Towards $C^0$ finite element methods for fourth-order elliptic equation. Part I: general boundary conditions

This paper is part of a series developing $C^0$ finite element methods for fourth-order elliptic equations on polygonal domains. Here, we investigate how boundary conditions influence the design of effective $C^0$ schemes, specifically focusing on equations without lower-order terms, namely the biharmonic equation. We propose a modified mixed formulation that decomposes the problem into a system of Poisson equations, where the number of equations depends on both the largest interior angle and the boundary conditions on its two adjacent sides. In contrast to the naive mixed formulation, which involves only two Poisson problems, the proposed approach guarantees convergence to the true solution for arbitrary polygonal domains and general boundary conditions, including Navier, Neumann, and mixed boundary conditions. $C^0$ finite element algorithms are developed, rigorous error estimates are established, and numerical experiments are presented to demonstrate the well-posedness and effectiveness of the proposed method.