Recovery Techniques for Finite Element Methods

TL;DR

This work surveys polynomial preserving recovery (PPR) for gradient and Hessian estimation within finite element methods, showing how recovery-based post-processing yields superconvergence and ultraconvergence under mild to translation-invariant mesh assumptions. It develops two analytical frameworks—supercloseness on mildly structured meshes and difference-quotient on translation-invariant meshes—and demonstrates that recovered quantities lead to asymptotically exact a posteriori error estimators for second- and fourth-order PDEs and interface problems. The results are supported by extensive numerical illustrations in second-order, fourth-order, and interface settings, including body-fitted and cut FEM, confirming the practical relevance for adaptive computation. The framework unifies post-processing across standard and nonstandard discretizations and highlights the broad applicability of recovery techniques in steering efficient, reliable adaptivity. Overall, polynomial preserving recovery provides a simple, robust, and widely applicable tool for both error estimation and pre-/post-processing in advanced FEM contexts.

Abstract

Post-processing techniques are essential tools for enhancing the accuracy of finite element approximations and achieving superconvergence. Among these, recovery techniques stand out as vital methods, playing significant roles in both post-processing and pre-processing. This paper provides an overview of recent developments in recovery techniques and their applications in adaptive computations. The discussion encompasses both gradient recovery and Hessian recovery methods. To establish the superconvergence properties of these techniques, two theoretical frameworks are introduced. Applications of these methods are demonstrated in constructing asymptotically exact {\it a posteriori} error estimators for second-order elliptic equations, fourth-order elliptic equations, and interface problems. Numerical experiments are performed to evaluate the asymptotic exactness of recovery type a posteriori error estimators.

Figures (20)

• Figure 1: Illustration of the setup for recovery techniques in one-dimensional space.
• Figure 2: Translation invariant meshes on the unit square: (a) uniform mesh with a regular pattern; (b) uniform mesh with a chevron pattern; (c) uniform mesh with a crisscross pattern; (d) uniform mesh with a Union Jack pattern.
• Figure 3: Illustration of the definition of $L(z_i, n)$: (a) Plot of $L(z_i, 0)$; (b) Plot of $L(z_i, 1)$; (c) Plot of $L(z_i, 2)$.
• Figure 4: Gradient recovery on uniform meshes: (a) regular pattern; (b) chevron pattern.
• Figure 5: Hessian recovery on uniform meshes with a chevron pattern.

Theorems & Definitions (67)

• Definition 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6
• Theorem 3.7
• proof