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.

In this paper the balancing domain decomposition method is extended to nonconforming plate elements. The condition number of the preconditioned system is shown to be bounded by 
$C\big[1+\ln(H/h)\big]^2$
 , where H measures the diameters of the subdomains, h is the mesh size of the triangulation, and the constant C is independent of H, h and the number of subdomains.

/pdf/balancing-domain-decomposition-for-nonconforming-plate-qbwxqjvxb0.pdf

Balancing domain decomposition for nonconforming plate elements

We investigate a quadratic $$C^0$$C0 interior penalty method for the approximation of isolated solutions of a von Karman plate. We prove that the discrete problem is uniquely solvable near an isolated solution and establish optimal order error estimates. Numerical results that illustrate the theoretical estimates are also presented.

A $$C^0$$C0 interior penalty method for a von Kármán plate

We consider finite element methods for elliptic distributed optimal control problems with pointwise state constraints on two and three dimensional convex polyhedral domains formulated as fourth order variational inequalities. We develop a new convergence analysis that is applicable to C-1 finite element methods, classical nonconforming finite element methods and discontinuous Galerkin methods.

A new convergence analysis of finite element methods for elliptic distributed optimal control problems with pointwise state constraints

We study a class of symmetric discontinuous Galerkin methods on graded meshes Optimal order error estimates are derived in both the energy norm and the L 2 norm, and we establish the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems Numerical results that confirm the theoretical results are also presented

Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

Three Maxwell eigensolvers are discussed in this paper. Two of them use classical nonconforming finite element approximations, and the other is an interior penalty type discontinuous Galerkin method. A main feature of these solvers is that they are based on the formulation of the Maxwell eigenproblem on the space H 0(curl;?)?H(div0;?). These solvers are free of spurious eigenmodes and they do not require choosing penalty parameters. Furthermore, they satisfy optimal order error estimates on properly graded meshes, and their analysis is greatly simplified by the underlying compact embedding of H 0(curl;?)?H(div0;?) in L 2(?). The performance and the relative merits of these eigensolvers are demonstrated through numerical experiments.

/pdf/nonconforming-maxwell-eigensolvers-2pilm5t1se.pdf

Susanne C. Brenner

Papers

Balancing domain decomposition for nonconforming plate elements

A $$C^0$$C0 interior penalty method for a von Kármán plate

A new convergence analysis of finite element methods for elliptic distributed optimal control problems with pointwise state constraints

Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

Nonconforming Maxwell Eigensolvers