Discontinuous Galerkin Methods for Isogeometric Analysis for Elliptic Equations on Surfaces

Abstract

We propose a method that combines isogeometric analysis (IGA) with the discontinuous Galerkin (DG) method for solving elliptic equations on 3-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted across the patch interfaces to glue the multiple patches, while the traditional IGA, which is very suitable for solving partial differential equations (PDEs) on (3D) surfaces, is employed within each patch. Our method takes advantage of both IGA and the DG method. Firstly, the time-consuming steps in mesh generation process in traditional finite element analysis (FEA) are no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed by knot insertion and order-elevation (Farin, in Curves and surfaces for CAGD, 2002). Secondly, our method can easily handle the cases with non-conforming patches and different degrees across the patches. Moreover, due to the geometric flexibility of IGA basis functions, especially the use of multiple patches, we can get more accurate modeling of more complex surfaces. Thus, the geometrical error is significantly reduced and it is, in particular, eliminated for all conic sections. Finally, this method can be easily formulated and implemented. We generally describe the problem and then present our primal formulation. A new ideology, which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain (the former is easier to get), is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form, and the error analysis under both the DG norm and the L 2 norm. The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the L 2 norm. Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.

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

“…2 (right). An analogous condition is required for v (4) . Examples of two-patch domains, which violate one of these three conditions, are visualized in Fig.…”

Foundations of the blended isogeometric discontinuous Galerkin (BIDG) method

“…The BIDG method seamlessly combines exact geometric design (CAD) with high-order accurate analysis (DGFEM), as schematically shown in figure 1. Some groundwork for BIDG-type methods is provided in [3,6,47,55,57], with patchbased methods recently being presented in [40,41,72] for elliptic problems. Also, please note the remark at the end of the appendix for a brief discussion of what is exactly meant by "isogeometric" throughout this paper.…”

Multipatch Discontinuous Galerkin Isogeometric Analysis

“…We will also consider multipatch dG IgA of diffusion problems of the form (1) on sufficiently smooth, open and closed surfaces Ω in R 3 , where the gradient ∇ must now be replaced by the surface gradient ∇ Ω , see, e.g., [18], and the patches Ω i , into which Ω is decomposed, are now images of the parameter domain Ω = (0, 1) 2 by the mapping Φ i : Ω ⊂ R 2 → Ω i ⊂ R 3 . The case of matching meshes was considered and analyzed in [28] by two of the co-authors, see also [35] for a similar work. It is clear that the results for non-matching spaces and for mesh grading presented here for the volumetric (2d and 3d) case can be generalized to diffusion problems on open and closed surfaces.…”

“…with t := min{s, k}, provided that the solution u of our surface diffusion problem (35) belongs to H 1+s (T H (Ω )) = W 1+s,2 (T H (Ω )) with some s > 1/2. In the case t = k, estimate (39) yields the convergence rate O(h k ) with respect to the dG norm, whereas the Aubin-Nitsche trick provides the faster rate O(h k+1 ) in the L 2 norm.…”