This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

Runge-Kutta discontinuous Galerkin methods for convection-dominated problems

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

/pdf/orthogonal-polynomials-of-several-variables-2cwkynqk1v.pdf

Orthogonal Polynomials of Several Variables

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

/pdf/the-development-of-discontinuous-galerkin-methods-17hxfhour7.pdf

The Development of Discontinuous Galerkin Methods

This article shows how to obtain multidimensional spectral methods as a warped product of one-dimensional spectral methods, thus generalizing direct (tensor) products. This generalization includes the fast Fourier transform. Applications are given for spectral approximation on a disk and on a triangle. The use of the disk spectral method for simulating the Navier-Stokes equations in a periodic pipe is detailed. The use of the triangle method in a spectral element scheme is discussed. The degree of approximation of the triangle method is computed in a new way, which favorably compares with the classical approximation estimates.

Spectral methods on triangles and other domains

Recently R. Kosaraju gave an O(nmo.75polylog(m)) step algorithm for tree pattern matching. We improve this result by designing a simple O(n6 polylog(m)) algorithm.

Faster tree pattern matching

We consider the problem of polynomial approximation to a real valued functionf defined on a compact set$\mathbb{X}$. An approximation theorem is proven in terms of the newly defined modulus of approximation. It is shown to imply a multidimensional Jackson type theorem which is stronger than previously known results even for the interval [−1, 1]. A strong multidimensional Bernstein type inverse theorem is also proven. We allow quite general approximation quasi-norms including$\mathcal{L}^{q} $ for 0<q≤∞.

The theory of multi-dimensional polynomial approximation

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulas of size $O(n^{\alpha+2})$ (where $\alpha=\log_{\sqrt{5}-1}2\simeq 3.27$) that amplify $(\psi-\frac{1}{n},\psi+\frac{1}{n})$ (where $\psi=(\sqrt{5}-1)/2\simeq0.62$) to $(2^{-n},1-2^{-n})$ and deduced as a consequence the existence of monotone formulas of the same size that compute the majority of $n$ bits. Boppana has shown that any monotone read-once formula that amplifies $(p-\frac{1}{n},p+\frac{1}{n})$ to $(\frac{1}{4},\frac{3}{4})$ (where $0We extend Boppana's results in two ways. We first show that his two lower bounds hold for general read-once formulas, not necessarily monotone, that may even include exclusive-or gates. We are then able to join his two lower bounds together and show that any read-once, not necessarily monotone, formula that amplifies $(p-\frac{1}{n},p+\frac{1}{n})$ to $(2^{-n},1-2^{-n})$ has size $\Omega(n^{\alpha+2})$. This result does not follow from Boppana's arguments, and it shows that the amount of amplification achieved by Valiant is the maximal achievable using read-once formulas.
In a companion paper we construct monotone read-once contact networks of size $O(n^{2.99})$ that amplify $(\frac{1}{2}-\frac{1}{n},\frac{1}{2}+\frac{1}{n})$ to $(\frac{1}{4},\frac{3}{4})$. This shows that Boppana's lower bound for the first amplification stage does not apply to contact networks, even if they are required to be both monotone and read-once.

Amplification by Read-Once Formulas

Let f be a de Morgan read-once function of n variables. Let f e be the random restriction obtained by independently assigning to each variable of f , the value 0 with probability (1 -e)/2, the value 1 with the same probability, and leaving it unassigned with probability e. We show that f e depends, on the average, on only O (e α n + e n 1/α ) variables, where . This result is asymptotically the tightest possible. It improves a similar result obtained recently by Hastad, Razborov and Yao.

How Do Read-Once Formulae Shrink?

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