Convergence of a full discretization for a second-order nonlinear elastodynamic equation in isotropic and anisotropic Orlicz spaces

Abstract

In this paper, we study a second-order, nonlinear evolution equation with damping arising in elastodynamics. The nonlinear term is monotone and possesses a convex potential but exhibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation. Moreover, we show uniqueness in a class of sufficiently smooth solutions and provide an a priori error estimate for the temporal semidiscretization.Mathematics Subject Classification (2010). 35L20; 47J35; 47H05; 65M12; 65M06; 35M60.for ε > 0, as well as for ε = 0 and presents numerical experiments [28]. The limit case ε = 0 constitutes the elastodynamic equationfor which one cannot expect smooth solutions even for smooth initial data (see [2,26]). For an excellent survey of the literature concerning equation (1.3) see [8]. The remainder of this paper is structured as follows: In Section 2, we introduce the necessary notation, give a brief introduction to Orlicz spaces and compare the growth condition (1.2) with the restrictive ∆ 2 -condition. The description of the numerical method we employ, the construction of the Galerkin scheme, the proof of existence and uniqueness of the numerical solution, and the derivation of a priori estimates for the fully discrete solution and the discrete time derivative follow in Section 3. Finally, in Section 4 we show convergence towards and, thus, existence of an exact solution, as well as its uniqueness (under additional regularity assumptions) and an error estimate for the temporal semidiscretization. The appendix contains an elementary lemma concerning the separability of the space for wich we want to construct the Galerkin scheme.

Convergence analysis of a Local Discontinuous Galerkin approximation for nonlinear systems with Orlicz-structure

“…There are only few numerical investigations for nonlinear problems with Orlicz-structure showing the convergence of discrete solutions to a weak solution. We are only aware of the studies [17,15,19,9,26]. Non of these contributions uses DG methods.…”

“…Steady problems with Orlicz-structure are treated, using finite element methods (FEM) in [17,15,9] and non-conforming methods in [5]. Unsteady problems are investigated in [19,26]. To the best of the author's knowledge, there are no studies of steady problems with Orlicz-structure using DG methods, except for [16,Remark 2.3], where the possibility of extending most of the results in [16] to balanced N-functions has already been mentioned.…”

Discrete solution for the nonlinear parabolic equations with diffusion terms in Museilak-spaces

“…In this paper, we consider the weak formulation of (1) and propose a convergent full discretization combining a piecewise constant finite element approximation with the backward Euler scheme, also there is constructed an approximation solution sequence for problem (1) and establish a priori estimation. Our study is done on the isotropic case and generalizes [12] and [13] where the authors studied only the case b(u) = u and K = 0.…”

Descret Solution for a Nonlinear Parabolic Equations with Diffusion Terms in Museilak-Spaces

“…In this paper, we consider the weak formulation of (1.1) and propose a convergent full discretization combining a piecewise constant finite element approximation with the backward Euler scheme, we construct an approximation solution sequence for problem (1.1) and establish a priori estimation. Our study is done on the isotropic case and generalizes [14] and [23] where the authors studied only the case b(u) = u and K = 0.…”