Convergence of a Full Discretization for a Second-Order Nonlinear Elastodynamic Equation in Isotropic and Anisotropic Orlicz Spaces

TL;DR

This paper analyzes a second-order nonlinear elastodynamic equation with damping in anisotropic Orlicz spaces, establishing global existence, uniqueness, and error estimates for a full discretization scheme.

Contribution

It introduces a convergence analysis of a full discretization for elastodynamic equations with anisotropic, nonpolynomial growth in Orlicz spaces, including error estimates.

Findings

Proves global existence of solutions in Orlicz spaces.

Establishes uniqueness for sufficiently smooth solutions.

Provides an a priori error estimate for the backward Euler scheme.

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.