A discontinuous Galerkin method for the Naghdi shell model

We propose a mixed discontinuous Galerkin method for the bending problem of Naghdi shell, and present an analysis for its accuracy. The error estimate shows that when components of the curvature tensor and Christoffel symbols are piecewise linear functions, 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 optimal rate of convergence, the triangulation should be properly refined in regions where the shell geometry changes dramatically. These are the results for a balanced method in which the primary displacement components and rotation components are approximated by discontinuous piecewise quadratic polynomials, while components of the scaled membrane stress tensor and shear stress vector are approximated by continuous piecewise linear functions. On elements that have edges on the free boundary of the shell, finite element space for displacement components needs to be enriched slightly, for stability purpose. Results on higher order finite elements are also included.