21 
A note on anisotropic interpolation error estimates for isoparametric quadrilateral finite elementsApel, Th. 30 October 1998 (has links) (PDF)
Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions.
The case of affine elements (parallelepipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multilinear map leads to estimates for certain classes of more general, isoparametric elements.
As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.

22 
Twopoint boundary value problems with piecewise constant coefficients: weak solution and exact discretizationWindisch, G. 30 October 1998 (has links) (PDF)
For twopoint boundary value problems in weak formulation with piecewise constant coefficients and piecewise continuous righthand side functions we derive a representation of its weak solution by local Green's functions. Then we use it to generate exact threepoint discretizations by Galerkin's method on essentially arbitrary grids. The coarsest possible grid is the set of points at which the piecewise constant coefficients and the right hand side functions are discontinuous. This grid can be refined to resolve any solution properties like boundary and interior layers much more correctly. The proper basis functions for the Galerkin method are entirely defined by the local Green's functions. The exact discretizations are of completely exponentially fitted type and stable. The system matrices of the resulting tridiagonal systems of linear equations are in any case irreducible Mmatrices with a uniformly bounded norm of its inverse.

23 
Variable preconditioning procedures for elliptic problemsJung, M., Nepomnyaschikh, S. V. 30 October 1998 (has links) (PDF)
For solving systems of grid equations approximating elliptic boundary value problems a method of constructing variable preconditioning procedures is presented. The main purpose is to discuss how an efficient preconditioning iterative procedure can be constructed in the case of elliptic problems with disproportional coefficients, e.g. equations with a large coefficient in the reaction term (or a small diffusion coefficient). The optimality of the suggested technique is based on fictitious space and multilevel decom position methods. Using an additive form of the preconditioners, we intro duce factors into the preconditioners to optimize the corresponding conver gence rate. The optimization with respect to these factors is used at each step of the iterative process. The application of this technique to twolevel $p$hierarchical precondi tioners and domain decomposition methods is considered too.

24 
A new method for computing the stable invariant subspace of a real Hamiltonian matrix or Breaking Van Loans curse?Benner, P., Mehrmann, V., Xu., H. 30 October 1998 (has links) (PDF)
A new backward stable, structure preserving method of complexity
O(n^3) is presented for computing the stable invariant subspace of
a real Hamiltonian matrix and the stabilizing solution of the
continuoustime algebraic Riccati equation. The new method is based
on the relationship between the invariant subspaces of the
Hamiltonian matrix H and the extended matrix /0 H\ and makes use
\H 0/
of the symplectic URVlike decomposition that was recently
introduced by the authors.

25 
Rankrevealing topdown ULV factorizationsBenhammouda, B. 30 October 1998 (has links) (PDF)
Rankrevealing ULV and URV factorizations are useful tools to determine the rank and to compute bases for nullspaces of a matrix. However, in the practical ULV (resp. URV ) factorization each left (resp. right) null vector is recomputed from its corresponding right (resp. left) null vector via triangular solves. Triangular solves are required at initial factorization, refinement and updating. As a result, algorithms based on these factorizations may be expensive, especially on parallel computers where triangular solves are expensive. In this paper we propose an alternative approach. Our new rankrevealing ULV factorization, which we call ¨topdown¨ ULV factorization ( TDULV factorization) is based on right null vectors of lower triangular matrices and therefore no triangular solves are required. Right null vectors are easy to estimate accurately using condition estimators such as incremental condition estimator (ICE). The TDULV factorization is shown to be equivalent to the URV factorization with the advantage of circumventing triangular solves.

26 
The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edgesApel, Th., Nicaise, S. 30 October 1998 (has links) (PDF)
This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones.

27 
Efficient time step parallelization of full multigrid techniquesWeickert, J., Steidten, T. 30 October 1998 (has links) (PDF)
This paper deals with parallelization methods for timedependent
problems where the time steps are shared out among the
processors. A Full Multigrid technique serves as solution
algorithm, hence information of the preceding time step and of
the coarser grid is necessary to compute the solution at each new
grid level. Applying the usual extrapolation formula to process
this information, the parallelization will not be very efficient.
We developed another extrapolation technique which causes a much
higher parallelization effect. Test examples show that no
essential loss of exactness appears, such that the method
presented here shall be wellapplicable.

28 
Local inequalities for anisotropic finite elements and their application to convectiondiffusion problemsApel, Thomas, Lube, Gert 30 October 1998 (has links) (PDF)
The paper gives an overview over local inequalities for anisotropic simplicial Lagrangian finite elements. The main original contributions are the estimates for higher derivatives of the interpolation error, the formulation of the assumptions on admissible anisotropic finite elements in terms of geometrical conditions in the threedimensional case, and an anisotropic variant of the inverse inequality. An application of anisotropic meshes in the context of a stabilized Galerkin method for a convectiondiffusion problem is given.

29 
NavierStokes equations as a differentialalgebraic systemWeickert, J. 30 October 1998 (has links) (PDF)
Nonsteady NavierStokes equations represent a differentialalgebraic system of strangeness index one after any spatial discretization. Since such systems are hard to treat in their original form, most approaches use some kind of index reduction. Processing this index reduction it is important to take care of the manifolds contained in the differentialalgebraic equation (DAE). We investigate for several discretization schemes for the NavierStokes equations how the consideration of the manifolds is taken into account and propose a variant of solving these equations along the lines of the theoretically best index reduction. Applying this technique, the error of the time discretisation depends only on the method applied for solving the DAE.

