Geodesic
Adaptive $hp$-finite element method for solving boundary value problems for the stationary reaction–diffusion equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 9, pp. 1512-1529 
N. D. Zolotareva; E. S. Nikolaev. Adaptive $hp$-finite element method for solving boundary value problems for the stationary reaction–diffusion equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 9, pp. 1512-1529. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_9_a7/
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     author = {N. D. Zolotareva and E. S. Nikolaev},
     title = {Adaptive $hp$-finite element method for solving boundary value problems for the stationary reaction{\textendash}diffusion equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_9_a7/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

An adaptive $hp$-finite element method with finite-polynomial basis functions is constructed for finding a highly accurate solution of a boundary value problem for the stationary reaction–diffusion equation. Adaptive strategies are proposed for constructing a sequence of finite-dimensional subspaces based on the use of correction indicators, i.e., quantities evaluating the degree to which a chosen characteristic of the approximate solution varies when the subspace is expanded by adding new test basis functions. Efficient algorithms for computing correction indicators are described. The method is intended for problems whose solutions have a local singularity, for example, for singularly perturbed boundary value problems.

[1] Babuška I., Rheinboldt W. S., “A posteriory error estimates for the finite element method”, Intern. J. Numer. Meth. Eng., 12:11 (1978), 1597–1615 | Zbl

[2] Babuška I., Rheinboldt W. C., “Error estimates for adaptive finite element computations”, SIAM J. Numer. Anal., 15:4 (1978), 736–754 | DOI | MR | Zbl

[3] Bank R. E., Weiser A., “Some a posteriory error estimates for elliptic partial differential equations”, Math. Comput., 44:170 (1985), 283–301 | DOI | MR | Zbl

[4] Ainsworth M., Oden J. T., “A procedure for a posteriory error estimation for $h-p$-finite element methods”, Comput. Meth. Appl. Mech. Eng., 101:1 (1992), 73–96 | DOI | MR | Zbl

[5] Ainsworth M., Oden J. T., “A unified approach to a posteriory error estimation using element residual methods”, Numer. Math., 65:1 (1993), 23–50 | DOI | MR | Zbl

[6] Bornemann F. A., Erdmann V., Kornhuber R., “A posteriory error estimates for elliptic problem in two and three space dimensions”, SIAM J. Numer. Anal., 33:3 (1996), 1188–1204 | DOI | MR | Zbl

[7] Verfurth R., “A posteriori error estimation and adaptive mesh-refinement techniques”, J. Comput. Appl. Math., 50:1–3 (1996), 67–83 | MR

[8] Ainsworth M., Oden J. T., “A posteriory error estimation in finite element analysis”, Comput. Meth. Appl. Mech. Eng., 142:1 (1997), 1–88 | DOI | MR | Zbl

[9] Zienkiewicz O. C., Kelly D. W., de Gago J. P., Babuška I., “Hierarchical finite element approaches, adaptive refinement and error estimates”, The Mathematics of Finite Elements and Applications, Academic Press, London, 1982, 313–346 | MR

[10] Zienkiewicz O. S., Craig A., “Adaptive refinement, error estimates, multigrid solution, and hierarchical finite element method concepts”, Accuracy Estimates and Adaptive Refinements in Finite Element Computations, John Wiley Sons, Chichester, 1986, 25–59 | MR

[11] Bank R. E., Smith R. K., “A posteriory error estimates based on hierarchical bases”, SIAM J. Numer. Anal., 30:4 (1993), 921–935 | DOI | MR | Zbl

[12] Nikolaev E. S., “Metod resheniya kraevykh zadach dlya sistem lineinykh ODU na adaptivno izmelchaemykh setkakh”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibern., 2000, no. 3, 11–19

[13] Nikolaev E. S., “Metod resheniya kraevoi zadachi dlya ODU 2-go poryadka na posledovatelnosti adaptivno izmelchaemykh i ukrupnyaemykh setok”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibern., 2004, no. 4, 5–16

[14] Nikolaev E. S., Shishkina O. V., “Metod resheniya zadachi Dirikhle ellipticheskogo uravneniya 2-go poryadka v poligonalnoi oblasti na posledovatelnosti adaptivno izmelchaemykh setok”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibern., 2005, no. 4, 3–15

[15] Zolotareva N. D., Nikolaev E. S., “Metod postroeniya setok, adaptiruyuschikhsya k resheniyu kraevykh zadach dlya obyknovennykh differentsialnykh uravnenii vtorogo i chetvertogo poryadkov”, Differents. up-niya, 45:8 (2009), 1165–1178 | MR | Zbl

[16] Zolotareva N. D., Nikolaev E. S., “Adaptivnaya $p$-versiya metoda konechnykh elementov resheniya kraevykh zadach dlya obyknovennykh differentsialnykh uravnenii vtorogo poryadka”, Differents. yp-niya, 49:7 (2013), 863–876 | Zbl

[17] Zolotareva N. D., Nikolaev E. S., “O stagnatsii v $p$-versii metoda konechnykh elementov”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibern., 2014, no. 3, 5–13