Convergence Analysis and Comparison of the H- and P- Extensions with Mixed Finite Element

A mixed formulation with two main variables, based on the Ciarlet-Raviart technique, with 0 C continuity shape functions is employed for the solution of some types of biharmonic equations in 1-D. The continuous and discrete Babuška-Brezzi inf-sup conditions are established. The formulation is numerically tested for both the hand pextensions. The model problems involve the standard biharmonic equation, with variable bending stiffness (with both regular and irregular exact solutions), as well as, a more general biharmonic equation with lower order term, constant coefficients and complicated boundary conditions (with smooth exact solution), resulting from a gradient elasticity problem. The standard, quasi-optimal finite element error rates of convergence are numerically confirmed in all cases. The basic conclusion of the numerical experimentation is that the p-extension provides much better accuracy than the h-extension, for both main variables. For the irregular exact solution, the observed rate of convergence corresponding to the second variable (displacements) is always higher than the error rate for the first variable (bending moment). The latter may be theoretically explained via the particular structure of the bilinear functionals of the given formulation.