Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces
The Isogeometric Analysis (IGA), that was introduced by Hughes et al.  and has since been developed intensively, see also monograph , is a very suitable framework for representing and discretizing Partial Differential Equations (PDEs) on surfaces. We refer the reader to the survey paper by Dziuk and Elliot  where different finite element approaches to the numerical solution of PDEs on surfaces are discussed. Very recently, Dedner et al.  have used and analyzed the Discontinuous Galerkin (DG) finite element method for solving elliptic problems on surfaces. The IGA of second-order PDEs on surfaces has been introduced and numerically studied by Dede and Quarteroni  for the single-patch case. Brunero  presented some discretization error analysis of the DG-IGA applied to plane (2d) diffusion problems that carries over to plane linear elasticity problems which have recently been studied numerically in . Evans and Hughes  used the DG technology in order to handle no-slip boundary conditions and multi-patch geometries for IGA of Darcy-Stokes-Brinkman equations. The efficient generation of the IGA equations, their fast solution, and the implementation of adaptive IGA schemes are currently hot research topics. The use of DG technologies will certainly facilitate the handling of the multi-patch case.
KeywordsDiscontinuous Galerkin Isogeometric Analysis Discontinuous Galerkin Scheme NURBS Basis Function Discontinuous Galerkin Solution
The authors gratefully acknowledge the financial support of the research project NFN S117-03 by the Austrian Science Fund. Furthermore, the authors want to thank their colleagues A. Mantzaflaris, S. Tomar and W. Zulehner for fruitful discussions as well as for their help in the implementation in G+SMO.
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