Multigrid Reduced in Time for Isogeometric Analysis

Abstract

Isogeometric Analysis [1] has become increasingly popular as an alternative to the Finite Element Method. Solving the resulting linear systems when adopting higher order B-spline basis functions remains a challenging task, as most (standard) iterative methods have a deteriorating preformance for higher values of the approximation order p.Recently, we succesfully applied p-multigrid methods to discretizations arising in IsogeometricAnalysis [2]. In contrast to h-multigrid methods, where each level of the multigrid hierarchycorresponds to a different mesh width h, the p-multigrid hierarchy is constructed based on different approximation orders. The residual equation is then solved at level p = 1, enabling the use of efficient solution techniques developed for low-order standard FEM. Numerical results show that the number of iterations needed for convergence is independent of both h and p when the p-multigrid method is enhanced with a smoother based on an Incomplete LU factorization with dual treshold (ILUT). However, a slight dependence on the number of patches has been observed for multipatch geometries.Since the resulting system matrix has a block structure in case of a multipatch geometry, weconsider the use of block ILUT as a smoother. Results indicate that the use of block ILUT can be an efficient alternative to ILUT on multipatch geometries within a heterogeneous HPC framework. Prelimenary results for p-multigrid methods adopting a block ILUT smoother will be presented in this talk. Furthermore, we investigate the use of alternative multigrid hierarchies, in particular when considering time-dependent problems.REFERENCES[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, Isogeometric Analysis: CAD, Finite Elements,NURBS, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanicsand Engineering, 194, 4135 - 4195, 2005[2] R.Tielen, M. Möller, D. Göddeke and C. Vuik, p-multigrid methods and their comparison toh-multigrid methods within Isogeometric Analysis, Computer Methods in Applied Mechanicsand Engineering, 372, 2020