Frontiers in Numerical Analysis - Durham 2010

herausgegeben von: James Blowey, Max Jensen

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Computational Science and Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book contains detailed lecture notes on four topics at the forefront of current research in computational mathematics. Each set of notes presents a self-contained guide to a current research area and has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results and techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described and directions for future research are given. This book is also suitable for professional mathematicians who require a succint and accurate account of recent research in areas parallel to their own, and graduates in mathematical sciences.

Inhaltsverzeichnis

Frontmatter

Some Remarks on Eigenvalue Approximation by Finite Elements

Abstract

The aim of this paper is to supplement the results of Boffi (Acta Numer. 19:1–120, 2010) with some additional remarks. In particular we deal with three distinct topics: we review some tutorial examples in one dimension and provide numerical codes for them; we analyze the case of multiple eigenvalues and show some numerical; we review a posteriori error analysis for eigenvalue problems.

Daniele Boffi, Francesca Gardini, Lucia Gastaldi

C 0 Interior Penalty Methods

Abstract

C ⁰ interior penalty methods are discontinuous Galerkin methods for fourth order problems. In this article we discuss various aspects of such methods including a priori error analysis, a posteriori error analysis and fast solution techniques.

Susanne C. Brenner

Introduction to Applications of Numerical Analysis in Time Domain Computational Electromagnetism

Abstract

We discuss two techniques for the solution of the time domain Maxwell system. The first is a partial differential equation based approach using conforming finite elements and implicit time stepping that is suitable when stiff problems are encountered, and where the medium is inhomogeneous. In particular we analyze the use of edge elements and certain A-stable schemes using the Fourier-Laplace transform. For a homogeneous medium, an integral equation approach can be used and we describe and analyze the convolution quadrature method applied to the electric field integral equation. In either case we emphasize that the convergence analysis depends on energy estimates for the continuous problem.

Qiang Chen, Peter Monk

Numerical Approximation of Large Contrast Problems with the Unfitted Nitsche Method

Abstract

These notes are concerned with the numerical treatment of the coupling between second order elliptic problems that feature large contrast between their characteristic coefficients. In particular, we study the application of Nitsche’s method to set up a robust approximation of interface conditions in the framework of the finite element method. The notes are subdivided in three parts. Firstly, we review the weak enforcement of Dirichlet boundary conditions with particular attention to Nitsche’s method and we discuss the extension of such technique to the coupling of Poisson equations. Secondly, we review the application of Nitsche’s method to large contrast problems, discretised on computational meshes that capture the interface of discontinuity between coefficients. Finally, we extend the previous schemes to the case of unfitted meshes, which occurs when the computational mesh does not conform with the interface between subproblems.

Erik Burman, Paolo Zunino

Backmatter

Titel: Frontiers in Numerical Analysis - Durham 2010
herausgegeben von: James Blowey
Max Jensen
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-23914-4
Print ISBN: 978-3-642-23913-7
DOI: https://doi.org/10.1007/978-3-642-23914-4