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Discretization of div–curl Systems by Weak Galerkin Finite Element Methods on Polyhedral Partitions

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Abstract

In this paper, the authors present a new discretization scheme for div–curl systems defined in connected domains with heterogeneous media by using the weak Galerkin finite element method. Two types of boundary value problems are considered in this study: (1) normal boundary condition, and (2) tangential boundary condition. A new variational formulation is developed for the normal boundary value problem by using the Helmholtz decomposition which avoids the computation of functions in the harmonic fields. Both boundary value problems are reduced to a general saddle-point problem involving the curl and divergence operators, for which the weak Galerkin finite element method is employed and analyzed. The novelty of the numerical technique lies in the discretization of the divergence operator applied to vector fields in heterogeneous media. Error estimates of optimal order are established for the corresponding finite element approximations in various discrete Sobolev norms.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Chunmei Wang

  2. Taizhou College, Nanjing Normal University, Taizhou, 225300, China

    Chunmei Wang

  3. Division of Mathematical Sciences, National Science Foundation, Arlington, VA, 22230, USA

    Junping Wang

Authors
  1. Chunmei Wang
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  2. Junping Wang
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Correspondence to Junping Wang.

Additional information

The research of Chunmei Wang was partially supported by National Science Foundation Award DMS-1522586, National Natural Science Foundation of China Award #11526113, Jiangsu Key Lab for NSLSCS Grant #201602, and by Jiangsu Provincial Foundation Award #BK20050538. The research of Junping Wang was supported by the NSF IR/D program, while working at National Science 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.

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Wang, C., Wang, J. Discretization of div–curl Systems by Weak Galerkin Finite Element Methods on Polyhedral Partitions. J Sci Comput 68, 1144–1171 (2016). https://doi.org/10.1007/s10915-016-0176-y

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  • DOI: https://doi.org/10.1007/s10915-016-0176-y

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