Convergence of an Adaptive Mixed Finite Element Method for Kirchhoff Plate Bending Problems

Some reliable and efficient a posteriori error estimators are produced for a mixed finite element method (the Hellan–Herrmann–Johnson (H-H-J) method) for Kirchhoff plate bending problems (cf. [K. Hellan, Acta Polytech. Scand. Civil Engrg. Ser., 46 (1967), pp. 1–28], [L. Herrmann, J. Eng. Mech. Div. ASCE, 93 (1967), pp. 13–26], [C. Johnson, Numer. Math., 21 (1973), pp. 43–62]). Based on these results with $k=0,1$, where k denotes the polynomial order of the discrete moment-field space, an adaptive mixed finite element method (AMFEM) is set up, and its convergence and complexity are studied thoroughly. The key points of the theoretical analysis include achieving a discrete Helmholtz decomposition and a discrete inf-sup condition, which serve as the main tools in deducing the quasi-orthogonality of the moment field and the discrete reliability of the estimator. It is shown that the AMFEM is a contraction for the sum of the moment-field error in an energy norm and the scaled error estimator between two consecutive adaptive loops. Moreover, an estimate for the AMFEM's complexity via the number of elements is developed.