Stabilized Continuous Linear Element Method for the Biharmonic Problems

Abstract

In this paper, we introduce a new stabilized continuous linear element method for solving biharmonic problems. Leveraging the gradient recovery operator, we reconstruct the discrete Hessian for piecewise continuous linear functions. By adding a stability term to the discrete bilinear form, we bypass the need for the discrete Poincaré inequality. We employ Nitsche's method for weakly enforcing boundary conditions. We establish well-posedness of the solution and derive optimal error estimates in energy and $L^2$ norms. Numerical results are provided to validate our theoretical findings.