Error estimates for completely discrete FEM in energy-type and weaker norms

Technical Summary of "Error estimates for completely discrete FEM in energy-type and weaker norms"

Problem Statement
The paper addresses the numerical analysis of Finite Element Methods (FEM) for boundary value problems involving linear second-order elliptic differential equations with a diffusion-convection-reaction structure. The problems are defined on bounded polyhedral Lipschitz domains in $\mathbb{R}^d$ ($d \in \{2, 3\}$) and feature mixed boundary conditions of the first, second, and third kinds. A central focus is on "completely discrete" formulations, where the discrete finite element spaces are not necessarily conforming (i.e., not embedded in the continuous solution space of the weak formulation) and the discrete problems do not necessarily satisfy standard consistency properties. The authors aim to provide a unified abstract framework for deriving error estimates in both energy-type norms and weaker norms (such as $L^2$) without assuming conformity or consistency a priori.

Methodology
The authors develop a general variational framework based on two real Banach spaces, $U$ and $V$, and their discrete counterparts $U_h$ and $V_h$. The methodology proceeds through the following steps:

1. Abstract Error Analysis:

   • Stability and Consistency: The paper generalizes Strang's second lemma to handle nonconforming and nonconsistent discretizations. It defines consistency on a subspace $W \subset U$ by requiring that the discrete bilinear form $a_h$ can be extended such that the continuous solution $u$ satisfies the discrete equation exactly (up to the discrete linear form $\ell_h$).
   • Energy Norm Estimates: Theorem 3 provides an error estimate in the discrete energy norm $\|u - u_h\|_{U_h}$. This estimate depends on the approximation error (best approximation in the discrete space), the stability constant (inf-sup condition), and a consistency error term measuring the deviation of the continuous solution from the discrete formulation.
   • Weaker Norm Estimates (Generalized Aubin-Nitsche): The paper extends the classical Aubin-Nitsche duality argument to the nonconforming/nonconsistent setting. Theorem 5 derives an error estimate in a weaker norm $\|u - u_h\|_{Z'}$ (where $Z$ is a reflexive Banach space). This bound decomposes the error into four terms involving:
     • The product of primal and dual approximation errors.
     • Consistency errors of the primal and dual problems.
     • Approximation errors arising from the difference between continuous and discrete bilinear/linear forms.
   • The analysis relies on the solvability and regularity of an adjoint problem and its discrete counterpart.

2. Application to Specific Discretizations:
   The abstract theory is applied to two concrete stabilized discretizations for the diffusion-convection-reaction equation:

   • Example I: Crouzeix-Raviart (CR1) Discretization: A nonconforming method using piecewise linear polynomials that are continuous at element face barycenters but discontinuous across faces. The method includes upwind stabilization terms for the convection part. The authors prove coercivity and derive optimal error estimates in the $H^1$-like norm and $L^2$-norm.
   • Example II: Interior Penalty Discontinuous Galerkin (IPG) Methods: A family of methods (including SIPG, NIPG, and IIPG) using discontinuous piecewise polynomials of degree $k$. The formulation includes penalty terms for jumps in the solution and fluxes, as well as upwind stabilization. The authors establish stability (coercivity) under specific conditions on the penalty parameter and mesh regularity, and prove convergence rates.

Key Contributions and Results

• Unified Framework: The paper presents a unified approach that does not require the discrete spaces to be subspaces of the continuous spaces (nonconforming) nor the discrete formulation to be consistent with the continuous one. This allows for the analysis of a broader class of stabilized methods.
• Generalized Aubin-Nitsche Technique: A significant contribution is the extension of the duality argument to completely discrete settings. Theorem 5 provides a rigorous bound for errors in weaker norms, explicitly accounting for consistency errors in both the primal and adjoint problems.
• Optimal Error Estimates:
  • For the Crouzeix-Raviart method, the paper proves optimal first-order convergence in the energy norm ($O(h)$) and second-order convergence in the $L^2$-norm ($O(h^2)$), assuming sufficient regularity of the solution and coefficients.
  • For IPG methods, optimal convergence rates are established. In the energy norm, the error is $O(h^m)$ for polynomial degree $k$ (with $m \le k$). In weaker norms, the paper demonstrates that for the Symmetric IPG (SIPG, $\theta = -1$), the adjoint problem remains consistent, allowing for optimal error estimates in negative norms ($O(h^{k+m+1})$). For non-symmetric variants ($\theta \neq -1$), the adjoint problem is shown to be nonconsistent, requiring additional estimation of the consistency error term.
• Stability Analysis: The authors provide detailed proofs of coercivity for the discrete bilinear forms, identifying necessary conditions on the penalty parameters and mesh geometry (shape and contact regularity) to ensure stability.

Significance and Claims
The authors state that while certain aspects of the exposition (such as their version of Strang's second lemma and variants of the duality argument) are not entirely new, the value of the work lies in the unified abstract framework. This framework clarifies and quantifies the interplay between stability, approximation errors, and consistency errors without relying on the restrictive assumptions of conformity or consistency.

The paper claims that this approach:

1. Allows for the analysis of more general finite element methods, including those that are inherently nonconforming or nonconsistent.
2. Provides a systematic way to derive optimal error estimates in weaker norms by generalizing the Aubin-Nitsche technique to these broader settings.
3. Offers guidelines for setting up and analyzing new stable and convergent schemes for diffusion-convection-reaction problems with mixed boundary conditions.

The authors modestly note that the theoretical results are applied to two specific examples (CR1 and IPG) to demonstrate the utility of the framework, but the abstract results are intended to be applicable to a wider range of models and numerical techniques. The work builds upon and extends previous literature (e.g., [DPD18], [DPD21], [DPE12]) by avoiding the inclusion of interpolation operators in the definition of consistency and by handling mixed boundary conditions and convection terms more generally.