Mixed finite element methods for elliptic obstacle problems

TL;DR

This work develops and analyzes mixed finite element methods for elliptic obstacle problems, covering both the membrane (second-order) and Kirchhoff–Love plate (fourth-order) cases. By introducing the obstacle reaction as an independent unknown, the authors formulate first-order systems whose discretization uses Raviart–Thomas-type spaces, enabling direct approximation of the reaction force and the complementarity condition. They establish well-posedness via inf-sup conditions, derive a priori error estimates (including weaker norms), and construct a posteriori error estimators based on postprocessed solutions, with regularized maximum functions employed for the plate problem. Extensive numerical experiments validate the theory, demonstrating reliable error control and the effectiveness of adaptive refinement in resolving contact regions and domain singularities.

Abstract

Mixed variational formulations for the first-order system of the elastic membrane obstacle problem and the second-order system of the Kirchhoff--Love plate obstacle problem are proposed. The force exerted by the rigid obstacle is included as a new unknown. A priori and a posteriori error estimates are derived for both obstacle problems. The a posteriori error estimates are based on conforming postprocessed solutions. Numerical experiments conclude this work.

Figures (8)

• Figure 1: Experimental rates of convergence for the errors $\|u - u_{h}\|_{\Omega}$ and $\|\boldsymbol{\sigma} - \boldsymbol{\sigma}_{h}\|_{\Omega}$ (left) and the error estimator $\rho$ with its individual contributions (right), with uniform refinement for the problem from Section \ref{['sec:membrane:num_ex_smooth']}.
• Figure 2: Error estimator $\rho$ and its individual contributions with uniform refinement (left) and adaptive refinement (right) for the problem from Section \ref{['sec:membrane:num_ex_unk_L']}.
• Figure 3: Meshes and solution component $u_h$ for the problem from Section \ref{['sec:membrane:num_ex_unk_L']}.
• Figure 4: Error and estimators for the problem from Section \ref{['sec:num_ex:plate1']}.
• Figure 5: Estimators on a sequence of uniformly (u) and adaptively (a) refined meshes for the problem from Section \ref{['sec:num_ex:plate2']}.

Theorems & Definitions (52)

• Proposition 2.1: auxiliary result
• proof
• Lemma 2.2: density result
• proof
• Lemma 2.3: stability estimate
• proof
• Theorem 2.4: well-posedness
• proof
• Theorem 2.5: well-posedness discrete problem
• Lemma 2.6