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We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch's framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble's framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches. [ABSTRACT FROM AUTHOR] more...

We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor–Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests. [ABSTRACT FROM AUTHOR] more...

In this paper, we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we derive continuous norms that allow the analysis of well-posedness and subsequently the derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth-order problem to be solved. For this reason, we use isogeometric analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss--Seidel smoother as well as a new macro Gauss--Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree. [ABSTRACT FROM AUTHOR] more...

In this paper we consider multiple saddle point problems with block tri diagonal Hessian in a Hilbert space setting. Well-posedness and the related issue of preconditioning are discussed. We give a characterization of all block structured norms which ensure well-posedness of multiple saddle point problems as a helpful tool for constructing block diagonal preconditioners. We subsequently apply our findings to a general class of PDE-constrained optimal control problems containing a regularization parameter a and derive a-robust preconditioners for the corresponding optimality systems. Finally, we demonstrate the generality of our approach with two optimal control problems related to the heat and the wave equation, respectively. Preliminary numerical experiments support the feasibility of our method. [ABSTRACT FROM AUTHOR] more...

The importance of Schur-complement-based preconditioners is well established for classical saddle point problems in |$\mathbb{R}^N \times \mathbb{R}^M$|⁠. In this paper we extend these results to multiple saddle point problems in Hilbert spaces |$X_1\times X_2 \times \cdots \times X_n$|⁠. For such problems with a block tridiagonal Hessian and a well-defined sequence of associated Schur complements, sharp bounds for the condition number of the problem are derived, which do not depend on the involved operators. These bounds can be expressed in terms of the roots of the difference of two Chebyshev polynomials of the second kind. If applied to specific classes of optimal control problems the abstract analysis leads to new existence results as well as to the construction of efficient preconditioners for the associated discretized optimality systems. [ABSTRACT FROM AUTHOR] more...