A Unified Variational Framework for Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions

This paper presents a unified weak formulation for elliptic partial differential equations of the form Formula: see text subject to mixed boundary conditions consisting of prescribed Dirichlet data on a portion of the boundary and Neumann data on the remainder. By multiplying the strong form with finite element test functions and applying integration by parts, we derive a solvable variational problem that naturally incorporates both types of boundary conditions. The proposed formulation provides a direct pathway to numerical discretization, improves implementation efficiency, and is fully compatible with standard finite element procedures. Numerical applicability on general polygonal domains is emphasized, ensuring accurate and reliable simulations for practical boundary value problems.