Rigorous A Posteriori Error Bounds for the Signorini Problem in Lp Norms

Core Concepts

This work derives rigorous a posteriori error bounds in Lp norms, for p ∈ (4,8), for the finite element approximation of the Signorini problem. The analysis relies on a novel sign- and bound-preserving interpolant and sharp dual stability results.

Abstract

The paper studies the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. The authors prove new rigorous a posteriori estimates of residual type in Lp, for p ∈ (4,8) in two spatial dimensions. The key highlights are: The 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 W2,p(4-ε)/3 for any ε ≪ 1. The authors make two key assumptions: Condition (A) on the structure of the contact set of the solution, and Condition (ah) on the structure of the contact set of the discrete solution. These assumptions are used to derive bounds on the positive and negative parts of the error, respectively. Under Condition (A) alone, the authors obtain h-robust a posteriori error control in Lp(Ω) for the positive part of the error. For the negative part of the error, the authors obtain an a posteriori bound that depends on the discrete solution satisfying Condition (ah), which can be verified a posteriori. The bound has an h-dependent regularity constant that remains small in numerical experiments. The authors summarise extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.

Stats

h2 ‖-∆U + U - f‖Lp(K) h2-1/q ‖J∇UK‖Lp(∂K)

Quotes

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Key Insights Distilled From

Duality based error control for the Signorini problem

by Ben S. Ashby... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.01251.pdf

Deeper Inquiries

How can the assumptions of Condition (A) and Condition (ah) be relaxed or generalized to broader classes of problems?

The assumptions of Condition (A) and Condition (ah) can be relaxed or generalized by considering more flexible criteria for the contact set and the boundary conditions. For Condition (A), instead of requiring the contact set to consist of finitely many connected components and isolated points, a more general condition could allow for a wider range of contact set structures, such as fractal contact sets or sets with accumulation points. This would broaden the applicability of the theory to a wider class of problems where the contact set may exhibit more complex geometries. Similarly, for Condition (ah), which imposes restrictions on the boundaries of discrete contact sets, a more relaxed condition could be formulated to accommodate different types of boundary configurations. This could involve considering different boundary conditions or contact set topologies that may not fit the specific criteria outlined in the original assumption. By generalizing these conditions, the theory could be applied to a more diverse set of problems with varying contact set characteristics and boundary conditions.

Can the a posteriori error bounds be extended to higher-order finite element methods or other discretization schemes?

Yes, the a posteriori error bounds derived in the context of the Signorini problem can be extended to higher-order finite element methods or other discretization schemes. The key lies in adapting the interpolation operators and error estimation techniques to accommodate the higher-order approximations. For higher-order finite element methods, the interpolation operators need to be adjusted to handle the increased complexity of the basis functions. By developing suitable interpolation schemes that preserve the properties of the solution, such as sign and bound preservation, the error bounds can be extended to higher-order approximations. Similarly, for other discretization schemes beyond finite element methods, such as spectral methods or finite volume methods, the error estimation framework can be adapted to suit the specific characteristics of the discretization. By formulating appropriate interpolation operators and error indicators tailored to the chosen scheme, the a posteriori error bounds can be generalized to these alternative methods.

What are the potential applications of these a posteriori error bounds in practical problems beyond the Signorini problem, such as in elasto-plasticity, fluid flow in porous media, or finance?

The a posteriori error bounds developed for the Signorini problem have broad applications in various practical problems beyond the specific domain of contact problems. Some potential applications include: Elasto-plasticity: In the field of elasto-plasticity, where materials exhibit both elastic and plastic behavior, the a posteriori error bounds can be utilized to assess the accuracy of numerical simulations and optimize the mesh refinement strategies. By providing reliable estimates of the discretization error, these bounds can enhance the efficiency and reliability of elasto-plasticity simulations. Fluid Flow in Porous Media: For modeling fluid flow in porous media, the a posteriori error bounds can aid in validating computational models and ensuring the accuracy of numerical simulations. By quantifying the error in the solution, these bounds enable researchers to refine the discretization and improve the predictive capabilities of porous media flow models. Finance: In the realm of finance, particularly in the numerical modeling of financial derivatives or risk management strategies, a posteriori error bounds can be valuable for verifying the correctness of computational algorithms and pricing models. By assessing the accuracy of numerical solutions, these bounds can instill confidence in financial calculations and support informed decision-making in the financial sector.