抛物方程的变网格连续时空有限体积元法

Variable Mesh Continuous Space-Time Finite Volume Element Method for Parabolic Equations

摘要: 从三次Lagrange插值最佳应力点的理论出发，本文提出一种变网格连续时空有限体积元格式，旨在解决抛物方程数值求解问题。通过耦合Legendre-Lobatto节点的Lagrange插值多项式与Gauss积分准则，在时空非匹配网格剖分条件下严格证明了数值解的存在唯一性，并建立$L^{\infty }\left(L^2\right)$ 和$L^{\infty }\left(H^1\right)$ 的最优阶误差估计理论。数值实验结果表明收敛数据与理论预测高度一致，验证了算法在非均匀网格环境中的计算优势与理论分析的有效性。

Abstract: Based on the theoretical framework of cubic Lagrange interpolation with optimal stress nodes, this study develops a variable mesh continuous space-time finite volume element scheme to resolve numerical challenges in solving parabolic equations. By integrating Lagrange interpolation polynomials at Legendre-Lobatto nodes with Gauss quadrature rules, we rigorously prove the existence and uniqueness of numerical solutions under spacetime non-matching grid partitions. Optimal-order error estimates in $L^{\infty }\left(L^2\right)$ and $L^{\infty }\left(H^1\right)$ norms are theoretically established. Numerical experiments demonstrate excellent agreement between convergence rates and theoretical predictions, confirming the computational advantages of the proposed algorithm in non-uniform grid environments and the validity of theoretical analysis.