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A weak Galerkin mixed finite element method for second order elliptic problems

Authors: Junping Wang and Xiu Ye
Journal: Math. Comp. 83 (2014), 2101-2126
MSC (2010): Primary 65N15, 65N30, 76D07; Secondary 35B45, 35J50
Published electronically: May 5, 2014
MathSciNet review: 3223326
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Abstract: 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 $ H^1$ and $ L^2$ norms are established for the corresponding weak Galerkin mixed finite element solutions.

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Additional Information

Junping Wang
Affiliation: Division of Mathematical Sciences, National Science Foundation, Arlington, Virginia 22230

Xiu Ye
Affiliation: Department of Mathematics, University of Arkansas at Little Rock, Little Rock, Arkansas 72204

Keywords: Weak Galerkin, finite element methods, discrete weak divergence, second order elliptic problems, mixed finite element methods
Received by editor(s): March 20, 2012
Received by editor(s) in revised form: November 23, 2012, and December 11, 2012
Published electronically: May 5, 2014
Additional Notes: The research of the first author was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation
This research was supported in part by National Science Foundation Grant DMS-1115097
Article copyright: © Copyright 2014 American Mathematical Society