Local Multigrid Methods in Adaptive Isogeometric Analysis with(T)HB-splines

Ludwig Mitter

Dec. 17, 2019, 3:30 p.m. S2 054

We present $h$-robust local multigrid solvers for Isogeometric Analysis (IgA) schemes based on (Truncated) Hierarchical Basis(THB) splines, which only use local relaxations.
Mesh refinement in IgA is more involved than in the finite element method. In particular, we use (T)HB-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., 2016), but an authoritative theoretical analysis of these tailored iterative solvers has been elusive. We adapt the multigrid method of (Hofreither et al.,2016) 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 (Xu, 1992) and a THB-quasi interpolant as proposed in (Speleers and Manni, 2016). We see that robust convergence is obtained, as soon as the relaxation operators $R_k$ on any(T)HB-spline level $k\in\{0,\ldots,N\},N\in\mathbb{N}$ satisfy the following norm equivalence for any tensor product B-Spline function $v_k$ of level $k\in\{0,\ldots,N\}$,\[ h^{-2}_k\lVert v_k\rVert_{L^2}\eqsim\lVert v_k\rVert^2_{R^{-1}_k}.\]We show, that the Jacobi-method satisfies this assumption and discuss the extension to the symmetric Gau\ss-Seidel method.