The Properties of the Weighted Space Hk2,α (Ω) and Weighted Set Wk2,α(Ω, δ)

V. A. Rukavishnikov 1 , E. V. Matveeva 2  and E. I. Rukavishnikova 1
  • 1 Computing Center of Far-Eastern Branch Russian Academy of Sciences, Kim-Yu-Chen Str. 65,, Khabarovsk, Russia
  • 2 Far Eastern State Transport University, Serisheva Str. 47,, Khabarovsk, Russia


We study the properties of the weighted space Hk(Ω) and weighted set Wk(Ω, δ)for boundary value problem with singularity.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] T. Apel, A-M. Sändig, J.R. Whiteman: 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.

  • [2] D. Arroyo, A. Bespalov, N. Heuer: On the finite element method for elliptic problems with degenerate and singular coefficients. Math. Comp. 76 (258) (2007) 509-537.

  • [3] F. Assous, J. Ciarlet, E. Garcia, J. Segré: Time-dependent Maxwell's equations with charges in singular geometries. Comput. Methods Appl. Mech. Engrg. 196 (1-3) (2006) 665-681.

  • [4] F. Assous, P. Ciarlet Jr, J. Segré: 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.

  • [5] T. Belytschko, R. Gracie, G. Ventura: A review of extended generalized finite element methods for material modeling. Model. Simul. Sci. Eng. 17 (4) (2009) 043001.

  • [6] S. Bordas, M. Duot, P. Le: A simple error estimator for extended finite elements. Int. J. Numer. Methods Eng. 24 (2008) 961-971.

  • [7] A. Byfut, A. Schrödinger: hp-Adaptive extended finite element method. Int. J. Numer. Methods Eng. 89 (2012) 1392-1418.

  • [8] M. Costabel, M. Dauge: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93 (2) (2002) 239-277.

  • [9] M. Costabel, M. Dauge, C. Schwab: Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci. 15 (4) (2005) 575-622.

  • [10] V. Ivannikov, C. Tiago, J.P. Moitinho de Almeida, P. Díez: Meshless methods in dual analysis: theoretical and implementation issues. In: Proceedings of the V International Conference on Adaptive Modeling and Simulation (ADMOS 2011), Paris, France. (2011) 291-308.

  • [11] H. Li, V. Nistor: Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM. J. Comput. Appl. Math. 224 (1) (2009) 320-338.

  • [12] G.R. Liu, T. Nguyen-Thoi: Smoothed finite elements methods. CRC Press/Taylor & Francis Group (2010).

  • [13] P. Morin, R.H. Hochetto, K.G. Siebert: Convergence of adaptive finite element methods. SIAM Rev. 44 (5) (2002) 631-658.

  • [14] H. Nguyen-Xuana, G.R. Liuc, S. Bordasd, S. Natarajane, T. Rabczukf: An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order. Comput. Methods Appl .Mech. Engrg. 253 (2013) 252-273.

  • [15] T. Nguyen-Thoi, H. Vu-Do, T. Rabczuk, H. Nguyen-Xuan: 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.

  • [16] V.P. Nguyen, T. Rabczuk, S. Bordas, M. Dufolt: Meshless methods: A review and computer implementation aspects. Math. Comput. Simulation. 79 (2008) 763-813.

  • [17] V.A. Rukavishnikov: On the difierential properties of Rfi-generalized solution of Dirichlet problem. Dokl. Akad. Nauk. 309 (6) (1989) 1318-1320.

  • [18] V.A. Rukavishnikov: On the existence and uniqueness of an Rι-generalized solution of a boundary value problem with uncoordinated degeneration of the input data. Dokl. Math. 90 (2) (2014) 562-564.

  • [19] V.A. Rukavishnikov, A.O. Mosolapov: New numerical method for solving time-harmonic Maxwell equations with strong singularity. J. Comput. Phys. 231 (6) (2012) 2438-2448.

  • [20] V.A Rukavishnikov, A.O Mosolapov: Weighted edge finite element method for Maxwell's equations with strong singularity. Dokl. Math. 87 (2) (2013) 156-159.

  • [21] V.A. Rukavishnikov, S.G. Nikolaev: On the Rfi-generalized solution of the Lamé system with corner singularity. Dokl. Math. 92 (1) (2015) 421-423.

  • [22] V.A. Rukavishnikov, S.G. Nikolaev: Weighted finite element method for an elasticity problem with singularity. Dokl. Math. 88 (3) (2013) 705-709.

  • [23] V. Rukavishnikov, E. Rukavishnikova: On the existence and uniqueness of Rι-generalized solution for Dirichlet problem with singularity on all boundary. Abstr. Appl. Anal. 2014 (2014) 568726.

  • [24] V.A. Rukavishnikov, E.I. Rukavishnikova: Dirichlet problem with degeneration of the input data on the boundary of the domain. Difier. Equ. 52 (5) (2016) 681-685.

  • [25] V.A. Rukavishnikov, H.I. Rukavishnikova: The finite element method for boundary value problem with strong singularity. J. Comput. Appl. Math. 234 (9) (2010) 2870-2882.

  • [26] V.A. Rukavishnikov, H.I. Rukavishnikova: 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.

  • [27] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu: The finite element method: its basis and fundamentals. Sixth edition. Elsevier (2005).


Journal + Issues