Geodesic
Mesh adaptation based on functional a posteriori estimates with Raviart–Thomas approximation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1277-1288 
M. E. Frolov; M. A. Churilova. Mesh adaptation based on functional a posteriori estimates with Raviart–Thomas approximation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1277-1288. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a9/
@article{ZVMMF_2012_52_7_a9,
     author = {M. E. Frolov and M. A. Churilova},
     title = {Mesh adaptation based on functional a posteriori estimates with {Raviart{\textendash}Thomas} approximation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1277--1288},
     year = {2012},
     volume = {52},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a9/}
}
TY  - JOUR
AU  - M. E. Frolov
AU  - M. A. Churilova
TI  - Mesh adaptation based on functional a posteriori estimates with Raviart–Thomas approximation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2012
SP  - 1277
EP  - 1288
VL  - 52
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a9/
LA  - ru
ID  - ZVMMF_2012_52_7_a9
ER  - 
%0 Journal Article
%A M. E. Frolov
%A M. A. Churilova
%T Mesh adaptation based on functional a posteriori estimates with Raviart–Thomas approximation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2012
%P 1277-1288
%V 52
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a9/
%G ru
%F ZVMMF_2012_52_7_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

Adaptive algorithms based on functional a posteriori estimates for the Dirichlet problem for the stationary diffusion equation with jump discontinuities in the equation coefficients are compared. The algorithms have been implemented in MATLAB with the use of both standard finite element approximations and the zero-order Raviart–Thomas approximation. The adaptation results are analyzed using indicators of the local error distribution. Specifically, sequences of finite-element partitions, effectivity indices of estimates, and relative errors of approximate solutions are compared.

[1] Repin S.I., A posteriori estimates for partial differential equations, Walter de Gruyter, Berlin, New York, 2008 | MR

[2] Babuska I., Whiteman J.R., Strouboulis T., Finite elements. An introduction to the method and error estimation, Oxford University Press, Oxford, 2011 | MR

[3] Verfurth R., A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley, Berlin; Teubner, Stuttgart, 1996

[4] Ainsworth M., Oden J.T., A posteriori error estimation in finite element analysis, John Wiley and Sons, New York, 2000 | MR

[5] Babuska I., Strouboulis T., The finite element method and its reliability, Oxford University Press, New York, 2001 | MR | Zbl

[6] Bangerth W., Rannacher R., Adaptive finite element methods for differential equations, Lect. Math., ETH Zurich, Birkhauser, Basel, 2003 | MR | Zbl

[7] Zienkiewicz O.C., Taylor R.L., Finite element method, v. 1, The basis, 5th ed., Butterworth–Heinemann, Oxford, 2000 | MR

[8] Neittaanmaki P., Repin S.I., Reliable methods for computer simulation – error control and a posteriori estimates, Elsevier, Amsterdam, 2004 | MR

[9] Mikhlin S.G., Variatsionnye metody v matematicheskoi fizike, Nauka, M., 1970

[10] Prager W., Synge J.L., “Approximations in elasticity based on the concept of function space”, Quart. Appl. Math., 5 (1947), 241–269 | MR | Zbl

[11] Ladeveze P., Pelle J.P., Mastering calculations in linear and nonlinear mechanics, Springer, New York, 2005 | MR | Zbl

[12] Roberts J.E., Thomas J.-M., “Mixed and hybrid methods”, Handbook of Numerical Analysis, v. II, Part I, Elsevier, Amsterdam, 1991, 527–637 | MR

[13] Braess D., Finite Elements. Theory, fast solvers and applications in solid mechanics, 3rd. ed., Cambridge Univ. Press, Cambridge, 2007 | Zbl

[14] Repin S.L, Sauter S., Smolianski A., “A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions”, Computing, 70:3 (2003), 205–233 | MR | Zbl

[15] Repin S.I., Sauter S., Smolianski A., “A posteriori error estimation for the Poisson equation with mixed Dirichlet–Neumann boundary conditions”, J. Comput. Appl. Math., 164–165 (2004), 601–612 | DOI | MR | Zbl

[16] Frolov M.E., Neittaanmaki P., Repin S.I., “On computational properties of a posteriori error estimates based upon the method of duality error majorants”, Numer. Math. Advanced Appl. (ENUMATH, 2003), Springer, Berlin, Heidelberg, 2004, 346–357 | DOI | MR | Zbl

[17] Frolov M.E., Aposteriornye otsenki tochnosti priblizhennykh reshenii variatsionnykh zadach dlya ellipticheskikh uravnenii divergentnogo tipa, Diss. $\dots$ kand. fiz.-matem. nauk, GPU, SPb, 2004

[18] Valdman J., “Minimization of functional majorant in a posteriori error analysis based on $H$(div) multigrid-preconditioned CG method”, Adv. Numer. Anal., 2009, Article ID 164519, 15 p. | MR | Zbl

[19] Lazarov R., Repin S.L, Tomar S.K., “Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems”, Numer. Methods Partial Differ. Equat., 25:4 (2009), 952–971 | DOI | MR | Zbl

[20] Ladyzhenskaya O.A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973

[21] Repin S.I., “A posteriori error estimation for variational problems with uniformly convex functionals”, Math. Comp., 69:230 (2000), 481–500 | DOI | MR | Zbl

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

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

[24] Shaidurov V.V., Mnogosetochnye metody konechnykh elementov, Nauka, M., 1989

[25] Boffi D., Brezzi F., Demkowicz L.F., Duran R.G., Falk R.S., Fortin M., Mixed finite elements, compatibility conditions and applications, Springer, Berlin, 2006 | MR

[26] Brenner S.G., Ridgway Scott L., The mathematical theory of finite element methods, Springer, New York, 2008 | MR

[27] Raviart P.A., Thomas J.M., “A mixed finite element method for second order elliptic problems”, Lect. Notes in Math., 606 (1977), 292–315, Springer, Berlin | DOI | MR

[28] Repin S.I., Frolov M.E., “Ob aposteriornykh otsenkakh tochnosti priblizhennykh reshenii kraevykh zadach dlya uravnenii ellipticheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 42:12 (2002), 1774–1787 | MR | Zbl