Optimal subspace correction methods for THB-splines

Ludwig Mitter

June 25, 2019, 3:30 p.m. S2 416-1

We present the analysis of $h$-robust subspace correction solvers for Isogeometric Analysis (IgA) schemes based on Truncated Hierarchical Basis (THB) splines.
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 method. 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. Doing so guarantees optimal computational complexity not only of this Local Multigrid Method, but of a general class of domain decomposition algorithms. We deduce a rigorous convergence analysis of abstract domain decomposition methods for THB-splines. Our work generalizes results of (Chen et al., 2012) and (Cho and Vazquez, 2018).