This paper focuses on the numerical solution of elliptic partial differential equations (PDEs) with Dirichlet and mixed boundary conditions, specifically addressing the challenges arising from irregular domains. Both finite element method (FEM) and finite difference method (FDM), face difficulties in dealing with arbitrary domains. The paper introduces a novel nodal symmetric ghost method based on a variational formulation, which combines the advantages of FEM and FDM. The method employs bilinear finite elements on a structured mesh and provides a detailed implementation description. A rigorous a priori convergence rate analysis is also presented. The convergence rates are validated with many numerical experiments, in both one and two space dimensions.

中文翻译：

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一种基于变分公式和正则方格的节点幽灵方法，用于两个空间维度中任意域上的椭圆问题

本文重点介绍了使用狄利克雷和混合边界条件对椭圆偏微分方程 （PDE） 的数值解，特别是解决了不规则域带来的挑战。有限元法 （FEM） 和有限差分法 （FDM） 在处理任意域时都面临困难。本文介绍了一种基于变分公式的新型节点对称鬼法，该方法结合了有限元法和 FDM 的优点。该方法在结构化网格上采用双线性有限元，并提供了详细的实现描述。还提出了严格的先验收敛率分析。收敛速率在一维和二维空间维度上通过许多数值实验得到验证。