HDG Methods for Naghdi Arches

In this paper, we introduce and analyze a class of hybridizable discontinuous Galerkin methods for Naghdi arches. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to approximations to the transverse and tangential displacement and bending moment at the element boundaries. The error analysis of the methods is based on the use of a projection especially designed to fit the structure of the numerical traces of the method. This property allows to prove in a very concise manner that the projection of the errors is bounded in terms of the distance between the exact solution and its projection. The study of the influence of the stabilization function on the approximation is then reduced to the study of how they affect the approximation properties of the projection in a single element. Consequently, we prove that when polynomials of degree \(k\) are used, the methods converge with the optimal order of \(k+1\) for all the unknowns and that they are free from shear and membrane locking. Finally, we show that all the numerical traces converge with order \(2k+1\). Numerical experiments validating these results are shown.