Geodesic
Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 3, pp. 463-480
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An abstract theory for discretizations of second-order quasilinear elliptic problems based on the mixed-hybrid discontinuous Galerkin method. Discrete schemes are formulated in terms of approximations of the solution to the problem, its gradient, flux, and the trace of the solution on the interelement boundaries. Stability and optimal error estimates are obtained under minimal assumptions on the approximating space. It is shown that the schemes admit an efficient numerical implementation. 
@article{ZVMMF_2014_54_3_a9,
     author = {R. Z. Dautov and E. M. Fedotov},
     title = {Abstract theory of hybridizable discontinuous {Galerkin} methods for second-order quasilinear elliptic problems},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {463--480},
     year = {2014},
     volume = {54},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a9/}
}
TY  - JOUR
AU  - R. Z. Dautov
AU  - E. M. Fedotov
TI  - Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2014
SP  - 463
EP  - 480
VL  - 54
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a9/
LA  - ru
ID  - ZVMMF_2014_54_3_a9
ER  - 
%0 Journal Article
%A R. Z. Dautov
%A E. M. Fedotov
%T Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2014
%P 463-480
%V 54
%N 3
%U http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a9/
%G ru
%F ZVMMF_2014_54_3_a9
R. Z. Dautov; E. M. Fedotov. Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 3, pp. 463-480. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_3_a9/

[1] Cockburn B., “Discontinuous Galerkin methods”, Z. Angew. Math. Mech., 83 (2003), 731–754 | DOI | MR | Zbl

[2] Arnold D. N., Brezzi F., Cockburn B., Marini L. D., “Unified analysis of discontinuous Galerkin methods for elliptic problems”, SIAM J. Numer. Anal., 39:5 (2002), 1749–1779 | DOI | MR | Zbl

[3] Cockburn B., Gopalakrishnan J., Lazarov R., “Unified hydridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems”, SIAM J. Numer. Anal., 47:2 (2009), 1319–1365 | DOI | MR | Zbl

[4] Arnold D. N., Brezzi F., “Mixed and non-conforming finite element methods: implementation, post-processing and error estimates”, Math. Modell. Numer. Anal., 19:1 (1985), 7–32 | MR | Zbl

[5] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Nauka, M., 1978

[6] S'yarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, M., 1980

[7] Brezzi F., Fortin M., Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991 | MR | Zbl

[8] Brenner S. C., “Poincare–Friedrichs inequalities for piecewise $H^1$ functions”, SIAM J. Numer. Anal., 41 (2003), 306–324 | DOI | MR | Zbl

[9] Lasis A., Süli E., Poincaré-type inequalities for broken Sobolev spaces, Technical Report 03/10, Oxford University Computing Laboratory, Oxford, England, 2003

[10] Pietro D., Ern A., “Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations”, Math. Comput., 79:271 (2010), 1303–1330 | DOI | MR | Zbl

[11] Brenner C., Wang K., Zhao J., “Poincare–Friedrichs inequalities for piecewise $H^2$ functions”, Numerical functional analysis and optimization, 25:5–6 (2004), 463–478 | DOI | MR | Zbl

[12] Raviart P.-A., Thomas J. M., “A mixed finite element method for 2-ond order elliptic problems”, Lecture Notes in Math., 606, 1977, 292–315 | DOI | MR | Zbl

[13] Brezzi F., Douglas J., Marini L. D., “Two families of mixed finite elements for second order elliptic problems”, Numer. Math., 47 (1985), 217–235 | DOI | MR | Zbl

[14] Karchevskii M. M., Lyashko A. D., Raznostnye skhemy dlya nelineinykh uravnenii matematicheskoi fiziki, Izd-vo KGU, Kazan, 1976

[15] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978 | MR