# A spectral method for an elliptic equation with a nonlinear Neumann boundary condition

A spectral method for an elliptic equation with a nonlinear Neumann boundary condition Numer Algor https://doi.org/10.1007/s11075-018-0550-y ORIGINAL PAPER A spectral method for an elliptic equation with a nonlinear Neumann boundary condition 1 2 2 Kendall Atkinson · David Chien · Olaf Hansen Received: 19 February 2018 / Accepted: 14 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 d d Abstract Let  be an open region in R , d ≥ 2, that is diffeomorphic to B . Consider solving −u + γu = 0on  with the Neumann boundary condition ∂u = b (·,u) over ∂. The function b is a nonlinear function of u. The problem is ∂n reformulated in a weak form, and then a spectral Galerkin method is used to create a sequence of finite dimensional nonlinear problems. An error analysis shows that under suitable assumptions, the solutions of the finite dimensional problems con- verge to those of the original problem. To carry out the error analysis, the original problem and the spectral method is converted to a nonlinear integral equation over 1/2 H () , and the reformulation is analyzed using tools for solving nonlinear inte- gral equations. Numerical examples are given to illustrate the method. In our error analysis, we assume the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Numerical Algorithms Springer Journals

# A spectral method for an elliptic equation with a nonlinear Neumann boundary condition

, Volume OnlineFirst – May 29, 2018
32 pages

/lp/springer_journal/a-spectral-method-for-an-elliptic-equation-with-a-nonlinear-neumann-tYtlqRc9Fu
Publisher
Springer Journals
Subject
Computer Science; Numeric Computing; Algorithms; Algebra; Theory of Computation; Numerical Analysis
ISSN
1017-1398
eISSN
1572-9265
D.O.I.
10.1007/s11075-018-0550-y
Publisher site
See Article on Publisher Site

### Abstract

Numer Algor https://doi.org/10.1007/s11075-018-0550-y ORIGINAL PAPER A spectral method for an elliptic equation with a nonlinear Neumann boundary condition 1 2 2 Kendall Atkinson · David Chien · Olaf Hansen Received: 19 February 2018 / Accepted: 14 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 d d Abstract Let  be an open region in R , d ≥ 2, that is diffeomorphic to B . Consider solving −u + γu = 0on  with the Neumann boundary condition ∂u = b (·,u) over ∂. The function b is a nonlinear function of u. The problem is ∂n reformulated in a weak form, and then a spectral Galerkin method is used to create a sequence of finite dimensional nonlinear problems. An error analysis shows that under suitable assumptions, the solutions of the finite dimensional problems con- verge to those of the original problem. To carry out the error analysis, the original problem and the spectral method is converted to a nonlinear integral equation over 1/2 H () , and the reformulation is analyzed using tools for solving nonlinear inte- gral equations. Numerical examples are given to illustrate the method. In our error analysis, we assume the

### Journal

Numerical AlgorithmsSpringer Journals

Published: May 29, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations