Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor


A virtual element method is introduced for the mixed approximation of a simple model problem for the Laplace operator on a polyhedron. The method is fully analysed when the meshes are made up of triangular right prisms, pyramids and tetrahedra. The local discrete spaces coincide with the lowest order Raviart–Thomas spaces on tetrahedral and triangular right prismatic elements, and extend them to pyramidal elements. The discrete scheme is well posed and optimal interpolation error estimates are proved on meshes which allow for anisotropic elements. In particular, local interpolation error estimates for the discrete element space are optimal and anisotropic on anisotropic right prisms. Furthermore, a discretization of the model problem in the presence of edge and vertex singularities is analysed for the proposed method on a family of suitably designed graded meshes, and optimal estimates for the approximation error are obtained, extending in this way the results of Farhloul et al. (ESAIM Math Model Numer Anal 35:907–920, 2001) where cylindrical domains with edge singularities were considered. © 2019, Istituto di Informatica e Telematica (IIT).


Documento: Artículo
Título:A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes
Autor:Jawtuschenko, A.B.; Lombardi, A.L.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, Buenos Aires, 1428, Argentina
Departamento de Matemática, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Av. Pellegrini 250, Rosario, 2000, Argentina
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), CCT Rosario, Argentina
Palabras clave:Anisotropic hybrid meshes; Edge and vertex singularities; Mixed finite element method; Raviart–Thomas spaces; Virtual element method
Título revista:Calcolo
Título revista abreviado:Calcolo


  • Acosta, G., Apel, T., Durán, R.G., Lombardi, A.L., Error estimates for Raviart–Thomas interpolation of any order on anisotropic tetrahedra (2011) Math. Comput., 80, pp. 141-163
  • Apel, T., (1999) Anisotropic Finite Elements: Local Estimates and Applications, , Series Advances Numerical Mathematics, Teubner, Stuttgart
  • Apel, T., Lombardi, A.L., Winkler, M., Anisotropic mesh refinement in polyhedral domains: error estimates with data in L 2 (Ω) (2014) ESAIM Math. Model Numer. Anal., 48, pp. 1117-1145
  • Apel, T., Nicaise, S., The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges (1998) Math. Models Appl. Sci., 21, pp. 519-549
  • Beirão Da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A., Basic principles of virtual element methods (2013) Math. Models Methods Appl. Sci., 23 (1), pp. 199-214
  • Beirão Da Veiga, L., Brezzi, F., Marini, L.D., Russo, A., Mixed virtual element methods for general second order elliptic problems on polygonal meshes (2016) ESAIM Math. Model. Numer. Anal., 50, pp. 727-747
  • Beirão Da Veiga, L., Mora, D., Rivera, G., Rodriguez, R., A virtual element method for the acoustic vibration problem (2017) Numer. Math., 136, pp. 725-763
  • Bergot, M., Cohen, G., Duruflé, M., Higher-order finite elements for hybrid meshes using new nodal pyramidal elements (2010) J. Sci. Comput., 42, pp. 345-381
  • Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M., (2008) Mixed Finite Elements, Compatibility Conditions and Applications, 1939. , Lecture Notes Mathematics, Springer, Berlin
  • Brezzi, F., Falk, R.S., Marini, L.D., Basic principles of mixed Virtual Element Methods (2014) ESAIM Math. Model. Numer. Anal., 48, pp. 1227-1240
  • Ciarlet, P., (1978) The Finite Element Method for Elliptic Problems, 4. , Studies Mathematics and Its Applications, North-Holland, Amsterdam
  • Farhloul, M., Nicaise, S., Paquet, L., Some mixed finite element methods on anisotropic meshes (2001) ESAIM Math. Model. Numer. Anal., 35, pp. 907-920
  • Gradinaru, V., Hiptmair, R., Whitney elements on pyramids (1999) Electron. Trans. Numer. Anal., 8, pp. 154-168
  • Grisvard, P., (1985) Elliptic Problems in Nonsmooth Domains, 24. , Monographs and Studies Mathematics, Pitman (Advanced Publishing Program), Boston
  • Jawtuschenko, A.B., (2018) Métodos mixtos con mallas híbridas para problemas elípticos en dominios poliedrales, ,, Tesis de Doctorado de la Universidad de Buenos Aires, Accessed 19 Mar 2019
  • Nédéléc, J.C., A new family of mixed finite elements in R 3 (1986) Numer. Math., 50, pp. 57-81
  • Nigam, N., Phillips, J., High-order conforming finite elements on pyramids (2012) IMA J. Numer. Anal., 32, pp. 448-483
  • Owen, S.J., Saigal, S., Formation of pyramids elements for hexahedra to tetrahedra transitions (2001) Comput. Methods Appl. Mech. Eng., 190, pp. 4505-4518
  • Raugel, G., Résolution numérique par une méthode d’éléments finis du problème Dirichlet pour le laplacien dans un polygone (1978) C. R. Acad. Sci. Paris Ser. A, 286, pp. A791-A794
  • Raviart, P.A., Thomas, J.-M., A mixed finite element method for second order elliptic problems (1977) Mathematical Aspects of the Finite Element Method, 606. , Galligani I, Magenes E, (eds), Lectures Notes Mathematics, Springer, Berlin


---------- APA ----------
Jawtuschenko, A.B. & Lombardi, A.L. (2019) . A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes. Calcolo, 56(2).
---------- CHICAGO ----------
Jawtuschenko, A.B., Lombardi, A.L. "A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes" . Calcolo 56, no. 2 (2019).
---------- MLA ----------
Jawtuschenko, A.B., Lombardi, A.L. "A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes" . Calcolo, vol. 56, no. 2, 2019.
---------- VANCOUVER ----------
Jawtuschenko, A.B., Lombardi, A.L. A mixed discretization of elliptic problems on polyhedra using anisotropic hybrid meshes. Calcolo. 2019;56(2).