Strengthened Cauchy-Schwarz and H\"older inequalities

We present some identities related to the Cauchy-Schwarz inequality in complex inner product spaces. A new proof of the basic result on the subject of Strengthened Cauchy-Schwarz inequalities is derived using these identities. Also, an analogous version of this result is given for Strengthened H\"older inequalities

Anisotropic finite elements for the Stokes problem: a posteriori error estimator and adaptive mesh

AbstractWe propose an a posteriori error estimator for the Stokes problem using the Crouzeix–Raviart/P0 pair. Its efficiency and reliability on highly stretched meshes are investigated. The analysis is based on hierarchical space splitting whose main ingredients are the strengthened Cauchy–Schwarz inequality and the saturation assumption. We give a theoretical proof of a method to enrich the Crouzeix–Raviart element so that the strengthened Cauchy constant is always bounded away from unity independently of the aspect ratio. An anisotropic self-adaptive mesh refinement approach for which the saturation assumption is valid will be described. Our theory is confirmed by corroborative numerical tests which include an internal layer, a boundary layer, a re-entrant corner and a crack simulation. A comparison of the exact error and the a posteriori one with respect to the aspect ratio will be demonstrated

Finite element approximation of multi-scale elliptic problems using patches of elements

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presente

A Bauer-Hausdorff Matrix Inequality

We present a biorthogonal process for two subspaces of C . Applying this process, we derive a matrix inequality, which generalizes the Bauer-Hausdorff inequality for vectors and includes the Wang-IP inequality for matrices. Meanwhile, we obtain its equivalent matrix inequality

Preconditioned discontinuous Galerkin method and convection-diffusion-reaction problems with guaranteed bounds to resulting spectra

This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of non-symmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection-)diffusion-reaction problems.Comment: 18 pages, 8 pages, and 1 figur

Finite element methods with patches and applications

Theoretical and numerical aspects of multi-scale problems are investigated. On one hand, mathematical analysis is done on a new method for numerically solving problems with multi-scale behavior using multiple levels of not necessarily nested grids. A particularly flexible multiplicative Schwarz method is presented, requiring no conformity between the meshes at the different scales. The relaxed iterative method consists in calculating successive corrections to the solution in regions where the variations of a problem are too strong to be captured by a coarse initial mesh. In these sub-domains patches of finite elements are applied. A priori and a posteriori error estimates are given and an exact spectral analysis of the iteration operator describing the algorithm is presented. Computational issues are addressed and numerical methods to obtain optimal convergence are given. Crucial implementation matters are discussed with special regard to usage of memory and CPU-time. On the other hand, the efficiency of the introduced correction method is demonstrated on Laplace model problems, either with changing Dirichlet-Neumann boundary conditions or in a polygonal domain with entrant corner. The regularity of the solutions is studied as well as the improvement of the convergence order in the mesh size using various sizes of patches. The correction algorithm is also applied to improve the accuracy in the simulation of the stress field in glacier modeling. A simple model to obtain the effective stress field in the ice mass of a glacier is presented and concluding results are obtained using patches in the regions where changes in the basal boundary conditions are involved

Adaptive algorithms for partial differential equations with parametric uncertainty

In this thesis, we focus on the design of efficient adaptive algorithms for the numerical approximation of solutions to elliptic partial differential equations (PDEs) with parametric inputs. Numerical discretisations are obtained using the stochastic Galerkin Finite Element Method (SGFEM) which generates approximations of the solution in tensor product spaces of finite element spaces and finite-dimensional spaces of multivariate polynomials in the random parameters. Firstly, we propose an adaptive SGFEM algorithm which employs reliable and efficient hierarchical a posteriori energy error estimates of the solution to parametric PDEs. The main novelty of the algorithm is that a balance between spatial and parametric approximations is ensured by choosing the enhancement associated with dominant error reduction estimates. Next, we introduce a two-level a posteriori estimate of the energy error in SGFEM approximations. We prove that this error estimate is reliable and efficient. Then we provide a rigorous convergence analysis of the adaptive algorithm driven by two-level error estimates. Four different marking strategies for refinement of stochastic Galerkin approximations are proposed and, in particular, for two of them, we prove that the sequence of energy errors computed by associated algorithms converges linearly. Finally, we use duality techniques for the goal-oriented error estimation in approximating linear quantities of interest derived from solutions to parametric PDEs. Adaptive enhancements in the proposed algorithm are guided by an innovative strategy that combines the error reduction estimates computed for spatial and parametric components of corresponding primal and dual solutions. The performance of all adaptive algorithms and the effectiveness of the error estimation strategies are illustrated by numerical experiments. The software used for all experiments in this work is available online

Multiskalen-Verfahren für Konvektions-Diffusions Probleme

In dieser Arbeit werden erstmalig über einen zur Nichtstandardform gehörenden Erzeugendensystem-Ansatz robuste Wavelet-basierte Multiskalen-Löser für allgemeine zweidimensionale stationäre Konvektions-Diffusions-Probleme entworfen und praktisch umgesetzt. Für Multiskalen-Verfahren, die lediglich direkte Unterraumzerlegungen verwenden, ist es im allgemeinen nicht mehr möglich, zugehörige Multiskalen-Glätter zu konstruieren, die im Grenzfall sehr starker Konvektion auf jeder Skala zu einem direkten Löser entarten. Als eine Möglichkeit zur Konstruktion robuster Multiskalen-Methoden bleibt die Wahl der Multiskalen-Zerlegungen selbst. Es ist sicherzustellen, dass man sowohl hinsichtlich der singulären Störung stabile Grobgitter- probleme als auch bezüglich der Maschenweite stabile Unterraum- zerlegungen erhält. Gleichzeitig muss der Aspekt der approximativen Gauss-Elimination beachtet werden, der durch das Zusammenspiel matrixabhängiger Prolongationen und Restriktionen mit einer hierarchischen Basis Zerlegung gegeben ist. Um alle diese Forderungen zu erfüllen, wird zunächst ausgehend von geometrischen Vergröberungen ein allgemeines Petrov--Galerkin Multiskalen-Konzept entwickelt, bei dem die Zerlegungen auf der Ansatz- und Testseite unterschiedlich sind. Es werden matrixabhängige Prolongationen, die von robusten Mehrgitter-Techniken her bekannt sind, verwendet, zusammen mit Wavelet-artigen und hierarchischen Multiskalen-Zerlegungen der Ansatz- und Testräume bezüglich des feinsten Gitters. Die Kernidee bei den vorgeschlagenen Verfahren ist, jeweils einen der Komplementräume auf der Ansatz- oder Testseite hierarchisch zu wählen, um zusammen mit einer problemabhängigen Vergröberung auf der anderen Seite physikalisch sinnvolle Grobgitter- diskretisierungen und gleichzeitig einen approximativen Eliminations- effekt zu erreichen. Die Komplementräume auf der entsprechend anderen Seite werden hingegen Wavelet-artig aufgespannt, was insbesondere zu einer Stabilisierung des Verfahrens bezüglich der Abhängigkeit von der Maschenweite der Diskretisierung führt. Mit den weiterhin entwickelten AMGlet-Zerlegungen, die auf rein algebraischen Prinzipien beruhen, gelingt es, geometrisch orientierte Tensorprodukt- Konstruktionen, die für separable Probleme erfolgreich sind, zu verlassen, um schwierige nichtseparable Aufgaben in unter Umständen kompliziert berandeten Gebieten behandeln zu können. Dies eröffnet darüberhinaus auch den Übergang von Modellproblemen hin zu praxisnahen Fragestellungen. Unterschiedliche numerische Beispiele zeigen, dass man durch die vorgeschlagenen Konstruktionen zu verallgemeinerten Hierarchische Basis Mehrgitter-Verfahren mit robusten Konvergenzeigenschaften gelangt