A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC PROBLEMS

A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H1 and L2 norms are established for the corresponding weak Galerkin mixed finite element solutions.

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

[2]  Shan Zhao,et al.  WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS. , 2013, Journal of computational physics.

[3]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[4]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[5]  Valmor F. de Almeida,et al.  Adaptive Hybrid Mesh Refinement for Multiphysics Applications , 2007 .

[6]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[7]  J. David Moulton,et al.  Convergence of mimetic finite difference discretizations of the diffusion equation , 2001, J. Num. Math..

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

[9]  Junping Wang,et al.  A weak Galerkin finite element method for the stokes equations , 2013, Adv. Comput. Math..

[10]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

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

[12]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[13]  Junping Wang,et al.  A weak Galerkin finite element method for second-order elliptic problems , 2011, J. Comput. Appl. Math..

[14]  Paola Causin,et al.  A Discontinuous Petrov-Galerkin Method with Lagrangian Multipliers for Second Order Elliptic Problems , 2005, SIAM J. Numer. Anal..

[15]  D. Arnold,et al.  Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates , 1985 .

[16]  Robert Eymard,et al.  A mixed finite volume scheme for anisotropic diffusion problems on any grid , 2006, Numerische Mathematik.

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

[18]  Franco Brezzi Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods (Springer Series in Computational Mathematics) , 1991 .

[19]  A. Ern,et al.  Mathematical Aspects of Discontinuous Galerkin Methods , 2011 .

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

[21]  Junping Wang,et al.  A computational study of the weak Galerkin method for second-order elliptic equations , 2011, Numerical Algorithms.

[22]  M. Fortin,et al.  Mixed finite elements for second order elliptic problems in three variables , 1987 .