On Kirchhoff's Model of Parabolic Type. (2nd June 2016)

• Record Type: Journal Article
• Authors: Kundu, Sudeep; Pani, Amiya K.; Khebchareon, Morrakot
• Is Part Of: Numerical functional analysis and optimization. Volume 37:Number 6(2016)
• Page Start: 719
• Page End: 752
• Publication Date: 2016-06-02
• Subjects: Backward Euler method -- discrete attractor -- existence of global attractor -- exponential decay -- finite element method -- Kirchhoff's model -- new regularity results -- numerical experiments -- parabolic type -- semidiscrete error analysis -- uniform in time estimates; 65M60 -- 65M12 -- 65M15; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2016.1176930
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: In this article, the existence of a global strong solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, the existence of a global attractor is shown to hold for the problem when the non-homogeneous forcing function is either independent of time or in L ∞ ( L 2 ). With the finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed, and it is also established that the semidiscrete system has a global discrete attractor. Optimal error estimates in L ∞ ( H 1 ) norm are derived which are valid uniformly in time. Further, based on a backward Euler method, a completely discrete scheme is analyzed and error estimates are derived. It is also further, observed that in cases where f = 0 or f = O ( e −γ0 t ) with γ0 > 0, the discrete solutions and error estimates decay exponentially in time. Finally, some numerical experiments are discussed which confirm our theoretical findings.