A comparison of some efficient numerical methods for a nonlinear elliptic problem

• Autorzy: Balázs Kovács
• Języki publikacji: EN

Abstrakty

EN

The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them and analyzed their behavior, efficiency and mesh independence. We also compare the theoretical results to the numerical ones, and finally we propose some improvements which we also investigate.

Słowa kluczowe

EN

Radiative cooling problem; Iterative methods; Stepwise variable preconditioning; Numerical experiments

Kategorie tematyczne

• 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
• 65J15: Equations with nonlinear operators (do not use )
• 65H10: Systems of equations

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 1
• Strony: 217-230

Daty

wydano

2012-02-01

online

2011-12-09

Twórcy

autor

Balázs Kovács

• Eötvös Loránd University

Bibliografia

• [1] Brezinski C., A classification of quasi-Newton methods, Numer. Algorithms, 2003, 33(1–4), 123–135 http://dx.doi.org/10.1023/A:1025551602679
• [2] Dennis J.E., Jr., Moré J.J., Quasi-Newton methods, motivation and theory, SIAM Rev., 1977, 19(1), 46–89 http://dx.doi.org/10.1137/1019005
• [3] Faragó I., Karátson J., Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators: Theory and Applications, Adv. Comput. Theory Pract., 11, Nova Science Publishers, Hauppauge, 2002
• [4] Karátson J., Faragó I., Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space, SIAM J. Numer. Anal., 2003, 41(4), 1242–1262 http://dx.doi.org/10.1137/S0036142901384277
• [5] Keller H.B., Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal., 1969, 35(5), 363–381 http://dx.doi.org/10.1007/BF00247683
• [6] Liao S., Su J., Chwang A.T., Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, International Journal of Heat and Mass Transfer, 2006, 49(15–16), 2437–2445 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2006.01.030
• [7] Schlichenmaier R., Bruls J.H.M.J., Schüssler M., Radiative cooling of a hot flux tube in the solar photosphere, Astronomy and Astrophysics, 1999, 349, 961–973
• [8] Sen M., Analytical Heat Transfer, lecture notes available at http://nd.edu/_msen/Teaching/IntHT/IntHTNotes.pdf
• [9] Vladimirov V.S., Equations of Mathematical Physics, Mir, Moscow, 1984
• [10] Zeidler E., Nonlinear Functional Analysis and its Applications. III, Springer, New York, 1985

Identyfikatory

DOI

10.2478/s11533-011-0071-6