There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

비만아 부모의 돌봄 경험: q 방법론

This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

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

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g., x(-->)(dagger) Mx(-->) for some matrix M. In this case, when A is sparse, N x N and has condition number kappa, the fastest known classical algorithms can find x(-->) and estimate x(-->)(dagger) Mx(-->) in time scaling roughly as N square root(kappa). Here, we exhibit a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa. Indeed, for small values of kappa [i.e., poly log(N)], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.

/pdf/quantum-algorithm-for-linear-systems-of-equations-3l9ipvg979.pdf

Quantum algorithm for linear systems of equations.

Preface(2nd ed.).- Preface(1st ed.).- Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element of Space.- Polynomial Approximation Theory in Sobolev Spaces.- n-Dimensional Variational Problems.- Finite Element Multigrid Methods.- Additive Schwarz Preconditioners.- Max-norm Estimates.- Adaptive Meshes.- Variational Crimes.- Applications to Planar Elasticity.- Mixed Methods.- Iterative Techniques for Mixed Methods.- Applications of Operator-Interpolation Theory.- References.- Index.

The Mathematical Theory of Finite Element Methods

Poincare--Friedrichs inequalities for piecewise H1 functions are established. They can be applied to classical nonconforming finite element methods, mortar methods, and discontinuous Galerkin methods.

Poincaré-Friedrichs Inequalities for Piecewise H 1 Functions

C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are analyzed in this paper. A post-processing procedure that can generate C1 approximate solutions from the C0 approximate solutions is presented. New C0 interior penalty methods based on the techniques involved in the post-processing procedure are introduced. These new methods are applicable to rough right-hand sides.

C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains

Korn's inequalities for piecewise H 1 vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.

/pdf/korn-s-inequalities-for-piecewise-h-1-vector-fields-zbw2mwnpzf.pdf

Korn's inequalities for piecewise $H^1$ vector fields

A linear nonconforming (conforming) displacement finite element method for the pure displacement (pure traction) problem in two-dimensional linear elasticity for a homogeneous isotropic elastic material is considered. In the case of a convex polygonal configuration domain, error estimates in the energy (L[sup 2]) norm are obtained. The convergence rate does not deteriorate for nearly incompressible material. Furthermore, the convergence analysis does not rely on the theory of saddle point problems. 22 refs.

https://www.ams.org/mcom/1992-59-200/S0025-5718-1992-1140646-2/S0025-5718-1992-1140646-2.pdf

Susanne C. Brenner

Papers

The Mathematical Theory of Finite Element Methods

Poincaré-Friedrichs Inequalities for Piecewise H 1 Functions

C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains

Korn's inequalities for piecewise $H^1$ vector fields

Linear finite element methods for planar linear elasticity