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A finite element method using singular functions for Poisson equations: Mixed boundary conditions. (English) Zbl 1124.65108
The purpose of this paper is to extend some results for Dirichlet boundary value problems of Poisson type in the framework of mixed boundary conditions. A singular function representation of the solution is provided for various boundary conditions and a variational problem for the regular part of the solution is derived. Next, the authors introduce a finite element approximation and develop the error analysis. Numerical results conclude this paper and illustrate the main abstract results.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65N15 Error bounds for boundary value problems involving PDEs
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##### References:
 [1] Babuska, I., Finite element method for domains with corners, Computing, 6, 264-273, (1970) · Zbl 0224.65031 [2] Babuska, I.; Kellogg, R.B.; Pitkäranta, J., Direct and inverse error estimates for finite elements with mesh refinements, Numer. math., 33, 447-471, (1979) · Zbl 0423.65057 [3] Babuska, I.; Miller, A., The post-processing approach in the finite element method—part 2: the calculation of stress intensity factors, Int. J. numer. methods engrg., 20, 1111-1129, (1984) · Zbl 0535.73053 [4] Babuska, I.; Suri, M., The h-p version of the finite element method with quasiuniform meshes, RAIRO modél. math. anal. numér., 21, 199-238, (1987) · Zbl 0623.65113 [5] Blum, H.; Dobrowolski, M., On finite element methods for elliptic equations on domains with corners, Computing, 28, 53-63, (1982) · Zbl 0465.65059 [6] H. Blum, Numerical treatment of corner and crack singularities, Dissertation, Bonn, 1981. · Zbl 0674.73070 [7] Bourlard, M.; Dauge, M.; Lubuma, M.-S.; Nicaise, S., Coefficients of the singularities for elliptic boundary value problems on domains with conical points III. finite element methods on polygonal domains, SIAM numer. anal., 29, 136-155, (1992) · Zbl 0794.35015 [8] Brenner, S., Multigrid methods for the computation of singular solutions and stress intensity factor I: corner singularities, Math. comput., 68, 226, 559-583, (1999) · Zbl 1043.65136 [9] Cai, Z.; Kim, S., A finite element method using singular functions for the Poisson equation: corner singularities, SIAM J. numer. anal., 39, 286-299, (2001) · Zbl 0992.65122 [10] Cai, Z.; Kim, S.; Shin, B., Solution methods for the Poisson equation: corner singularities, SIAM J. sci. comput., 39, 286-299, (2001) [11] Cai, Z.; Kim, S.; Woo, G., A finite element method using singular functions for the Poisson equation: crack singularities, Numer. linear algebra appl., 9, 445-455, (2002) · Zbl 1071.65152 [12] Dauge, M., Elliptic boundary value problems on corner domains, () · Zbl 0668.35001 [13] Destuynder, P.; Djaoua, M., Estimation de l’erreur sur le coefficient de la singularité de la solution d’un probléme elliptique sur un ouvert avec coin, RAIRO ser. rouge, 14, 239-248, (1980) · Zbl 0456.65062 [14] M. Djaoua, Equations Intégrales pour un Probleme Singulier dans le Plan, These de Troisieme Cycle, Universite Pierre et Marie Curie, Paris, 1977. [15] Dobrowolski, M., Numerical approximation of elliptic interface and corner problems, (1981), Habilitation-schrift Bonn [16] Fix, G.J.; Gulati, S.; Wakoff, G.I., On the use of singular functions with finite elements approximations, J. comput. phys., 13, 209-228, (1973) · Zbl 0273.35004 [17] Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman Boston, MA · Zbl 0695.35060 [18] Rüde, U., Mathematical and computational techniques for multilevel adaptive methods, Frontiers in applied mathematics, vol. 13, (1993), SIAM Philadelphia · Zbl 0857.65127 [19] Schatz, A.; Wahlbin, L.; Schatz, A.; Wahlbin, L., Maximum norm estimates in the finite element method on plane polygonal domains, part 2 (refinements), Math. comput., Math. comput., 33, 146, 465-492, (1979) · Zbl 0417.65053 [20] Yserentant, H., The convergence of multi-level methods for solving finite element equations in the presence of singularities, Math. comput., 47, 399-409, (1986) · Zbl 0615.65115 [21] Zhang, S.; Zhang, S.; Zhang, S., Optimal-order nonnested multigrid methods for solving finite element equations. III: on degenerate meshes, Math. comput., Math. comput., Math. comput., 64, 23-49, (1995) · Zbl 0824.65124
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