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- Veronika Pillwein, Stefan Takacs, Bruno Buchberger, Bert Jüttler, Ulrich Langer, Esther Klann +7 others
- 2010

The numerical treatment of systems of partial differential equations (PDEs) is of great interest as many problems from applications belong to that class. Also the optimality system of optimal control problems that is discussed in this work has such a structure. These problems are not elliptic and therefore both the construction of an efficient numerical… (More)

In this paper we consider multigrid methods for solving saddle point problems. The choice of an appropriate smoothing strategy is a key issue in this case. Here we focus on the widely used class of collective point smoothers. These methods are constructed by a point-wise grouping of the unknowns leading to, e.g., collective Richardson, Jacobi or… (More)

- Stefan Takacs, Walter Zulehner, Michael Kuhn, Michael Jung, Ulrich Langer, Sergei V. Nepomnyaschikh +2 others
- 1997

For iterative solvers, besides convergence proofs that may state qualitative results for some classes of problems, straightforward methods to compute (bounds for) convergence rates are of particular interest. A widely-used straightforward method to analyze the convergence of numerical methods for solving discretized systems of partial differential equations… (More)

In this paper we consider the convergence theory for an all-at-once multigrid method for a distributed optimal control problem. Such an analysis has been recently done, see [11]. Here, we give a new proof which is based on a more straightforward approach. The main benefit of this new approach is the possibility to extend the analysis to domains where full… (More)

- Stefan Takacs, Thomas Takacs, Michael Jung, Ulrich Langer, Sergei V. Nepomnyaschikh, Ralf Pfau +1 other
- 2015

In this paper, we will give approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the… (More)

The fast solution of linear systems arising from an isogeometric dis-cretization of a partial differential equation is of great importance for the practical use of Isogeometric Analysis. For classical finite element discretizations, multigrid methods are well known to be fast solvers showing optimal convergence behavior. However, if a geometric multigrid… (More)

- Clemens Hofreither, Stefan Takacs, Walter Zulehner, Michael Jung, Ulrich Langer, Sergei V. Nepomnyaschikh +2 others
- 2015

The fast solution of linear systems arising from an isogeometric dis-cretization of a partial differential equation is of great importance for the practical use of Isogeometric Analysis. For classical finite element discretizations, multigrid methods are well known to be fast solvers showing optimal convergence behavior. However, if a geometric multigrid… (More)