全离散FDM-FEM格式求解热-弹性-损伤耦合非线性偏微分方程组的分析
Analysis of a fully discretized FDM-FEM scheme for solving thermo-elastic-damage coupled nonlinear PDE systems
AI总结：
本文针对热-弹性-损伤耦合非线性偏微分方程组，提出基于低阶Galerkin有限元与隐式有限差分的全离散格式，并严格证明其收敛性及给出收敛阶。
摘要：
本文考虑一个非线性偏微分方程组，该方程组由一个抛物型热方程与一个描述位移场行为的双曲型动量方程非线性耦合，并与一个基于内部损伤变量的非线性椭圆方程耦合。我们提出了一种数值格式，该格式基于低阶Galerkin有限元方法（FEM）对时间相关非线性偏微分方程组进行空间离散，并采用隐式有限差分方法（FDM）对时间变量方向进行离散。此外，我们给出了精确解和离散解在逐点时间$L^2$范数下的先验估计。基于这些先验估计，我们严格证明了全离散系统的解收敛到精确解。通过刻画内部参数的性质，我们找到了关于离散化参数的收敛阶。
英文摘要：
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.
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