C0 Interior Penalty Methods (C0IPM) are discontinuous Galerkin methods that can overcome the shortcomings of the classical approaches, such as C1 finite elements, nonconforming finite elements, and a mixed formulation. They are effective for fourth order elliptic boundary value problems and use k-th polynomial Lagrange triangular finite element for k ≥ 2. They are also useful for capturing smooth solutions and are much simpler than C1 finite elements.  

• S.C. Brenner, J., L.-Y. Sung, and Z. Tan. 'C0 interior penalty methods for an elliptic distributed optimal control problem with general tracking and pointwise state constraints.' Computers & Mathematics with Applications (2024)

The least-squares finite element (LSFEM) is to minimize some norm of the residual of a problem. A least-squares based method always has a unique minimizer. LSFEM approach provides the independent choice of finite elements for each variable. It also always produces a symmetric positive definite linear discrete system. Moreover, once the existence and uniqueness of the solution for the continuous problem is shown, that of the discrete solution for the discrete problem is guaranteed. However, the transformed first-order system has the drawback of enlarging the size of the system.

Solving a second order problem directly with the least squares approach results in a fourth-order linear system that requires far more smoothness of the solution than first order problems. Thus, the second order problem is transformed  to a first-order system by proposing new variable.

To handle domain singularity, we apply the weighted-norm minimization technique to overcome the difficulties in dealing with the edge and corner singularities in a polyhedral domain. A weighted Poincaré-type inequality has been proved with edge and corner singularities on the domain.

• J. and E. Lee, 'Weighted norm least squares finite element method for Poisson equation in a polyhedral domain.' Journal of Computational and Applied Mathematics (2016)