Rot-div mixed finite element method of two dimensional Hodge Laplacian problem

We develop a novel mixed method for addressing two-dimensional Laplacian problem with Dirichlet boundary conditions, which is recast as a rot-div system of three first-order equations. We have established the well-posedness of this new method and presented the a priori error estimates. The numerical applications of Bercovier-Engelman and Ruas test cases are developed, assessing the effectiveness of the proposed rot-div mixed method. Additionally, the efficiency of the proposed mixed method is demonstrated for typical finite elements, testing the optimal convergence rate and comparing the results with analytical solutions for all unknowns and the rotation and divergence of u. Our mixed method easily generalizes to electric and magnetic boundary conditions, and mixed boundary conditions.