Discontinuous Galerkin Isogeometric Analysis of elliptic problems on segmentations with non-matching interfaces

Isogeometric analysis on non-matching segmentation: discontinuous Galerkin techniques and efficient solvers

“…As a next step, we need to assign the points x g2 ∈ F g2 to the points x g1 ∈ F g1 . We follow the same ideas as in [16] and [17]. Since F g2 is an simple face and F g1 is a B-spline surface, due to the fact that it is the the image of some face ∂ Ω under the mapping Φ * 1 , we construct a parametrization of F g1 , lets say Φ g21 : F g2 → F g1 , which is one-to-one and defined as 20) where n Fg2 is the unit normal vector on F g2 , see Fig.…”

“…1(c), and ζ g is a B-spline function. The parametrization Φ g21 defined in (2.20) helps us to assign the diametrically opposite points located on ∂Ω g12 , see discussion in [16] and [17]. We are only interested in small gap regions, see below (2.23), and, thereby, if n Fg1 is the unit normal vector on F g1 , we can suppose that n Fg2 ≈ −n Fg1 .…”

“…Although dG IgA methods for decompositions including gap regions have been presented by the authors in [16] and [17], we repeat here the main parts for the completeness of the current paper. We present our analysis for the case of two patches described in Subsection 2.7, see also Fig.…”

“…We need to derive new conditions on the interior faces of the gap/overllaping regions and in order to construct the numerical fluxes and to ensure communication of the discrete problems patch-wise problems. We extent the ideas presented in [16,17], where discontinuous Galerkin (dG) methods for IgA have been developed on decompositions including only gaps. In particular, Taylor expansions are also used on arXiv:1610.03634v1 [math.NA] 12 Oct 2016 the interior faces of the overlapping regions, to approximate the normal fluxes on the overlapping boundaries.…”

Discontinuous Galerkin isogeometric analysis for segmentations generating overlapping regions

“…The estimate (3.38) can be shown using trace inequality and the estimates (3.37), see details in Lemma 10 in [22]. See also [16] and [14]. Main error estimate The estimate given in (3.39) concerns the distance between the DG-IGA solution u * h ∈ V B and the solution u * ∈ H (T * H (Ω)) of the problems in (3.4).…”