Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods

Yahya Alnashri, Jerome Droniou

Research output: Contribution to journalArticleResearchpeer-review

5 Citations (Scopus)

Abstract

Gradient schemes is a framework which enables the unified convergence analysis of many different methods–such as finite elements (conforming, non-conforming and mixed) and finite volumes methods–for 2nd order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.

Original languageEnglish
Pages (from-to)2788-2807
Number of pages20
JournalComputers and Mathematics with Applications
Volume72
Issue number11
DOIs
Publication statusPublished - 1 Dec 2016

Keywords

  • Elliptic variational inequalities
  • Error estimates
  • Gradient schemes
  • Hybrid mimetic mixed methods
  • Obstacle problem
  • Signorini boundary conditions

Cite this

@article{13683f7b23d74f3c8dda3d6889d5d535,
title = "Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods",
abstract = "Gradient schemes is a framework which enables the unified convergence analysis of many different methods–such as finite elements (conforming, non-conforming and mixed) and finite volumes methods–for 2nd order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.",
keywords = "Elliptic variational inequalities, Error estimates, Gradient schemes, Hybrid mimetic mixed methods, Obstacle problem, Signorini boundary conditions",
author = "Yahya Alnashri and Jerome Droniou",
year = "2016",
month = "12",
day = "1",
doi = "10.1016/j.camwa.2016.10.004",
language = "English",
volume = "72",
pages = "2788--2807",
journal = "Computers and Mathematics with Applications",
issn = "0898-1221",
publisher = "Elsevier",
number = "11",

}

Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods. / Alnashri, Yahya; Droniou, Jerome.

In: Computers and Mathematics with Applications, Vol. 72, No. 11, 01.12.2016, p. 2788-2807.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods

AU - Alnashri, Yahya

AU - Droniou, Jerome

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Gradient schemes is a framework which enables the unified convergence analysis of many different methods–such as finite elements (conforming, non-conforming and mixed) and finite volumes methods–for 2nd order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.

AB - Gradient schemes is a framework which enables the unified convergence analysis of many different methods–such as finite elements (conforming, non-conforming and mixed) and finite volumes methods–for 2nd order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.

KW - Elliptic variational inequalities

KW - Error estimates

KW - Gradient schemes

KW - Hybrid mimetic mixed methods

KW - Obstacle problem

KW - Signorini boundary conditions

UR - http://www.scopus.com/inward/record.url?scp=85002734980&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2016.10.004

DO - 10.1016/j.camwa.2016.10.004

M3 - Article

VL - 72

SP - 2788

EP - 2807

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 11

ER -