In this work, we study the approximation properties of multi-patch dG-IgA methods, that apply the multipatch Isogeometric Analysis (IgA) discretization concept and the discontinuous Galerkin (dG) technique on the interfaces between the patches, for solving linear diffusion problems with diffusion coefficients that may be discontinuous across the patch interfaces. The computational domain is divided into non-overlapping sub-domains, called patches in IgA, where B-splines, or NURBS finite dimensional approximations spaces are constructed. The solution of the problem is approximated in every sub-domain without imposing any matching grid conditions and without any continuity requirements for the discrete solution across the interfaces. Numerical fluxes with interior penalty jump terms are applied in order to treat the discontinuities of the discrete solution on the interfaces. We provide a rigorous a priori discretization error analysis for problems set in 2d-and 3d-dimensional domains, with solutions belonging to W l,p , l ≥ 2, p ∈ (2d/(d + 2(l − 1)), 2]. In any case, we show optimal convergence rates of the discretization with respect to the dG -norm.

Isogeometric analysis (IgA) uses the same class of basis functions for both, representing the geometry of the computational domain and approximating the solution. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This multi-patch representation corresponds to a decomposition of the computational domain into non-overlapping subdomains also called patches in the geometrical framework. We will present discontinuous Galerkin (dG) methods that allow for discontinuities across the subdomain (patch) boundaries. The required interface conditions are weakly imposed by the dG terms associated with the boundary of the sub-domains. The construction and the corresponding discretization error analysis of such dG multi-patch IgA schemes will be given for heterogeneous diffusion model problems in volumetric 2d and 3d domains as well as on open and closed surfaces. The theoretical results are confirmed by numerous numerical experiments which have been performed in G+ + +SMO. The concept and the main features of the IgA library G+ + +SMO are also described.

We propose a new discontinuous Galerkin Isogeometric Analysis (IgA) technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with non-matching interfaces. 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-patch 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. Finally, the solution of this modified problem is approximated by a special discontinuous Galerkin IgA 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 rate of the method. The theoretical estimates are supported by numerical examples in two-and three-dimensional computational domains. Introduction.In the numerical solution of many realistic problems by means of Isogeometric Analysis (IgA), the whole computational (physical) domain Ω can often not be represented by a single volume patch that is the image of the parameter domain by a single, smooth and regular B-spline or NURBS map. In this case, it is necessary to perform a decomposition of the computational domain Ω into subdomains, in other words, to describe the domain Ω by multiple patches. Typical examples are complicated 3d domains, different diffusion coefficients, or even different mathematical models in different parts of the domain. Superior B-splines (NURBS, T-spline etc) finite dimensional spaces are used, in order to construct parametrizations for these subdomains [6]. It is typical for IgA that the same basis functions are used to approximate the solution of the problem under consideration, see [11] and [3]. Despite the advantages, that B-splines (NURBS etc) offer for the parametrization of the subdomains, some serious difficulties can arise, especially, when the subdomains topologically differ a lot from a cube. The segmentation procedure, that starts from the geometrical description of the corresponding surface patches, can lead us to non-compatible parametrizations of the geometry, meaning that the parametrized interfaces of adjusting subdomains are not identical after the volume segmentation, see, e.g., [12,21,23] for the discussion of isogeometric segmentation pipeline. In this paper, we call a non-watertight isogeometric segmentation also non-matching interface parametrization. The result of this phenomenon is the creation o...

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