A virtual element method with arbitrary regularity

and Gianmarco Manzini∗ Los Alamos National Laboratory, Plasma Physics and Applied Mathematics Group, T-5, Theoretical Division, MS B284, Los Alamos, NM 87544, USA and Centro di Simulazione Numerica Avanzata (CeSNA)–IUSS Pavia, v.le Lungo Ticino Sforza 56, I 27100 Pavia, Italy and Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) – CNR, via Ferrata 1, I 27100 Pavia, Italy ∗Corresponding author: marco.manzini@imati.cnr.it gmanzini@lanl.gov

[1]  D. W. Scharpf,et al.  The TUBA Family of Plate Elements for the Matrix Displacement Method , 1968, The Aeronautical Journal (1968).

[2]  K. Bell A refined triangular plate bending finite element , 1969 .

[3]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[4]  P. Grisvard Elliptic Problems in Nonsmooth Domains , 1985 .

[5]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[6]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[7]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[8]  Miguel Ángel Martínez,et al.  Overview and recent advances in natural neighbour galerkin methods , 2003 .

[9]  N. Sukumar,et al.  Conforming polygonal finite elements , 2004 .

[10]  Konstantin Lipnikov,et al.  Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes , 2005, SIAM J. Numer. Anal..

[11]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[12]  N. Sukumar,et al.  Archives of Computational Methods in Engineering Recent Advances in the Construction of Polygonal Finite Element Interpolants , 2022 .

[13]  Lourenço Beirão da Veiga,et al.  A residual based error estimator for the Mimetic Finite Difference method , 2007, Numerische Mathematik.

[14]  Gianmarco Manzini,et al.  Flux reconstruction and solution post-processing in mimetic finite difference methods , 2008 .

[15]  Gianmarco Manzini,et al.  An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems , 2008 .

[16]  Annalisa Buffa,et al.  Mimetic finite differences for elliptic problems , 2009 .

[17]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[18]  Gianmarco Manzini,et al.  Convergence Analysis of the Mimetic Finite Difference Method for Elliptic Problems , 2009, SIAM J. Numer. Anal..

[19]  Gianmarco Manzini,et al.  Mimetic finite difference method for the Stokes problem on polygonal meshes , 2009, J. Comput. Phys..

[20]  Gianmarco Manzini,et al.  Convergence analysis of the high-order mimetic finite difference method , 2009, Numerische Mathematik.

[21]  L. B. D. Veiga,et al.  A Mimetic discretization method for linear elasticity , 2010 .

[22]  T. Belytschko,et al.  A generalized finite element formulation for arbitrary basis functions: From isogeometric analysis to XFEM , 2010 .

[23]  T. Belytschko,et al.  The extended/generalized finite element method: An overview of the method and its applications , 2010 .

[24]  Gianmarco Manzini,et al.  Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems , 2011, J. Comput. Phys..

[25]  N. Sukumar,et al.  Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons , 2011 .

[26]  Gianmarco Manzini,et al.  Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes , 2011, SIAM J. Numer. Anal..

[27]  E. Wachspress,et al.  A Rational Finite Element Basis , 1975 .

[28]  Lourenço Beirão da Veiga,et al.  Virtual Elements for Linear Elasticity Problems , 2013, SIAM J. Numer. Anal..

[29]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[30]  Gianmarco Manzini,et al.  Mimetic finite difference method , 2014, J. Comput. Phys..