Block ILU smoothers for p-multigrid methods in Isogeometric Analysis

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Isogeometric Analysis (IgA) is an extension of the more well known Finite Element Method (FEM). It allows for more accurate descriptions of boundary value problems on irregular domains. However, many of the traditional iterative solution strategies that are known to work well in FEM do not show the same behavior in IgA, especially for increasing order of basis functions p.

A method shown to have fast convergence in this situation is a p-multigrid method with a smoother based on a Block ILUT factorization. Most of the blocks of this factorization are efficiently calculated. The same holds for the smoothing steps.

It is therefore our objective to make changes to the Block ILUT smoother. Inspiration is taken from methods where ILU factorizations are constructed using a fixed-point iteration. We combine these with the existing Block ILUT smoother.
This ultimately leads to two new proposed methods we will call Block Fixed-point ILU and Block ParILUT.

The existing methods as well as the newly suggested methods are tested and compared on computational costs of the factorization, the number of nonzero entries in this factorization and the number of multigrid iterations needed
to reach convergence, if these factorizations are to be used as smoother. The benchmark used for these tests is a convection diffusion reaction (CDR) equation on a multipatch geometry with 4, 16 or 64 patches.