Previous |  Up |  Next

Article

MSC: 46E35
Keywords:
weighted functional spaces; weighted functional sets; weighted Sobolev spaces.
Summary:
We study the properties of the weighted space $H_{2,\alpha }^k(\Omega )$ and weighted set $W_{2,\alpha }^k(\Omega ,\delta )$ for boundary value problem with singularity.
References:
[1] Apel, T., Sändig, A-M., Whiteman, J.R.: Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci., 19, 1, 1996, 63-85, DOI 10.1002/(SICI)1099-1476(19960110)19:1<63::AID-MMA764>3.0.CO;2-S | MR 1365264
[2] Arroyo, D., Bespalov, A., Heuer, N.: On the finite element method for elliptic problems with degenerate and singular coefficients. Math. Comp., 76, 258, 2007, 509-537, DOI 10.1090/S0025-5718-06-01910-7 | MR 2291826
[3] Assous, F., Ciarlet, J., Garcia, E., Segré, J.: Time-dependent Maxwell's equations with charges in singular geometries. Comput. Methods Appl. Mech. Engrg., 196, 1--3, 2006, 665-681, DOI 10.1016/j.cma.2006.07.007 | MR 2270148
[4] Assous, F., Jr, P. Ciarlet, Segré, J.: Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method. J. Comput. Phys., 161, 1, 2000, 218-249, DOI 10.1006/jcph.2000.6499 | MR 1762079
[5] Belytschko, T., Gracie, R., Ventura, G.: A review of extended generalized finite element methods for material modeling. Model. Simul. Sci. Eng., 17, 4, 2009, 043001, DOI 10.1088/0965-0393/17/4/043001
[6] Bordas, S., Duflot, M., Le, P.: A simple error estimator for extended finite elements. Int. J. Numer. Methods Eng., 24, 2008, 961-971, DOI 10.1002/cnm.1001 | MR 2474664
[7] Byfut, A., Schrödinger, A.: hp-Adaptive extended finite element method. Int. J. Numer. Methods Eng., 89, 2012, 1392-1418, DOI 10.1002/nme.3293 | MR 2899574
[8] Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math., 93, 2, 2002, 239-277, DOI 10.1007/s002110100388 | MR 1941397
[9] Costabel, M., Dauge, M., Schwab, C.: Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci., 15, 4, 2005, 575-622, DOI 10.1142/S0218202505000480 | MR 2137526
[10] Ivannikov, V., Tiago, C., Almeida, J.P. Moitinho de, Díez, P.: Meshless methods in dual analysis: theoretical and implementation issues. Proceedings of the V International Conference on Adaptive Modeling and Simulation (ADMOS 2011), Paris, France, 2011, 291-308,
[11] Li, H., Nistor, V.: Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM. J. Comput. Appl. Math., 224, 1, 2009, 320-338, DOI 10.1016/j.cam.2008.05.009 | MR 2474235
[12] Liu, G.R., Nguyen-Thoi, T.: Smoothed finite elements methods. 2010, CRC Press/Taylor & Francis Group,
[13] Morin, P., Hochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev., 44, 5, 2002, 631-658, DOI 10.1137/S0036144502409093 | MR 1980447
[14] Nguyen-Xuana, H., Liuc, G.R., Bordasd, S., Natarajane, S., Rabczukf, T.: An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order. Comput. Methods Appl .Mech. Engrg., 253, 2013, 252-273, DOI 10.1016/j.cma.2012.07.017 | MR 3002792
[15] Nguyen-Thoi, T., Vu-Do, H., Rabczuk, T., Nguyen-Xuan, H.: A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes. Comput. Methods Appl. Mech. Engrg., 199, 2010, 3005-3027, DOI 10.1016/j.cma.2010.06.017 | MR 2740774
[16] Nguyen, V.P., Rabczuk, T., Bordas, S., Dufolt, M.: Meshless methods: A review and computer implementation aspects. Math. Comput. Simulation., 79, 2008, 763-813, DOI 10.1016/j.matcom.2008.01.003 | MR 2477562
[17] Rukavishnikov, V.A.: On the differential properties of $R_\nu $-generalized solution of Dirichlet problem. Dokl. Akad. Nauk., 309, 6, 1989, 1318-1320, MR 1045325
[18] Rukavishnikov, V.A.: On the existence and uniqueness of an $R_\nu $-generalized solution of a boundary value problem with uncoordinated degeneration of the input data. Dokl. Math., 90, 2, 2014, 562-564, DOI 10.1134/S1064562414060155 | MR 3408915
[19] Rukavishnikov, V.A., Mosolapov, A.O.: New numerical method for solving time-harmonic Maxwell equations with strong singularity. J. Comput. Phys., 231, 6, 2012, 2438-2448, DOI 10.1016/j.jcp.2011.11.031 | MR 2881024
[20] Rukavishnikov, V.A, Mosolapov, A.O: Weighted edge finite element method for Maxwell's equations with strong singularity. Dokl. Math., 87, 2, 2013, 156-159, DOI 10.1134/S1064562413020105 | MR 2881024
[21] Rukavishnikov, V.A., Nikolaev, S.G.: On the $R_\nu $-generalized solution of the Lamé system with corner singularity. Dokl. Math., 92, 1, 2015, 421-423, DOI 10.1134/S1064562415040080 | MR 3443988
[22] Rukavishnikov, V.A., Nikolaev, S.G.: Weighted finite element method for an elasticity problem with singularity. Dokl. Math., 88, 3, 2013, 705-709, DOI 10.1134/S1064562413060215 | MR 3203317
[23] Rukavishnikov, V., Rukavishnikova, E.: On the existence and uniqueness of $R_\nu $-generalized solution for Dirichlet problem with singularity on all boundary. Abstr. Appl. Anal. 2014, 2014, 568726, MR 3230527
[24] Rukavishnikov, V.A., Rukavishnikova, E.I.: Dirichlet problem with degeneration of the input data on the boundary of the domain. Differ. Equ., 52, 5, 2016, 681-685, DOI 10.1134/S0012266116050141 | MR 3541462
[25] Rukavishnikov, V.A., Rukavishnikova, H.I.: The finite element method for boundary value problem with strong singularity. J. Comput. Appl. Math., 234, 9, 2010, 2870-2882, DOI 10.1016/j.cam.2010.01.020 | MR 2652132
[26] Rukavishnikov, V.A., Rukavishnikova, H.I.: On the error estimation of the finite element method for the boundary value problems with singularity in the Lebesgue weighted space. Numer. Funct. Anal. Optim., 34, 12, 2013, 1328-1347, DOI 10.1080/01630563.2013.809582 | MR 3175621
[27] Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The finite element method: its basis and fundamentals. Sixth edition. 2005, Elsevier, MR 3292660
Partner of
EuDML logo