Duality based error control for the Signorini problem

TL;DR

This work derives rigorous a posteriori error bounds in $L^{p}(\Omega)$ for conforming piecewise linear FE approximations to the Signorini problem in 2D, with $p\in(4,\infty)$. The authors employ a duality-based framework, separating the positive and negative parts of the discretization error and introducing a novel bound-preserving bilateral interpolation operator to handle the non-smooth contact set. They prove residual-based estimates for the positive part and, under verifiable a posteriori conditions on the discrete contact set, analogous bounds for the negative part, achieving optimal $L^{p}$ convergence rates when combined with adaptive mesh refinement. Numerical experiments in 2D and 3D validate the theory, showing that adaptivity recovers optimal rates that are hindered by suboptimal uniform refinement, though the theory in 3D requires further development due to geometric and regularity challenges.

Abstract

In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in $L^{p}$, for $p \in (4,\infty)$ in two spatial dimensions. This new analysis treats the positive and negative parts of the discretisation error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in $W^{2,(4 - \varepsilon)/3}$ for any $\varepsilon \ll 1$. We summarise extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.

Figures (10)

• Figure 1: An illustration to show how the trace of a function which violates \ref{['struct_of_contact_set']} might behave.
• Figure 2: An illustration of three piecewise linear operators, black, applied to the same function, blue, that satisfies $z>0$. The Lagrange interpolant, $\mathcal{I}\xspace(z) \geqslant 0$, the nonlinear interpolant $\Pi\xspace(z) \leqslant z$ and the bilateral approximation $0 \leqslant \widehat{\Pi}\xspace(z) \leqslant z$.
• Figure 3: Example 1 §\ref{['sec:l4:ex_1']}, contour plot and various iterations of the adaptive mesh $\mathcal{T}^i$ of an approximation to a function $u \in \operatorname{H}^{2}(\Omega\xspace)$. Figure \ref{['fig:strategy_comparison']} shows a double-logarithmic plot comparing the error in $\operatorname{L}^{4}(\Omega\xspace)$ (black line) with the error estimate (red line).
• Figure 4: Comparison of $\operatorname{L}^{32}(\Omega\xspace)$ error computed against exact solution for uniform and adaptive meshes for example 1. Uniform mesh refinement delivers suboptimal convergence of order $h^{\tfrac{25}{16}-\varepsilon}$ as predicted by the a priori results of christof1754finite. Optimal rates are recovered using adaptive mesh refinement.
• Figure 5: Example 1 §\ref{['sec:l4:ex_1']}, visualisation of the discrete contact set $\mathcal{A}_U$. Meshes $\mathcal{T}^{6}$, $\mathcal{T}^{11}$, $\mathcal{T}^{15}$. Solid line is contact (i.e. $U=0$), dashed line indicates non-contact. Critical points are shown with red dots. A boundary degree of freedom is in $\mathcal{A}_U$ if it is zero. For this example the discrete contact set varies on coarse meshes, and once sufficient resolution is attained we have two critical points where the boundary condition changes (note that this matches the known exact solution).

Theorems & Definitions (38)

• Proposition 2.1: Regularity of the primal problem Grisvard:1985
• Remark 2.2
• Definition 2.3: Contact Set
• Remark 2.4: \ref{['struct_of_contact_set']}
• Proposition 2.5: Improved Regularity of the Primal Problem christof1754finite
• Proposition 2.6: The discrete problem is well posed kinderlehrer2000introduction
• Definition 2.7: Discrete contact Set
• Definition 2.8: \ref{['def:condition_A_h_full']} christof1754finite
• Remark 2.9: A priori error bounds christof1754finite
• Proposition 3.1: Trace theorem in $\operatorname{L}^{s}$