Strictly equivalent a~posteriori error estimators for quasi-optimal nonconforming methods

TL;DR

The paper develops strictly equivalent a posteriori error estimators for quasi-optimal nonconforming finite element methods solving symmetric elliptic problems of second and fourth order. It introduces a residual-based framework that splits into conforming and nonconforming parts, localizes the conforming residual with a partition of unity and local projections, and separates data oscillation into a computable component and a case-dependent oscillatory part, thereby achieving error-dominated oscillation and computability. The methodology is demonstrated across Poisson and biharmonic problems, including CR, dG, C0-IP, and Morley-type methods, with explicit estimators that are computable via local projections and averaging operators. The paper also provides numerical benchmarks for a Poisson problem with rough data, illustrating the estimators’ effectiveness and adaptivity, particularly when the data exhibits singular behavior along a geometric feature. Overall, the results supply a unified, practically implementable pathway for strictly equivalent a posteriori control of a broad class of nonconforming schemes in both second- and fourth-order elliptic settings.

Abstract

We devise a posteriori error estimators for quasi-optimal nonconforming finite element methods approximating symmetric elliptic problems of second and fourth order. These estimators are defined for all source terms that are admissible to the underlying weak formulations. More importantly, they are equivalent to the error in a strict sense. In particular, their data oscillation part is bounded by the error and, furthermore, can be designed to be bounded by classical data oscillations. The estimators are computable, except for the data oscillation part. Since even the computation of some bound of the oscillation part is not possible in general, we advocate to handle it on a case-by-case basis. We illustrate the practical use of two estimators obtained for the Crouzeix-Raviart method applied to the Poisson problem with a source term that is not a function and its singular part with respect to the Lebesgues measure is not aligned with the mesh.

Figures (4)

• Figure 2.1: Commutative diagram summarizing the construction of a fully stable nonconforming method $M_h$.
• Figure 6.1: Distribution in space of the indicators in the estimator $\mathrm{est}_{\widetilde{\mathsf{CR}},4}$, with oscillation replaced by surrogate, on the mesh $\mathcal{T}_4$ obtained by uniform refinement. Lower and higher values correspond to cold and warm colors, respectively.
• Figure 6.2: Initial mesh $\mathcal{T}_0$ and subsequent meshes $\mathcal{T}_5$ and $\mathcal{T}_{10}$ generated by adaptive refinement.
• Figure 6.3: Error $\mathrm{err}_k$ (blue) and estimator $\mathrm{est}_{\widetilde{\mathsf{CR}},k}$ (red) for uniform ($\bigcirc$) and adaptive ($\triangle$) mesh refinement versus $\#\mathcal{T}_k$. The dashed line indicates the decay rate $(\#\mathcal{T}_k)^{-0.5}$.

Theorems & Definitions (88)

• Proposition 2.1: Quasi-optimality
• proof
• Remark 2.2: Overconsistency
• Lemma 3.1: Error, residual and their decompositions
• proof
• Remark 3.2: Extended weak formulation
• Remark 3.3: Decomposition features
• Proposition 3.4: Approximating the distance to conformity
• proof
• Remark 3.5: Computable realization of the distance to conformity