30 
Numerical methods for solving openloop non zerosum differential Nash games / Numerische Methoden zur Lösung von OpenLoopNichtNullsummenDifferentialNashSpielenCalà Campana, Francesca January 2021 (has links) (PDF)
This thesis is devoted to a theoretical and numerical investigation of methods to solve openloop non zerosum differential Nash games. These problems arise in many applications, e.g., biology, economics, physics, where competition between different agents appears. In this case, the goal of each agent is in contrast with those of the others, and a competition game can be interpreted as a coupled optimization problem for which, in general, an optimal solution does not exist. In fact, an optimal strategy for one player may be unsatisfactory for the others. For this reason, a solution of a game is sought as an equilibrium and among the solutions concepts proposed in the literature, that of Nash equilibrium (NE) is the focus of this thesis. The building blocks of the resulting differential Nash games are a dynamical model with different control functions associated with different players that pursue noncooperative objectives. In particular, the aim of this thesis is on differential models having linear or bilinear statestrategy structures. In this framework, in the first chapter, some wellknown results are recalled, especially for noncooperative linearquadratic differential Nash games. Then, a bilinear Nash game is formulated and analysed. The main achievement in this chapter is Theorem 1.4.2 concerning existence of Nash equilibria for noncooperative differential bilinear games. This result is obtained assuming a sufficiently small time horizon T, and an estimate of T is provided in Lemma 1.4.8 using specific properties of the regularized NikaidoIsoda function. In Chapter 2, in order to solve a bilinear Nash game, a semismooth Newton (SSN) scheme combined with a relaxation method is investigated, where the choice of a SSN scheme is motivated by the presence of constraints on the players’ actions that make the problem nonsmooth. The resulting method is proved to be locally convergent in Theorem 2.1, and an estimate on the relaxation parameter is also obtained that relates the relaxation factor to the time horizon of a Nash equilibrium and to the other parameters of the game. For the bilinear Nash game, a Nash bargaining problem is also introduced and discussed, aiming at determining an improvement of all players’ objectives with respect to the Nash equilibrium. A characterization of a bargaining solution is given in Theorem 2.2.1 and a numerical scheme based on this result is presented that allows to compute this solution on the Pareto frontier. Results of numerical experiments based on a quantum model of two spinparticles and on a population dynamics model with two competing species are presented that successfully validate the proposed algorithms. In Chapter 3 a functional formulation of the classical homicidal chauffeur (HC) Nash game is introduced and a new numerical framework for its solution in a timeoptimal formulation is discussed. This methodology combines a Hamiltonian based scheme, with proximal penalty to determine the time horizon where the game takes place, with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed endtime. The resulting numerical optimization scheme has a bilevel structure, which aims at decoupling the computation of the endtime from the solution of the pursuitevader game. Several numerical experiments are performed to show the ability of the proposed algorithm to solve the HC game. Focusing on the case where a collision may occur, the time for this event is determined. The last part of this thesis deals with the analysis of a novel sequential quadratic Hamiltonian (SQH) scheme for solving openloop differential Nash games. This method is formulated in the framework of Pontryagin’s maximum principle and represents an efficient and robust extension of the successive approximations strategy in the realm of Nash games. In the SQH method, the HamiltonPontryagin functions are augmented by a quadratic penalty term and the NikaidoIsoda function is used as a selection criterion. Based on this fact, the key idea of this SQH scheme is that the PMP characterization of Nash games leads to a finitedimensional Nash game for any fixed time. A class of problems for which this finitedimensional game admits a unique solution is identified and for this class of games theoretical results are presented that prove the wellposedness of the proposed scheme. In particular, Proposition 4.2.1 is proved to show that the selection criterion on the NikaidoIsoda function is fulfilled. A comparison of the computational performances of the SQH scheme and the SSNrelaxation method previously discussed is shown. Applications to linearquadratic Nash games and variants with control constraints, weighted L1 costs of the players’ actions and tracking objectives are presented that corroborate the theoretical statements. / Diese Dissertation handelt von eine theoretischen und numerischen Untersuchung von Methoden zur Lösung von OpenLoopNichtNullsummenDifferentialNashSpielen. Diese Probleme treten in vielen Anwendungen auf, z.B., Biologie, Wirtschaft, Physik, in denen die Konkurrenz zwischen verschiedenen Wirkstoffen bzw. Agenten auftritt. In diesem Fall steht das Ziel jedes Agenten im Gegensatz zu dem der anderen und ein Wettbewerbsspiel kann als gekoppeltes Optimierungsproblem interpretiert werden. Im Allgemeinen gibt es keine optimale Lösung für ein solches Spiel. Tatsächlich kann eine optimale Strategie für einen Spieler für den anderen unbefriedigend sein. Aus diesem Grund wird ein Gle ichgewicht eines Spiels als Lösung gesucht, und unter den in der Literatur vorgeschlagenen Lösungskonzepten steht das NashGleichgewicht (NE) im Mittelpunkt dieser Arbeit. ...

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