TY - JOUR

T1 - Domain decomposition solvers for nonlinear multiharmonic finite element equations

AU - Copeland, D. M.

AU - Langer, U.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: This publication is partially based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), and by the Austrian Science Fund 'Fonds zur Forderung der wissenschaftlichen Forschung (FWF)' under the grant P19255.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2010/1

Y1 - 2010/1

N2 - In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure. © 2010 de Gruyter.

AB - In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure. © 2010 de Gruyter.

UR - http://hdl.handle.net/10754/598010

UR - https://www.degruyter.com/doi/10.1515/jnum.2010.008

UR - http://www.scopus.com/inward/record.url?scp=77958580743&partnerID=8YFLogxK

U2 - 10.1515/JNUM.2010.008

DO - 10.1515/JNUM.2010.008

M3 - Article

SN - 1570-2820

VL - 18

SP - 157

EP - 175

JO - Journal of Numerical Mathematics

JF - Journal of Numerical Mathematics

IS - 3

ER -