[Paper Review] Quasi-optimal nonconforming methods for symmetric elliptic problems. III -- DG and other interior penalty methods

Andreas Veeser, Pietro Zanotti|arXiv (Cornell University)|Oct 10, 2017

Advanced Numerical Methods in Computational Mathematics16 references3 citations

TL;DR

This paper introduces quasi-optimal nonconforming finite element methods for symmetric elliptic problems by modifying the right-hand side discretization via a computationally feasible $H^1_0$-smoothing operator. The new variants achieve quasi-optimality with constants uniformly bounded for shape regular meshes and tending to 1 as the penalty parameter increases, improving stability and consistency over standard interior penalty methods.

ABSTRACT

We devise new variants of the following nonconforming finite element methods: DG methods of fixed arbitrary order for the Poisson problem, the Crouzeix-Raviart interior penalty method for linear elasticity, and the quadratic $C^0$ interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to $1$ as the penalty parameter increases.

Motivation & Objective

• To address the lack of full stability and quasi-optimality in standard symmetric interior penalty (SIP) and other nonconforming methods for symmetric elliptic problems.
• To improve consistency and stability of discontinuous Galerkin and $C^0$ interior penalty methods by modifying the right-hand side discretization via a smoothing operator.
• To ensure that the resulting methods are quasi-optimal with respect to an extended energy norm, even for non-conforming discrete spaces.
• To construct a computationally feasible $H^1_0$-smoothing operator that preserves face averages and enhances consistency.
• To achieve uniformly bounded quasi-optimality constants across shape-regular meshes, with constants approaching 1 as the penalty parameter increases.

Proposed method

• Introduces a novel smoothing operator $E: S^{0}_1 \to H^1_0(\Omega)$ that maps discontinuous piecewise affine functions to conforming $H^1_0$ functions while preserving face averages.
• Modifies the discrete variational formulation by replacing $\int_\Omega f\sigma$ with $\langle f, E\sigma \rangle$ to improve consistency and stability.
• Employs a HCT-type averaging operator and a higher-order bubble function to construct the smoothing operator $E$, ensuring bounded operator norm in terms of mesh shape regularity.
• Defines a new bilinear form $b_{\mathrm{BS}}$ for the biharmonic problem that ensures coercivity and symmetry, using jumps in second normal derivatives and penalty terms.
• Applies abstract quasi-optimality theory from prior work to establish convergence bounds in an extended energy norm $|\cdot|_{1;\eta}$ or $|\cdot|_{2;\eta}$.
• Proves that the quasi-optimality constant is bounded by $\sqrt{1 + C_{\gamma_{\mathcal{M}}} \big(\alpha(\eta_* / \eta) \eta\big)^{-1}}$, which tends to 1 as $\eta \to \infty$.

Experimental results

Research questions

• RQ1 Can standard interior penalty methods for symmetric elliptic problems be made quasi-optimal by modifying the right-hand side discretization?
• RQ2 How can a smoothing operator be constructed to improve consistency and stability of nonconforming methods without sacrificing computational feasibility?
• RQ3 What is the dependence of the quasi-optimality constant on the penalty parameter and mesh shape regularity?
• RQ4 Can the proposed framework be extended to higher-order methods, such as $C^0$ interior penalty methods for the biharmonic problem?
• RQ5 Does the modified method achieve optimal convergence rates without requiring augmentation terms in the error estimate?

Key findings

• The proposed method $M_{\mathrm{C0}}$ for the biharmonic problem is $|\cdot|_{2;\eta}$-quasi-optimal with a constant bounded by $\sqrt{1 + C_{\gamma_{\mathcal{M}}} \big(\alpha(\eta_* / \eta) \eta\big)^{-1}}$.
• The quasi-optimality constant tends to 1 as the penalty parameter $\eta$ increases, indicating improved conditioning and convergence behavior.
• The smoothing operator $E_{\mathrm{C0}}$ is computationally feasible and ensures that $\left\{\!\!\left\{ \nabla E_{\mathrm{C0}}\sigma \right\}\!\!\right\} = 0$ on internal faces, preserving consistency.
• The method achieves optimal convergence rates without requiring the augmentation term $\mathrm{AG}(u-s)$, which is not bounded in general for $H^1_0$ solutions.
• The abstract theory from the first part of the series is successfully applied to prove quasi-optimality under mild assumptions on mesh regularity and penalty parameter.
• The framework extends to higher-order methods, such as $p \geq 3$ $C^0$ interior penalty methods, by constructing higher-order smoothing operators.