双调和柯西问题的最小二乘弱Galerkin方法
A Least-Squares Weak Galerkin Method for the Biharmonic Cauchy Problem
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中文总结 AI 辅助
该研究针对双调和柯西问题，提出LS-WG有限元方法，将四阶方程重构为二阶耦合系统，消除inf-sup条件、适配多边形网格，理论上保证解的唯一性与最优误差，数值实验验证了方法的准确性与有效性。
中文摘要 AI 辅助
本文针对双调和方程的柯西问题，开发了一种最小二乘弱Galerkin（LS-WG）有限元方法。所提方法将四阶方程重构为两个二阶方程的耦合系统，采用弱有限元空间上的离散弱拉普拉斯算子对其进行离散化。得到的最小二乘格式产生对称正定线性系统，从而消除了混合有限元方法所需的离散inf-sup条件，同时避免了全局$C^1$协调有限元空间的构造。此外，弱Galerkin框架自然适配一般多边形网格，在网格生成和逼近方面提供了极大灵活性。在连续双调和柯西问题存在唯一解的假设下，本文证明了离散LS-WG解的唯一性，并推导了离散能量范数下的最优阶误差估计。数值实验验证了理论收敛速率，证明了所提方法的准确性、鲁棒性和有效性。
英文摘要
We develop a least-squares weak Galerkin (LS-WG) finite element method for the Cauchy problem of the biharmonic equation. The proposed approach reformulates the fourth-order equation as a coupled system of two second-order equations, which are discretized using discrete weak Laplacian operators on weak finite element spaces. The resulting least-squares formulation yields a symmetric positive definite linear system, thereby eliminating the discrete inf-sup condition required by mixed finite element methods while avoiding the construction of globally $C^1$-conforming finite element spaces. Furthermore, the weak Galerkin framework naturally accommodates general polygonal meshes, offering considerable flexibility in mesh generation and approximation. Under the assumption that the continuous biharmonic Cauchy problem admits a unique solution, we establish the uniqueness of the discrete LS-WG solution and derive optimal-order error estimates in a discrete energy norm. Numerical experiments confirm the theoretical convergence rates and demonstrate the accuracy, robustness, and effectiveness of the proposed method.
