Reparameterization and Adaptive Quadrature for the Isogeometric Discontinuous Galerkin Method.

We use the Poisson problem with Dirichlet boundary conditions to illustrate the complications that arise from using non-matching interface parameterizations within the framework of Isogeometric Analysis on a multi-patch domain, using discontinuous Galerkin (dG) techniques to couple terms across the interfaces. The dG-based discretization of a partial differential equation is based on a modified variational form, where one introduces additional terms that measure the discontinuity of the values and normal derivatives across the interfaces between patches. Without matching interface parameterizations, firstly, one needs to identify pairs of associated points on the common interface of the two patches for correctly evaluating the additional terms. We will use reparameterizations to perform this task. Secondly, suitable techniques for numerical integration are needed to evaluate the quantities that occur in the discretization with the required level of accuracy. We explore two possible approaches, which are based on subdivision and adaptive refinement, respectively.