On the quadratic finite element approximation to the obstacle problem

2002 ◽  
Vol 92 (4) ◽  
pp. 771-778 ◽  
Author(s):  
Lie-heng Wang
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Finite Element Approximation ◽  
Element Approximation

scholarly journals Finite element approximation of an obstacle problem for a class of integro–differential operators

2020 ◽  
Vol 54 (1) ◽  
pp. 229-253 ◽  
Author(s):  
Andrea Bonito ◽  
Wenyu Lei ◽  
Abner J. Salgado
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Fractional Laplacian ◽  
Differential Operators ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Finite Element Scheme ◽  
Order Elliptic Operator ◽  
Nonlocal Part ◽  
Differential Part

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 273-292 ◽  
Author(s):  
Thirupathi Gudi ◽  
Papri Majumder
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Time Discretization ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Fully Discrete ◽  
Backward Euler ◽  
Space Discretization ◽  
Speed Of Propagation

AbstractWe introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order {\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity {u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

scholarly journals Morley finite element method for the von Karman obstacle problem

2021 ◽  
Author(s):  
Carsten Carstensen ◽  
Sharat Gaddam ◽  
Neela Nataraj ◽  
Amiya K Pani ◽  
Devika Shylaja
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
A Priori ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Small Data ◽  
Polygonal Domain ◽  
Optimal Convergence ◽  
Von Karman ◽  
Von Kármán

This paper focusses on the  von Karman equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Karman obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Karman obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.

On Finite Element Formulations for the Obstacle Problem – Mixed and Stabilised Methods

2017 ◽  
Vol 17 (3) ◽  
pp. 413-429 ◽  
Author(s):  
Tom Gustafsson ◽  
Rolf Stenberg ◽  
Juha Videman
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Numerical Examples ◽  
Finite Element Formulations

AbstractWe discuss the differences between the penalty, mixed and stabilised methods for the finite element approximation of the obstacle problem. The theoretical properties of the methods are discussed and illustrated through numerical examples.

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