[Paper Review] Robust A Posteriori Error Estimates for Stabilized Finite Element Methods

Lutz Tobiska, R. Verfürth|arXiv (Cornell University)|Feb 24, 2014

Advanced Numerical Methods in Computational Mathematics17 references3 citations

TL;DR

This paper establishes robust a posteriori error estimates for a broad class of stabilized finite element methods applied to convection-diffusion equations. It demonstrates that streamline diffusion, local projection, subgrid-scale, and continuous interior penalty methods all yield the same error indicators with constants independent of viscosity, convection, or reaction terms, ensuring uniform reliability and efficiency across all schemes.

ABSTRACT

There is a wide range of stabilized finite element methods for stationary and non-stationary convection-diffusion equations such as streamline diffusion methods, local projection schemes, subgrid-scale techniques, and continuous interior penalty methods to name only a few. We show that all these schemes give rise to the same robust a posteriori error estimates, i.e. the multiplicative constants in the upper and lower bounds for the error are independent of the size of the convection or reaction relative to the diffusion. Thus, the same error indicator can be used modulo higher order terms caused by data approximation.

Motivation & Objective

• To unify a posteriori error estimation for multiple stabilized finite element methods in convection-diffusion problems.
• To prove that error indicators derived from different stabilized schemes yield equivalent robust error bounds.
• To ensure the error estimates remain uniform with respect to the viscosity ε and reaction parameter β, even in convection-dominated regimes.
• To derive computable upper and lower bounds for the residual that are independent of the relative size of convection or reaction terms.
• To extend the analysis to both stationary and non-stationary problems using a common framework based on residual and consistency error control.

Proposed method

• Uses the general framework of [39] to establish generic robust equivalence between error and residual.
• Derives explicit, computable upper bounds for consistency errors in each stabilized scheme via Lemmas 2.3–2.6.
• Applies energy norms and dual norms tailored to the symmetric part of the bilinear form to ensure robustness.
• Employs a residual-based error estimator with local contributions η, θ, and Θ for different schemes, including a parameter σ_cip for continuous interior penalty.
• Introduces time-discrete error estimators for non-stationary problems using time-averaged data and discrete time steps.
• Establishes global upper and local lower bounds for the error in both space and time, with constants independent of ε and β.

Experimental results

Research questions

• RQ1 Can a single a posteriori error estimator be robustly applied across multiple stabilized finite element methods for convection-diffusion problems?
• RQ2 Are the error estimates derived from different stabilized schemes uniformly reliable and efficient, regardless of the relative size of convection or reaction terms?
• RQ3 Can the consistency error introduced by each stabilization method be bounded uniformly and explicitly in terms of mesh parameters?
• RQ4 Do the same error indicators apply to both stationary and non-stationary convection-diffusion equations with the same robustness properties?
• RQ5 Can the residual-based error estimator be made robust with respect to the viscosity ε and reaction parameter β across all stabilized schemes?

Key findings

• All considered stabilized finite element methods—streamline diffusion, local projection, subgrid-scale, and continuous interior penalty—produce the same robust a posteriori error estimates.
• The upper and lower bounds for the error are uniform with respect to ε and β, meaning the constants in the bounds are independent of the convection or reaction strength.
• For the stationary case, the residual norm is bounded above and below by the error estimator terms η, θ, and Θ, with constants c♭ and c♭ independent of ε and β.
• For the non-stationary case, the global error is bounded above by a sum of estimators over time steps, with constants c* and c* independent of T, ε, and β.
• The time-discrete error estimator includes contributions from initial data, time evolution, and data approximation, all bounded uniformly.
• The parameter σ_cip equals 1 for the continuous interior penalty method and 0 for others, reflecting scheme-specific consistency error contributions.