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.

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Quantum algorithm for linear systems of equations.

V-cycle, F-cycle and W-cycle multigrid algorithms for interior penalty methods for second order elliptic boundary value problems are studied in this paper. It is shown that these algorithms converge uniformly with respect to all grid levels if the number of smoothing steps is sufficiently large, and that the contraction numbers decrease as the number of smoothing steps increases, at a rate determined by the elliptic regularity of the problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

https://www.onlinelibrary.wiley.com/doi/pdf/10.1002/anac.200410019

Convergence of Multigrid Algorithms for Interior Penalty Methods

An optimal order multigrid method is developed for the pure displacement problem in two-dimensional linear elasticity. It is based on a nonconforming mixed formulation, where the displacement is approximated by weakly continuous piecewise linear vector functions, and the pressure is approximated by piecewise constants. The full multigrid convergence is proved. The performance of the multigrid method does not deteriorate as the material becomes nearly incompressible.

A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity

Poincare–Friedrichs inequalities are derived for piecewise H 2 functions on two dimensional domains. These inequalities can be applied to classical non-conforming finite element methods, mortar methods and discontinuous Galerkin methods.

Poincaré–Friedrichs Inequalities for Piecewise H 2 Functions

A new numerical method for computing the divergence-free part of the solution of the time-harmonic Maxwell equations is studied in this paper. It is based on a discretization that uses the locally divergence-free Crouzeix-Raviart nonconforming P 1 vector fields and includes a consistency term involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive e) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

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A locally divergence-free nonconforming finite element method for the time-harmonic maxwell equations

An optimal order multigrid method for the lowest-order Raviart–Thomas mixed triangular finite element is developed. The algorithm and the convergence analysis are based on the equivalence between Raviart–Thomas mixed methods and certain nonconforming methods. Both the Dirichlet and singular Neumann boundary value problems for second-order elliptic equations are discussed.

Susanne C. Brenner

Papers

Convergence of Multigrid Algorithms for Interior Penalty Methods

A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity

Poincaré–Friedrichs Inequalities for Piecewise H 2 Functions

A locally divergence-free nonconforming finite element method for the time-harmonic maxwell equations

A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method