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.

A new additive Schwarz preconditioner for the Finite Element Tearing and Interconnecting (FETI) method is analyzed in this paper. This preconditioner has the unique feature that the coefficient matrix of its ``coarse grid'' problem is mesh independent. For a model second order heterogeneous elliptic boundary value problem in two dimensions, the condition number of the preconditioned system is shown to be bounded by C[1+ln(H/h)]2, where h is the mesh size, H is the typical diameter of the subdomains, and the constant C is independent of h, H, the number of subdomains and the coefficients of the boundary value problem.

An additive Schwarz preconditioner for the FETI method

A partition of unity method for a class of fourth order elliptic variational inequalities

We extend the Hodge decomposition approach for the cavity problem of two-dimensional time-harmonic Maxwell's equations to include the impedance boundary condition, with anisotropic electric permittivity and sign-changing magnetic permeability. We derive error estimates for a P1 finite element method based on the Hodge decomposition approach and present results of numerical experiments that involve metamaterials and electromagnetic cloaking. The well-posedness of the cavity problem when both electric permittivity and magnetic permeability can change sign is also discussed. Copyright © 2015 John Wiley & Sons, Ltd.

Hodge decomposition for two-dimensional time-harmonic Maxwell's equations: impedance boundary condition

A nonconforming finite element method for a two-dimensional curl–curl problem is studied in this paper. It uses weakly continuous P1 vector fields and penalizes the local divergence. Two consistency terms involving the jumps of the vector fields across element boundaries are also included to ensure the convergence of the scheme. Optimal convergence rates (up to an arbitrary positive ∊) in both the energy norm and the L2 norm are established on graded meshes. This scheme can also be used in the computation of Maxwell eigenvalues without generating spurious eigenmodes. The theoretical results are confirmed by numerical experiments.

A nonconforming penalty method for a two-dimensional curl–curl problem

We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order Hh−1, where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods are also presented.

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Susanne C. Brenner

Papers

An additive Schwarz preconditioner for the FETI method

A partition of unity method for a class of fourth order elliptic variational inequalities

Hodge decomposition for two-dimensional time-harmonic Maxwell's equations: impedance boundary condition

A nonconforming penalty method for a two-dimensional curl–curl problem

Schwarz Methods for a Preconditioned WOPSIP Method for Elliptic Problems