[Paper Review] Reliable and efficient a posteriori error estimates of DG methods for a frictional contact problem

Fei Wang, Weimin Han|arXiv (Cornell University)|Jan 20, 2014

Advanced Numerical Methods in Computational Mathematics33 references3 citations

TL;DR

This paper presents reliable and efficient a posteriori error estimators for discontinuous Galerkin (DG) methods applied to a frictional contact problem, formulated as an elliptic variational inequality of the second kind. By relating the error in the variational inequality to that of an associated linear problem, the authors derive residual-type estimators and prove both reliability and efficiency, with bounds depending on local residuals, solution jumps, and data oscillations.

ABSTRACT

A posteriori error estimators are studied for discontinuous Galerkin methods for solving a frictional contact problem, which is a representative elliptic variational inequality of the second kind. The estimators are derived by relating the error of the variational inequality to that of a linear problem. Reliability and efficiency of the estimators are shown.

Motivation & Objective

• To develop a posteriori error estimators for discontinuous Galerkin (DG) methods applied to a frictional contact problem governed by an elliptic variational inequality of the second kind.
• To establish both reliability and efficiency of the proposed error estimators, ensuring they accurately reflect the true error in adaptive finite element methods.
• To extend the framework of a posteriori error analysis from linear problems to variational inequalities by leveraging duality and residual techniques.
• To provide computable error indicators that guide mesh refinement in adaptive algorithms, particularly for problems with non-smooth or non-differentiable solutions.
• To ensure the error estimators are robust with respect to mesh irregularities, including hanging nodes, due to the DG method's flexibility.

Proposed method

• Derive a posteriori error estimators by relating the error in the variational inequality to the error in an associated linear elliptic problem.
• Construct residual-type error indicators based on element residuals, jump terms across element interfaces, and data oscillations.
• Use a recovery technique involving local projections of the residual and numerical fluxes to approximate the true error.
• Apply the Cauchy-Schwarz inequality and inverse estimates to bound the error in terms of local residuals and solution jumps.
• Employ a dual problem approach and stability estimates to control the consistency and reliability of the error indicators.
• Prove efficiency by bounding the local error indicators from below using the true error, data oscillations, and jumps in the dual solution.

Experimental results

Research questions

• RQ1 Can reliable and efficient a posteriori error estimators be constructed for discontinuous Galerkin methods applied to frictional contact problems?
• RQ2 How can the error in a variational inequality of the second kind be related to the error in an associated linear problem to enable error estimation?
• RQ3 What are the key components (residuals, jumps, data oscillations) that contribute to the local error indicators in DG methods for such problems?
• RQ4 To what extent do the proposed estimators reflect the true error in adaptive mesh refinement?
• RQ5 Can the efficiency of the estimators be theoretically proven under standard finite element assumptions?

Key findings

• The proposed a posteriori error estimators are reliable, meaning the global error is bounded above by a constant times the sum of local error indicators.
• The estimators are efficient, as the local error indicators are bounded below by a constant times the true error, ensuring no over-refinement.
• The efficiency bound includes terms involving the energy norm of the error, jumps in the dual solution, data oscillations, and local residual terms.
• The error estimator satisfies the bound $ \eta_K \leq C\left(|u - u_h|_{\omega_K} + \sum_{e \in \mathcal{E}(K) \cap \mathcal{E}_2} |\lambda - \lambda_h|_{*,e} + h_K \|f - f_h\|_{\omega_K} + \sum_{e \in \mathcal{E}(K) \cap \mathcal{E}_2} h_e \|\lambda_h - \overline{\lambda}_h\|_e^2 \right) $, with $ C $ independent of mesh size.
• The analysis confirms that the error estimators are robust under mesh refinement and applicable to general meshes with hanging nodes due to the DG method’s flexibility.