Analysis of a fully discretized FDM-FEM scheme for solving thermo-elastic-damage coupled nonlinear PDE systems

TL;DR

This paper develops and analyzes a fully discretized numerical scheme combining finite element and finite difference methods for solving complex nonlinear PDE systems modeling thermo-elastic damage, providing convergence proofs and error estimates.

Contribution

It introduces a novel fully discretized FDM-FEM scheme for coupled nonlinear PDEs and proves its convergence with error order analysis.

Findings

Proved convergence of the numerical scheme to the exact solution.

Derived a priori estimates for both exact and discrete solutions.

Established the order of convergence based on discretization parameters.

Abstract

In this paper, we consider a nonlinear PDE system governed by a parabolic heat equation coupled in a nonlinear way with a hyperbolic momentum equation describing the behavior of a displacement field coupled with a nonlinear elliptic equation based on an internal damage variable. We present a numerical scheme based on a low-order Galerkin finite element method (FEM) for the space discretization of the time-dependent nonlinear PDE system and an implicit finite difference method (FDM) to discretize in the direction of the time variable. Moreover, we present a priori estimates for the exact and discrete solutions for the pointwise-in-time $L^2$-norm. Based on the a priori estimates, we rigorously prove the convergence of the solutions of the fully discretized system to the exact solutions. Denoting the properties of the internal parameters, we find the order of convergence concerning the discretization parameters.