Multipatch Discontinuous Galerkin Isogeometric Analysis

Abstract

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.

Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries

“…We recall results from [18] concerning basis functions and rewrite them in a way which is convenient for their extension to the multi-patch case. Amongst others, we present the specific basis from [18] for bi-degree (4, 4) with respect to spline instead of Bézier coefficients and introduce a new similar specific basis for bi-degree (3,3). In addition, we present modified functions near to the vertices of the common interface of the two-patch domain.…”

Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces

“…Alternatively, multi-patches can also be coupled via interior penalty Galerkin methods. In our earlier articles, see e.g., [13,12,15], we analyzed the multi-patch discontinuous Galerkin IGA (dGIGA) for diffusion and biharmonic problems and presented several convincing numerical results for conforming domains with matching meshes. However, in this paper, we will generalize the analysis to include non-matching meshes with jumping diffusion coefficients across patch boundaries and present a priori error estimates for diffusion problems.…”

Space-Time Multipatch Discontinuous Galerkin Isogeometric Analysis for Parabolic Evolution Problems

“…In most industrial applications, the computational domains involve several domains or patches. Several assembling strategies involving multiple patches have been presented in IgA including mortar methods [6] and discontinuous Galerkin (dG) methods [18].…”

“…In [18], the dG-IgA was presented where the approximation estimates are valid for each patch including non-matching meshes. Using these approximation results for B-splines or NURBS together with the elliptic-ity of the discrete bilinear form a h (·, ·) with respect to a discrete norm · h with boundedness and consistency results, asymptotically optimal discretization error estimate in the discrete dG-IgA norm · h were obtained.…”