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.

翻译：本文聚焦于具有狄利克雷和混合边界条件的椭圆偏微分方程数值求解，特别针对不规则区域带来的挑战。有限元法和有限差分法在处理任意区域时均面临困难。本文提出了一种基于变分公式的新型节点对称鬼点法，该方法融合了有限元法和有限差分法的优势。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。同时给出了严格的先验收敛速率分析。通过大量一维和二维空间数值实验验证了收敛速率。