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 ${\mathcal {L}}^{2}$ 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 ${\mathcal {L}}^{2}$ norm. Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.

Keywords

Elliptic equation on (3D) surface / Isogeometric analysis / Discontinuous Galerkin method / Multiple patches / Non-conforming patches / The optimal convergence rate