Stabilized Continuous Linear Element Method for the Biharmonic Problems (Communications in Computational Physics, 2025)
In this paper, we introduce a new stabilized continuous linear elementmethod for solving biharmonic problems. Leveraging the gradient recovery operator, we reconstruct the discrete Hessian for piecewise continuous linear functions. Byadding a stability term to the discrete bilinear form, we bypass the need for the discretePoincaré inequality. We employ Nitsche's method for weakly enforcing boundary conditions. We establish well-posedness of the solution and derive optimal error estimatesin energy and $L^2$ norms. Numerical results are provided to validate our theoreticalfindings.
Keywords: Biharmonic problems; gradient recovery; superconvergence; linear finite element.