Spectral Element Methods a Priori and a Posteriori Error Estimates for Penalized Unilateral Obstacle Problem
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Spectral Element Methods a Priori and a Posteriori Error Estimates for Penalized Unilateral Obstacle Problem Bochra Djeridi1 · Radouen Ghanem1 · Hocine Sissaoui1 Received: 14 August 2019 / Revised: 1 August 2020 / Accepted: 26 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre–Gauss–Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates. Keywords Obstacle problem · Penalty approximation · Spectral method · Finite element method · A priori error estimate · A posteriori error estimate Mathematics Subject Classification 49J20 · 65M60 · 35R35
1 Introduction The theory of variational inequalities and related obstacle problems is an important field of mathematics which was studied by many researchers. Its applications are found for example in engineering, economics and finance. Such problems usually involve finding equilibrium points of systems or shape optimization problems subject to given constraints called obstacles. More details on variational inequalities and their applications may be found, for example, in [2,12,20,25]. In this paper, the obstacle problem under interest is a prototype for a class of elliptic unilateral obstacle problems that involve free boundaries modelling many phenomena such
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Radouen Ghanem [email protected] Numerical Analysis, Optimization and Statistical Laboratory (LANOS), Badji-Mokhtar, Annaba University, P.O. Box 12, 23000 Annaba, Algeria 0123456789().: V,-vol
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as phase transition, jet flow and gas expansion in a porous medium; see for example [12]. Moreover, the study of the relevant obstacle problem in a such setting leads to challenging new directions and is motivated by problems in mechanical engineering and mathematical finance, see for instance, [16,19]. Therefore, for the resolution of this type of problems, numerical methods are required. Thus, in the mathematical literature, finite element methods seem to be the most popular for the discretisation and resolution of obstacle problems see for example [8,11,29]. Besides, other numerical methods can also be found, such as
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