Analysis of a discontinuous Galerkin method for the bending problem of Koiter shell

We present an analysis for a mixed finite element method for the bending problem of Koiter shell. We derive an error estimate showing that when the geometrical coefficients of the shell mid-surface satisfy certain conditions the finite element method has the optimal order of accuracy, which is uniform with respect to the shell thickness. Generally, the error estimate shows how the accuracy is affected by the shell geometry and thickness. It suggests that to achieve the optimal rate of convergence, the triangulation should be properly refined in regions where the shell geometry changes dramatically. The analysis is carried out for a balanced method in which the normal component of displacement is approximated by discontinuous piecewise cubic polynomials, while the tangential components are approximated by discontinuous piecewise quadratic polynomials, with some enrichments on elements affixed to the shell free boundary. Components of the membrane stress are approximated by continuous piecewise linear functions. We also include a theory of balanced higher order elements and a theory of a simpler lower order method.