Multigrid Convergence

Let $ D:w>0$ be a bounded domain in $ \mathbb{R}^d$, described by a smooth positive weight function with $ \operatorname{grad}w_{\vert\partial D}\ne 0$ and denote by $ B^h_i$, $ i\in I$, the weighted extended b-spline basis for $ D$. Then, the Ritz-Galerkin system

$\displaystyle \int_D \operatorname{grad}u^h\,\operatorname{grad}B^h_i = \int_D f\,B^h_i,\quad i\in I, $

for the finite element approximation $ u^h = \sum_k c_k B^h_k$ of Poisson's equation can be solved by a multigrid iteration based on b-spline subdivision.

The convergence rate of the $ w$-cycle is independent of $ h$ if sufficiently many smoothing steps are used.


$ \square$ with U. Reif and J. Wipper, Numerische Mathematik 91 No. 2 (2002), 237-256.

$ \square$ Finite Element Methods with B-Splines, SIAM, 2003.

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