Convergence and performance of the h- and p-extensions with mixed finite element C0-continuity formulations, for tension and buckling of a gradient elastic beam

Mixed formulations with C0-continuity basis functions are employed for the solution of some types of one-dimensional fourth- and sixth-order equations, resulting from axial tension and buckling of gradient elastic beams, respectively. A basic characteristic of gradient elasticity type equations is the appearance of boundary layers in the higher-order derivatives of the displacements (e.g., in the stress fields). This is due to the small parameters (related to the size of the microstructure) entering the governing equations. The proposed mixed formulations are based on generalizations of the well-known Ciarlet–Raviart mixed method, where the new main variables are related to second-order (or fourth order, for the buckling problem) derivatives of the displacement field. The continuous and discrete Babusˇka–Brezzi inf–sup conditions are established. The mixed formulations are numerically tested for both the uniform h- and p-extensions. With regard to the axial tension problem, the standard quasi-optimal rates of convergence are numerically verified in all cases (i.e., algebraic rate of convergence for the h-extension and exponential rate for the p-extension). On the other hand, the h-extension observed convergence rates of the critical (buckling) load for the second model problem are slightly higher than the theoretical ones found in the literature (especially for polynomial order p = 1). The respective observed rates of convergence of the buckling load for the p-extension are still exponential.