Geodesic
On the error estimation of the finite element method for the third boundary value problem with singularity in the space $L_{2,\nu+\gamma}^*$
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 2, pp. 177-185.

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The paper analyzes the finite element method for the third boundary value problem for non-self-conjugate second order elliptic equation with coordinated degeneration of initial data and with strong singularity of solution. The scheme of the finite element method is constructed on the basis of the definition of $R_{\nu}$-generalized solution to the problem, and the finite element space contains singular power functions. It is established that the rate of convergence of an approximate solution to the exact $R_{\nu}$-generalized solution in the norm of the Lebesgue weight space $L_{2,\nu+\gamma}^*\Omega$ has second order. 
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V. A. Rukavishnikov; E. I. Rukavishnikova. On the error estimation of the finite element method for the third boundary value problem with singularity in the space $L_{2,\nu+\gamma}^*$. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 2, pp. 177-185. http://geodesic.mathdoc.fr/item/SJVM_2004_7_2_a7/

[1] Rukavishnikov V. A., Rukavishnikova E. I., “Metod konechnykh elementov dlya pervoi kraevoi zadachi s soglasovannym vyrozhdeniem iskhodnykh dannykh”, Dokl. RAN, 338:6 (1994), 731–733 | Zbl

[2] Rukavishnikov V. A., Rukavishnikova E. I., “The finite element method for the third boundary value problem with strong singularity of solution”, ENUMATH 1997: Proceedings of Second European Conference on Numerical Mathematics and Advanced Applications, World Scientific Publishing Company, Singapore, 1998, 540–548 | MR | Zbl

[3] Rukavishnikov V. A., Rukavishnikova E. I., Tretya kraevaya zadacha s silnoi singulyarnostyu, Preprint, Vychislitelnyi tsentr DVO RAN, Vladivostok, 1997

[4] Rukavishnikov V. A., Ereklintsev A. G., Koertsitivnye otsenki $R_\nu$-obobschennogo resheniya dlya pervoi i tretei kraevykh zadach s singulyarnostyu resheniya, Preprint, Vychislitelnyi tsentr DVO RAN, Vladivostok, 2003

[5] Aubin J. P., “Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin's and finite difference methods”, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 599–637 | MR | Zbl

[6] Nitsche J. A., “Ein Kriterium fur die quasi-optimalitat des Ritzchen Verfahrens”, J. Numer. Math., 11 (1968), 346–348 | DOI | MR | Zbl

[7] Syarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR