A Parallel Space-Time Domain Decomposition Method for Some
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Optimal Additive Schwarz Methods for the Hp-BEM: the Hypersingular Integral Operator in 3D on Locally Refined Meshes
Computers and Mathematics with Applications 70 (2015) 1583–1605 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Optimal additive Schwarz methods for the hp-BEM: The hypersingular integral operator in 3D on locally refined meshes T. Führer a, J.M. Melenk b,∗, D. Praetorius b, A. Rieder b a Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Vicuña Mackenna 4860, Santiago, Chile b Technische Universität Wien, Institut für Analysis und Scientific Computing, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria article info a b s t r a c t Article history: We propose and analyze an overlapping Schwarz preconditioner for the p and hp boundary Available online 6 August 2015 element method for the hypersingular integral equation in 3D. We consider surface trian- gulations consisting of triangles. The condition number is bounded uniformly in the mesh Keywords: size h and the polynomial order p. The preconditioner handles adaptively refined meshes hp-BEM and is based on a local multilevel preconditioner for the lowest order space. Numerical Hypersingular integral equation Preconditioning experiments on different geometries illustrate its robustness. Additive Schwarz method ' 2015 Elsevier Ltd. All rights reserved. 1. Introduction Many elliptic boundary value problems that are solved in practice are linear and have constant (or at least piecewise constant) coefficients. In this setting, the boundary element method (BEM, [1–4]) has established itself as an effective alter- native to the finite element method (FEM). Just as in the FEM applied to this particular problem class, high order methods are very attractive since they can produce rapidly convergent schemes on suitably chosen adaptive meshes. -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
Family Name Given Name Presentation Title Session Code
Family Name Given Name Presentation Title Session Code Abdoulaev Gassan Solving Optical Tomography Problem Using PDE-Constrained Optimization Method Poster P Acebron Juan Domain Decomposition Solution of Elliptic Boundary Value Problems via Monte Carlo and Quasi-Monte Carlo Methods Formulations2 C10 Adams Mark Ultrascalable Algebraic Multigrid Methods with Applications to Whole Bone Micro-Mechanics Problems Multigrid C7 Aitbayev Rakhim Convergence Analysis and Multilevel Preconditioners for a Quadrature Galerkin Approximation of a Biharmonic Problem Fourth-order & ElasticityC8 Anthonissen Martijn Convergence Analysis of the Local Defect Correction Method for 2D Convection-diffusion Equations Flows C3 Bacuta Constantin Partition of Unity Method on Nonmatching Grids for the Stokes Equations Applications1 C9 Bal Guillaume Some Convergence Results for the Parareal Algorithm Space-Time ParallelM5 Bank Randolph A Domain Decomposition Solver for a Parallel Adaptive Meshing Paradigm Plenary I6 Barbateu Mikael Construction of the Balancing Domain Decomposition Preconditioner for Nonlinear Elastodynamic Problems Balancing & FETIC4 Bavestrello Henri On Two Extensions of the FETI-DP Method to Constrained Linear Problems FETI & Neumann-NeumannM7 Berninger Heiko On Nonlinear Domain Decomposition Methods for Jumping Nonlinearities Heterogeneities C2 Bertoluzza Silvia The Fully Discrete Fat Boundary Method: Optimal Error Estimates Formulations2 C10 Biros George A Survey of Multilevel and Domain Decomposition Preconditioners for Inverse Problems in Time-dependent -
The Spectral Projection Decomposition Method for Elliptic Equations in Two Dimensions∗
SIAM J. NUMER.ANAL. c 1997 Society for Industrial and Applied Mathematics Vol. 34, No. 4, pp. 1616–1639, August 1997 015 THE SPECTRAL PROJECTION DECOMPOSITION METHOD FOR ELLIPTIC EQUATIONS IN TWO DIMENSIONS∗ P. GERVASIO† , E. OVTCHINNIKOV‡ , AND A. QUARTERONI§ Abstract. The projection decomposition method (PDM) is invoked to extend the application area of the spectral collocation method to elliptic problems in domains compounded of rectangles. Theoretical and numerical results are presented demonstrating the high accuracy of the resulting method as well as its computational efficiency. Key words. domain decomposition, spectral methods, elliptic problems AMS subject classifications. 65N55, 65N35 PII. S0036142994265334 Introduction. Spectral methods represent a relatively new approach to the nu- merical solution of partial differential equations which appeared more recently than the widely used finite difference and finite element methods (FEMs). The first at- tempt of a systematic analysis was undertaken by D. Gottlieb and S. Orszag in their celebrated monograph [16], while a more comprehensive study appeared only 10 years later in [7], where spectral methods were shown to be a viable alternative to other numerical methods for a broad variety of mathematical equations, and particularly those modeling fluid dynamical processes. As a matter of fact, spectral methods fea- ture the property of being accurate up to an arbitrary order, provided the solution to be approximated is infinitely smooth and the computational domain is a Carte- sian one. Moreover, if the first condition is not fulfilled, the method automatically adjusts to provide an order of accuracy that coincides with the smoothness degree of the solution (measured in the scale of Sobolev spaces). -
Parallel Lines Cut by a Transversal
Parallel Lines Cut by a Transversal I. UNIT OVERVIEW & PURPOSE: The goal of this unit is for students to understand the angle theorems related to parallel lines. This is important not only for the mathematics course, but also in connection to the real world as parallel lines are used in designing buildings, airport runways, roads, railroad tracks, bridges, and so much more. Students will work cooperatively in groups to apply the angle theorems to prove lines parallel, to practice geometric proof and discover the connections to other topics including relationships with triangles and geometric constructions. II. UNIT AUTHOR: Darlene Walstrum Patrick Henry High School Roanoke City Public Schools III. COURSE: Mathematical Modeling: Capstone Course IV. CONTENT STRAND: Geometry V. OBJECTIVES: 1. Using prior knowledge of the properties of parallel lines, students will identify and use angles formed by two parallel lines and a transversal. These will include alternate interior angles, alternate exterior angles, vertical angles, corresponding angles, same side interior angles, same side exterior angles, and linear pairs. 2. Using the properties of these angles, students will determine whether two lines are parallel. 3. Students will verify parallelism using both algebraic and coordinate methods. 4. Students will practice geometric proof. 5. Students will use constructions to model knowledge of parallel lines cut by a transversal. These will include the following constructions: parallel lines, perpendicular bisector, and equilateral triangle. 6. Students will work cooperatively in groups of 2 or 3. VI. MATHEMATICS PERFORMANCE EXPECTATION(s): MPE.32 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. -
An Abstract Theory of Domain Decomposition Methods with Coarse Spaces of the Geneo Family Nicole Spillane
An abstract theory of domain decomposition methods with coarse spaces of the GenEO family Nicole Spillane To cite this version: Nicole Spillane. An abstract theory of domain decomposition methods with coarse spaces of the GenEO family. 2021. hal-03186276 HAL Id: hal-03186276 https://hal.archives-ouvertes.fr/hal-03186276 Preprint submitted on 31 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. An abstract theory of domain decomposition methods with coarse spaces of the GenEO family Nicole Spillane ∗ March 30, 2021 Keywords: linear solver, domain decomposition, coarse space, preconditioning, deflation Abstract Two-level domain decomposition methods are preconditioned Krylov solvers. What sep- arates one and two- level domain decomposition method is the presence of a coarse space in the latter. The abstract Schwarz framework is a formalism that allows to define and study a large variety of two-level methods. The objective of this article is to define, in the abstract Schwarz framework, a family of coarse spaces called the GenEO coarse spaces (for General- ized Eigenvalues in the Overlaps). This is a generalization of existing methods for particular choices of domain decomposition methods. -
Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method
Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method James W. Lottes¤ and Paul F. Fischery February 4, 2004 Abstract We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are con- sidered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Sev- eral of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids. 1 Introduction The availability of fast elliptic solvers is essential to many areas of scientific computing. For unstructured discretizations in three dimensions, iterative solvers are generally optimal from both work and storage standpoints. Ideally, one would like to have computational complexity that scales as O(n) for an n-point grid problem in lRd, implying that the it- eration count should be bounded as the mesh is refined. Modern iterative methods such as multigrid and Schwarz-based domain decomposition achieve bounded iteration counts through the introduction of multiple representations of the solution (or the residual) that allow efficient elimination of the error at each scale. The theory for these methods is well established for classical finite difference (FD) and finite element (FE) discretizations, and order-independent convergence rates are often attained in practice. For spectral element (SE) methods, there has been significant work on the development of Schwarz-based methods that employ a combination of local subdomain solves and sparse global solves to precondition conjugate gradient iteration. -
Geometry by Its History
Undergraduate Texts in Mathematics Geometry by Its History Bearbeitet von Alexander Ostermann, Gerhard Wanner 1. Auflage 2012. Buch. xii, 440 S. Hardcover ISBN 978 3 642 29162 3 Format (B x L): 15,5 x 23,5 cm Gewicht: 836 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 The Elements of Euclid “At age eleven, I began Euclid, with my brother as my tutor. This was one of the greatest events of my life, as dazzling as first love. I had not imagined that there was anything as delicious in the world.” (B. Russell, quoted from K. Hoechsmann, Editorial, π in the Sky, Issue 9, Dec. 2005. A few paragraphs later K. H. added: An innocent look at a page of contemporary the- orems is no doubt less likely to evoke feelings of “first love”.) “At the age of 16, Abel’s genius suddenly became apparent. Mr. Holmbo¨e, then professor in his school, gave him private lessons. Having quickly absorbed the Elements, he went through the In- troductio and the Institutiones calculi differentialis and integralis of Euler. From here on, he progressed alone.” (Obituary for Abel by Crelle, J. Reine Angew. Math. 4 (1829) p. 402; transl. from the French) “The year 1868 must be characterised as [Sophus Lie’s] break- through year. -
Math 1312 Sections 2.3 Proving Lines Parallel. Theorem 2.3.1: If Two Lines
Math 1312 Sections 2.3 Proving Lines Parallel. Theorem 2.3.1: If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel. Example 1: If you are given a figure (see below) with congruent corresponding angles then the two lines cut by the transversal are parallel. Because each angle is 35 °, then we can state that a ll b. 35 ° a ° b 35 Theorem 2.3.2: If two lines are cut by a transversal so that the alternate interior angles are congruent, then these lines are parallel. Example 2: If you are given a pair of alternate interior angles that are congruent, then the two lines cut by the transversal are parallel. Below the two angles shown are congruent and they are alternate interior angles; therefore, we can say that a ll b. a 75 ° ° b 75 Theorem 2.3.3: If two lines are cut by a transversal so that the alternate exterior angles are congruent, then these lines are parallel. Example 3: If you are given a pair of alternate exterior angles that are congruent, then the two lines cut by the transversal are parallel. For example, the alternate exterior angles below are each 105 °, so we can say that a ll b. 105 ° a b 105 ° Theorem 2.3.4: If two lines are cut by a transversal so that the interior angles on one side of the transversal are supplementary, then these lines are parallel. Example 3: If you are given a pair of consecutive interior angles that add up to 180 °(i.e. -
CHAPTER 3 Parallel and Perpendicular Lines Chapter Outline
www.ck12.org CHAPTER 3 Parallel and Perpendicular Lines Chapter Outline 3.1 LINES AND ANGLES 3.2 PROPERTIES OF PARALLEL LINES 3.3 PROVING LINES PARALLEL 3.4 PROPERTIES OF PERPENDICULAR LINES 3.5 PARALLEL AND PERPENDICULAR LINES IN THE COORDINATE PLANE 3.6 THE DISTANCE FORMULA 3.7 CHAPTER 3REVIEW In this chapter, you will explore the different relationships formed by parallel and perpendicular lines and planes. Different angle relationships will also be explored and what happens to these angles when lines are parallel. You will continue to use proofs, to prove that lines are parallel or perpendicular. There will also be a review of equations of lines and slopes and how we show algebraically that lines are parallel and perpendicular. 114 www.ck12.org Chapter 3. Parallel and Perpendicular Lines 3.1 Lines and Angles Learning Objectives • Identify parallel lines, skew lines, and parallel planes. • Use the Parallel Line Postulate and the Perpendicular Line Postulate. • Identify angles made by transversals. Review Queue 1. What is the equation of a line with slope -2 and passes through the point (0, 3)? 2. What is the equation of the line that passes through (3, 2) and (5, -6). 3. Change 4x − 3y = 12 into slope-intercept form. = 1 = − 4. Are y 3 x and y 3x perpendicular? How do you know? Know What? A partial map of Washington DC is shown. The streets are designed on a grid system, where lettered streets, A through Z run east to west and numbered streets 1st to 30th run north to south. -
Parallel Lines
Madhav Kaushish Parallel Lines Learning Outcomes • Defining Objects Precisely • Comparing and Evaluating Definitions T: What are parallel lines? S: They are lines which do not meet. T: Are these lines parallel? S: No. They are segments, not lines. Lines go on forever. T: So are you saying segments cannot be parallel. Are the two segments below parallel? S: Yes they are. How about if we say that 2 lines or segments are parallel if they do not meet even when extended. T: Okay. How about the following. Are they parallel lines? S: No, they are curves, not lines. T: I’m guessing that when you say line, you mean straight line and when you say curve, you mean a non-straight path. Is that right? S: Yes. Madhav Kaushish T: The words are a little confusing since we seem to have two commonly used terms for the same thing: line and straight line. Just for the purposes of this session, let us use the following classification: Straight Lines Lines Non-Straight Lines A line is something you can trace with your finger without lifting it (of course in the geometry we are working in, you cannot actually trace a line since it has no width). It could be finite or not. We can call segments finite lines. S: Does that mean that when you ask us for what parallel lines are, you want us to include non- straight lines? T: That is your decision. We would like the definition to be as general as possible. So, if you can come up with aa definition which covers both straight and non-straight lines and leads to interesting theorems, then you should do that. -
Math 135 Notes Parallel Postulate .Pdf
Euclidean verses Non Euclidean Geometries Euclidean Geometry Euclid of Alexandria was born around 325 BC. Most believe that he was a student of Plato. Euclid introduced the idea of an axiomatic geometry when he presented his 13 chapter book titled The Elements of Geometry. The Elements he introduced were simply fundamental geometric principles called axioms and postulates. The most notable are Euclid’s five postulates which are stated in the next passage. 1) Any two points can determine a straight line. 2) Any finite straight line can be extended in a straight line. 3) A circle can be determined from any center and any radius. 4) All right angles are equal. 5) If two straight lines in a plane are crossed by a transversal, and sum the interior angle of the same side of the transversal is less than two right angles, then the two lines extended will intersect. According to Euclid, the rest of geometry could be deduced from these five postulates. Euclid’s fifth postulate, often referred to as the Parallel Postulate, is the basis for what are called Euclidean Geometries or geometries where parallel lines exist. There is an alternate version to Euclid fifth postulate which is usually stated as “Given a line and a point not on the line, there is one and only one line that passed through the given point that is parallel to the given line. This is a short version of the Parallel Postulate called Fairplay’s Axiom which is named after the British math teacher who proposed to replace the axiom in all of the schools textbooks.