P-multigrid methods for Isogeometric analysis

Isogeometric Analysis is a methodology that bridges the gap between Computer Aided Design (CAD) and the Finite Element Method (FEM) by adopting the building blocks used in CAD, namely Non-UniformRational B-Splines and B-splines, as a basis for FEM. The use of these high-order spline functions does not only lead to an accurate representation of the geometry, but has shown to be advantageous in many different fields of research. In order to obtain accurate numerical solutions, sufficiently fine meshes have to be considered which results in very large linear systems of equations. Furthermore, the condition numbers of the system matrices grow exponentially in the spline degree p, making the use of standard iterative solvers less efficient. Direct methods, on the other hand, might not be a viable alternative for large problem sizes, due to memory constraints and difficulties to parallelize. In recent years, the development of efficient iterative solvers for Isogeometric Analysis has therefore become an active field of research. For standard FEM, multigrid methods are known to be among the most efficient solvers for elliptic partial differential equations. The direct application of these methods to linear systems arising in Isogeometric Analysis results, however, in multigrid methods with deteriorating performance for higher values of the spline degree p, since the multigrid smoother becomes less and less effective in damping the error. This has led to the development of multigrid methods with non-standard smoothers. In this dissertation,we propose the use of p-multigridmethods as an alternative solution strategy. Within our p-multigrid method, the coarse grid correction is obtained at level p Æ 1, enabling the use of well-known solution methods for standard Lagrangian FEM (in particular h-multigrid methods). Furthermore, the support of the basis functions significantly reduces at level p Æ 1, thereby reducing the number of non-zero entries in the coarse grid operators. We analyze the performance of our p-multigrid method, adopting different smoothers, for single patch and multipatch geometries. In particular, we perform a spectral analysis to investigate the interplay between the coarse grid correction and smoothing procedure and obtain the asymptotic convergence rate of the p-multigrid method for a representative scenario. Numerical results (i.e., iteration numbers and CPU timings) are obtained for a variety of two- and three-dimensional benchmarks and compared to (state-of-the-art) h-multigrid methods to show the potential of p-multigrid methods in the context of Isogeometric Analysis. For time-dependent partial differential equations, we apply Multigrid Reduced in Time (MGRIT), which is a parallel-in-time method, on discretizations arising in Isogeometric Analysis. Here, MGRIT is successfully combined with a p-multigrid method to obtain an overall efficient method.