

Abstract

We propose a novel universal construction of two-level overlapping Schwarz preconditioners for $2m$th-order elliptic boundary value problems, where $m$ is a positive integer. The word "universal" here signifies that the coarse space construction can be applied to any finite element discretization for any $m$ that satisfies some common assumptions. We present numerical results for conforming, nonconforming, and discontinuous Galerkin-type finite element discretizations for high-order problems to demonstrate the scalability of the proposed two-level overlapping Schwarz preconditioners.

Figures (7)

• Figure 1: Construction of a collection $\{ \phi_i \}_{i \in \mathcal{I}_H}$ satisfying \ref{['coarse_basis']} by using the sixth-degree Argyris finite element space AFS:1968 on a triangular coarse grid $\mathcal{T}_H$ ($d = 2$, $m = 2$).
• Figure 1: Reference elements of finite element discretizations used in \ref{['Sec:Applications']}: (a) Bogner--Fox--Schmit BFS:1965Valdman:2020, (b) Adini HS:2013LL:1975, (c)$C^0$ interior penalty BW:2005EGHLMT:2002, and (d) Adini-type sixth-order nonconforming elements proposed by Jin and Wu JW:2023.
• Figure 2: Construction of a collection $\{ \phi_i \}_{i \in \mathcal{I}_H}$ satisfying \ref{['coarse_basis']} by using the $C^2$-$Q_3$ finite element space HZ:2015 on a rectangular coarse grid $\mathcal{T}_H$ ($d = 2$, $m = 3$).
• Figure 2: Relative residual error \ref{['error']} of the conjugate gradient method preconditioned by the proposed two-level additive Schwarz preconditioner for solving the fourth-order Bogner--Fox--Schmit finite element discretization. (a) Comparison with the cases of no preconditioner and the one-level preconditioner ($h = 2^{-7}$, $H = 2^{-3}$, $\delta = 2^{-5}$). (b, c) Various mesh sizes $h$ ($H/h = 2^4$).
• Figure 3: Relative residual error \ref{['error']} of the conjugate gradient method preconditioned by the proposed two-level additive Schwarz preconditioner for solving the fourth-order Adini finite element discretization. (a) Comparison with the cases of no preconditioner and the one-level preconditioner ($h = 2^{-7}$, $H = 2^{-3}$, $\delta = 2^{-5}$). (b, c) Various mesh sizes $h$ ($H/h = 2^4$).

Theorems & Definitions (21)

• Remark 2.2
• Remark 2.3
• Lemma 3.1
• Remark 3.2
• Lemma 4.2
• Proof 1
• Remark 4.3
• Remark 4.4
• Lemma 5.1
• Proof 2