二阶双相滞后方程的有限元方法及误差估计

Finite Element Methods and Error Estimates for Second-Order Dual-Phase-Lag Equations

摘要: 本文针对数学形式复杂的二阶双相滞后热传导方程的研究，其包含三阶时间导数项，先引入两个辅助变量将原高阶方程降阶为一阶耦合方程组；随后采用有限元方法和Crank-Nicolson格式进行离散，构建了二阶双相滞后模型的全离散有限元格式，通过构造离散能量泛函完成全离散格式的稳定性证明，同时分析截断误差与投影误差的叠加效应，证明全离散格式在$L^2$ 范数下的收敛阶为$O\left(h^{k+1}+\Delta t^2\right)$ 。最后，通过对二维区域上的二阶双相滞后方程的数值实验，验证了本文所构造有限元格式的有效性与收敛性。本文的研究为双相滞后方程提供了一种有限元数值求解方法，加强了该类方程有限元格式的稳定性与收敛性理论研究。

Abstract: In this paper, we investigate the second-order dual-phase-lag heat conduction equation, which is mathematically complex due to the presence of a third-order time derivative term. Two auxiliary variables are first introduced to reduce the original high-order equation to a first-order coupled system. Subsequently, the finite element method combined with the Crank-Nicolson scheme is employed for discretization, leading to the construction of a fully discrete finite element scheme for the second-order dual-phase-lag model. By constructing a discrete energy functional, we prove the stability of the fully discrete scheme. Furthermore, the combined effect of the truncation error and the projection error is analyzed, and it is proved that the convergence order of the fully discrete scheme in the $L^2$ -norm is $O\left(h^{k+1}+\Delta t^2\right)$ . Finally, numerical experiments on the second-order dual-phase-lag equation in a two-dimensional domain are carried out in this paper to verify the effectiveness and convergence of the finite element scheme constructed in this paper. This study provides a finite element numerical solution method for the dual-phase-lag equation and strengthens the theoretical research on the stability and convergence of the finite element scheme for such equations.