Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces

摘要

We have developed and analyzed a new method for the numerical approximation of diffusion problems on open and closed surfaces by combining the discontinuous Galerkin technique with isogeometric analysis. We refer to our approach as the Discontinuous Galerkin Isogeometric Analysis (DG-IGA). In our DG approach we allow discontinuities only across the boundaries of the patches, into which the computational domain is decomposed, and enforce the interface conditions in the DG framework. For simplicity of presentation, we assume that the meshes are matching across the patches, and the solution u is at least patch-wise in H~(1+s), i.e. u ∈ H~(1+s)(T_H), with some s ＞ 1/2. The cases of non-matching meshes and low-regularity solution, that are technically more involved and that were investigated, e.g., by Di Pietro and Ern, will be considered in a forthcoming paper. The parallel solution of the DG-IGA equations can efficiently be performed by Domain Decomposition (DD) solvers like the IETI technique proposed by Kleiss et al. , see also for other DD solvers. The construction and analysis of efficient solution strategies is currently a hot research topic since, beside efficient generation techniques, the solvers are the efficiency bottleneck in large-scale IGA computations.