A parallel multigrid solver for multi-patch Isogeometric Analysis

Isogeometric Analysis (IgA) is a framework for setting up spline-based discretizations of partial differential equations, which has been introduced around a decade ago and has gained much attention since then. If large spline degrees are considered, one obtains the approximation power of a high-order method, but the number of degrees of freedom behaves like for a low-order method. One important ingredient to use a discretization with large spline degree, is a robust and preferably parallelizable solver. While numerical evidence shows that multigrid solvers with standard smoothers (like Gauss Seidel) does not perform well if the spline degree is increased, the multigrid solvers proposed by the authors and their co-workers proved to behave optimal both in the grid size and the spline degree. In the present paper, the authors want to show that those solvers are parallelizable and that they scale well in a parallel environment.Comment: The first author would like to thank the Austrian Science Fund (FWF) for the financial support through the DK W1214-04, while the second author was supported by the FWF grant NFN S117-0

Local multigrid solvers for adaptive isogeometric analysis in hierarchical spline spaces

We propose local multigrid solvers for adaptively refined isogeometric discretizations using (truncated) hierarchical B-splines ((T)HB-splines). Smoothing is only performed in or near the refinement areas on each level, leading to a computationally efficient solving strategy. We prove robust convergence of the proposed solvers with respect to the number of levels and the mesh sizes of the hierarchical discretization space under the assumption that the hierarchical mesh satisfies an admissibility condition, i.e., the number of interacting mesh levels is uniformly bounded. We also provide several numerical experiments. The main analytical tools are quasi-interpolators for THB-splines and the abstract convergence theory of subspace correction methods

Experiences with Fertilizer Taxes in Europe

A levy on nitrogen fertilizer is evaluated, using real life experiences in Austria, Finland and Sweden until 1995. In these countries such a levy system was introduced in 1986, 1976 and 1985, respectively. Rates varied from between 10% and 72% of the price of fertilizer. Price elasticity in these situations was estimated to vary between ‐ 0.1 and ‐ 0.5. In Austria the fertilizer industry at first did not pass the costs of the levy on to the farmers, but fertilizer use decreased by 15%, indicating that not only economic stimuli but also other aspects, such as a growing environmental awareness, influence fertilizer use as well. The decrease in fertilizer use in the three countries has probably led to a reduction of nitrogen load to the environment. Levy revenues were mainly used to support grain exports. Administration costs were low, about 0.75% of the tax revenues. It is concluded that a fertilizer tax as a policy instrument is not a perfect method, but that such economic instruments can be part of an effective policy mix to solve nitrogen problems.