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.

翻译：本文研究由抛物型热方程非线性耦合双曲型动量方程（描述位移场行为）及基于内部损伤变量的非线性椭圆方程所控制的非线性偏微分方程组。我们提出一种数值格式：采用低阶Galerkin有限元法（FEM）对时变非线性偏微分方程组进行空间离散，并运用隐式有限差分法（FDM）沿时间变量方向进行离散。进一步，我们给出精确解和离散解在时间逐点$L^2$范数下的先验估计。基于该先验估计，严格证明了全离散系统解收敛于精确解。通过刻画内部参数性质，我们获得了关于离散参数的收敛阶。