Isogeometric Analysis and Schwarz non-matching overlapping additive domain decomposition methods

Michel Bercovier

Jan. 23, 2015, 9 a.m. S2 416-2

The IGA paradigm for the discretisation of Partial Differential Equations (PDEs) leads to a rich interchange between Computer Aided Geometry for Design , Computational Geometry and Numerical Modelling based on PDEs.

In this talk we will review briefly the IGA paradigm and detail some of the interactions.

More specifically many objects are defined as CSG free form volumes constructs. Hence it is natural to consider Domain Decomposition methods as candidate solvers for ‘real life’ domains. Each atom of the CSG construct can be considered as a simple domain, where IGA is directly implemented. Thus based on CSG trees we can apply, for the solution of large PDEs problems on the global domain, the simplest Schwarz Additive Domain Decomposition Method (SADDM). We suppose that our primitive patches Ωi, i = 1, ..., n, are overlapping ( i.e. there is always a pair (i, j) such Ωi ∩ Ωj has a non void interior) and that the respective isoparametric transformations are NON-MATCHING: the reference grid and knots defining each physical domain are not related.

We give several examples illustrating the power of this approach: direct use of CGS primitives, local zooming instead of refinements, and parallelization for large problems. We show that there is no degradation of the powerful approximation properties of IGA when using non matching meshes.