Local THB-Multigrid

Ludwig Mitter

Jan. 15, 2019, 3:30 p.m. S2 054

We present a $h$-robust multigrid solver for Isogeometric Analysis (IgA)
schemes based on Truncated Hierarchical Basis (THB) splines, which only
uses local relaxations.

Mesh refinement in IgA is more involved than in the finite element
method. In particular, we use THB-splines for localized meshes in our
adaptive IgA scheme. So far, the solution of the emergent large scale,
uniformly sparse linear systems has indeed been addressed (Hofreither et
al., 2017), but an authoritative theoretical analysis of these tailored
iterative solvers has been elusive.

We adapt the multigrid method of (Hofreither et al., 2017) inasmuch
relaxations are performed on a smaller number of degrees of freedom,
which are related to the local features of the adaptive scheme. We
deduce a rigorous convergence analysis of this Local Multigrid Method
based on the celebrated Xu-Zikatanov formula and a new THB-quasi
interpolant as proposed in (Buffa and Giannelli, 2015) and (Speleers and
Manni,2018). The topics of computational complexity, marking, refinement
and an application to the nested iteration scheme and the Stokes problem
will be addressed.