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.

We consider the Poisson equation $-\Delta u = f$ with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain $\O$. We develop a finite element multigrid method on quasi-uniform grids that obtains ${\cal{O}}(h^{m+1-\epsilon})$ convergence in the $H^1(\O)$ norm for any positive $\epsilon$ when $f \in H^m(\O)$. The cost of the method is proportional to the number of elements in the triangulation. The results of this paper can be generalized to other equations and other boundary conditions.

Overcoming Corner Singularities Using Multigrid Methods

We develop new multigrid methods for a class of saddle point problems that include the Stokes system in fluid flow and the Lame system in linear elasticity as special cases. The new smoothers in the multigrid methods involve optimal preconditioners for the discrete Laplace operator. We prove uniform convergence of the $$W$$ W -cycle algorithm in the energy norm and present numerical results for $$W$$ W -cycle and $$V$$ V -cycle algorithms.

Multigrid methods for saddle point problems: Stokes and Lamé systems

Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

In this paper we develop isoparametric C 0 interior penalty methods for plate bending problems on smooth domains. The orders of convergence of these methods are shown to be optimal in the energy norm. We also consider the convergence of these methods in lower order Sobolev norms and discuss subparametric C 0 interior penalty methods. Numerical results that illustrate the performance of these methods are presented.

Isoparametric C0 interior penalty methods for plate bending problems on smooth domains

We investigate numerically a triquadratic $C^0$ interior penalty method for elliptic distributed optimal control problems in three dimensions with pointwise state constraints, which is based on the formulation of these problems as fourth order variational inequalities. We obtain numerical results that are similar to the ones reported in [7, 8] for fourth order variational inequalities in two dimensions. The deal.II library [1, 2] is used for the numerical experiments.

Susanne C. Brenner

Papers

Overcoming Corner Singularities Using Multigrid Methods

Multigrid methods for saddle point problems: Stokes and Lamé systems

Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

Isoparametric C0 interior penalty methods for plate bending problems on smooth domains

A $$\varvec{C}^0$$ Interior Penalty Method for Elliptic Distributed Optimal Control Problems in Three Dimensions with Pointwise State Constraints