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An adaptive superconvergent finite element method based on local residual minimization

Ignacio Muga, Sergio Rojas, Patrick Vega

TL;DR

This work addresses diffusion-dominated PDEs where the flux satisfies $\boldsymbol q = -\boldsymbol \nabla u$ and proposes an adaptive finite element method by marrying a superconvergent postprocessing of the primal variable with residual-minimization–driven adaptivity. The key idea is to perform residual minimization on a local postprocessing scheme, producing a superconvergent scalar postprocessed solution and a local error representation that robustly guides mesh refinement, while preserving a cheap elementwise computation through a local saddle-point formulation. The paper develops two a posteriori estimators: a built-in residual-based estimator and an enhanced estimator that includes flux mismatch and interelement jumps, both demonstrated to be reliable and, under mild assumptions, efficient. Numerical experiments in two dimensions with Brezzi-Douglas-Marini elements confirm the theoretical findings, showing sharp error indicators, optimal convergence on refined meshes, and effective resolution of singularities and boundary layers, with clear potential for extension to other mixed methods and convection–diffusion–reaction systems. Overall, the approach provides a PDE-independent framework for accurate, adaptive mixed FEMs with a provably sharp a posteriori error control built into the postprocessing step, enabling efficient computation and parallelization.

Abstract

We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal variable with an adaptive finite element method via residual minimization. Such a residual minimization procedure is performed on a local postprocessing scheme, commonly used in the context of mixed finite element methods. Given the local nature of that approach, the underlying saddle point problems associated with residual minimizations can be solved with minimal computational effort. We propose and study a posteriori error estimators, including the built-in residual representative associated with residual minimization schemes; and an improved estimator which adds, on the one hand, a residual term quantifying the mismatch between discrete fluxes and, on the other hand, the interelement jumps of the postprocessed solution. We present numerical experiments in two dimensions using Brezzi-Douglas-Marini elements as input for our methodology. The experiments perfectly fit our key theoretical findings and suggest that our estimates are sharp.

An adaptive superconvergent finite element method based on local residual minimization

TL;DR

This work addresses diffusion-dominated PDEs where the flux satisfies and proposes an adaptive finite element method by marrying a superconvergent postprocessing of the primal variable with residual-minimization–driven adaptivity. The key idea is to perform residual minimization on a local postprocessing scheme, producing a superconvergent scalar postprocessed solution and a local error representation that robustly guides mesh refinement, while preserving a cheap elementwise computation through a local saddle-point formulation. The paper develops two a posteriori estimators: a built-in residual-based estimator and an enhanced estimator that includes flux mismatch and interelement jumps, both demonstrated to be reliable and, under mild assumptions, efficient. Numerical experiments in two dimensions with Brezzi-Douglas-Marini elements confirm the theoretical findings, showing sharp error indicators, optimal convergence on refined meshes, and effective resolution of singularities and boundary layers, with clear potential for extension to other mixed methods and convection–diffusion–reaction systems. Overall, the approach provides a PDE-independent framework for accurate, adaptive mixed FEMs with a provably sharp a posteriori error control built into the postprocessing step, enabling efficient computation and parallelization.

Abstract

We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal variable with an adaptive finite element method via residual minimization. Such a residual minimization procedure is performed on a local postprocessing scheme, commonly used in the context of mixed finite element methods. Given the local nature of that approach, the underlying saddle point problems associated with residual minimizations can be solved with minimal computational effort. We propose and study a posteriori error estimators, including the built-in residual representative associated with residual minimization schemes; and an improved estimator which adds, on the one hand, a residual term quantifying the mismatch between discrete fluxes and, on the other hand, the interelement jumps of the postprocessed solution. We present numerical experiments in two dimensions using Brezzi-Douglas-Marini elements as input for our methodology. The experiments perfectly fit our key theoretical findings and suggest that our estimates are sharp.
Paper Structure (23 sections, 11 theorems, 75 equations, 8 figures)

This paper contains 23 sections, 11 theorems, 75 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A review of related literature
  3. Outline of the paper
  4. Theoretical background
  5. Notation
  6. Model problem
  7. A mixed FEM
  8. A superconvergent postprocessing
  9. Main results
  10. Postprocessing based on local residual minimization
  11. A posteriori error analysis
  12. A built-in a posteriori error estimator
  13. An improved a posteriori error estimator
  14. Numerical validation
  15. A smooth solution
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $(u,\boldsymbol q)\in\mathcal{V}\times\boldsymbol{\mathcal{H}}$ be the solution of eq:continuous_formulation and $(u_h,\boldsymbol q_h)\in\mathcal{V}_h^{p-1}\times\boldsymbol{\mathcal{H}}_h^p$ be the solution of eq:discrete_formulation. Let us suppose that $u$ lies in $H^s(\Omega)$ with $s>3/2$. where $|\cdot|_{s,\Omega}$ is the standard $H^s(\Omega)$-seminorm adams_fournier_2003a. Moreover, i

Figures (8)

  • Figure 1: Convergence rates for a smooth solution scenario under uniform mesh refinements.
  • Figure 2: Effectivity index and saturation constant for a smooth solution scenario.
  • Figure 3: Convergence curves for a low-regularity solution scenario.
  • Figure 4: Sequence of adapted meshes for a low-regularity solution scenario, with $p=3$.
  • Figure 5: Increment in order of convergence.
  • ...and 3 more figures

Theorems & Definitions (29)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Corollary 3.2: a priori error analysis
  • Remark 3.3: General test spaces
  • ...and 19 more