Discontinuous Galerkin Isogeometric Analysis of Elliptic Problems on Non-matching Interfaces

We propose a discontinuous Galerkin Isogeometric Analysis method for the numerical solution of elliptic diffusion problems on decompositions into volumetric patches with non-matching interfaces. Indeed, due to an incorrect segmentation procedure, it may happen that the interfaces of adjacent subdomains don’t coincide. In this way, gap regions, which are not present in the original physical domain, are created. In this paper, the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains. We apply a multi-domain approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem. The last formulation is further modified by deriving interface conditions without the presence of the solution in the gap. In particular, the gap terms in the interface conditions are replaced by Taylor expansions with respect the adjacent subdomain solutions. Finally, the solution of this modified problem is approximated by developing a discontinuous Galerkin Isogeometric Analysis technique. The ideas are illustrated on a model diffusion problem with discontinuous diffusion coefficients. We develop a rigorous theoretical framework for the proposed method clarifying the influence of the gap size onto the convergence of the method. The theoretical estimates are supported by numerical examples in twoand three-dimensional computational domains.