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A new discontinuous Galerkin method for Kirchhoff plates. (English) Zbl 1231.74416
Summary: A general framework of constructing ${C}^{0}$ discontinuous Galerkin (CDG) methods is developed for solving the Kirchhoff plate bending problem, following some ideas in P. Castillo et al. [SIAM J. Numer. Anal. 38, No. 5, 1676–1706 (2000; Zbl 0987.65111)] and B. Cockburn [ZAMM, Z. Angew. Math. Mech. 83, No. 11, 731–754 (2003; Zbl 1036.65079)]. The numerical traces are determined based on a discrete stability identity, which lead to a class of stable CDG methods. A stable CDG method, called the LCDG method, is particularly interesting in our study. It can be viewed as an extension to fourth-order problems of the LDG method studied in P. Castillo (2000; Zbl 0987.65111) and C. Cockburn (2003; Zbl 1036.65079). For this method, optimal order error estimates in certain broken energy norm and ${H}^{1}$-norm are established. Some numerical results are reported, confirming the theoretical convergence orders.
##### MSC:
 74S05 Finite element methods in solid mechanics 74K20 Plates (solid mechanics)