A Weak Galerkin Mixed Finite Element Method for Biharmonic Equations

  • Lin Mu
  • Junping WangEmail author
  • Yanqiu Wang
  • Xiu Ye
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 45)


This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in (Wang et al., Comput. Appl. Math. 241:103–115, 2013) for second-order elliptic problems, is based on the concept of discrete weak gradients. The method uses completely discrete finite element functions, and, using certain discrete spaces and with stabilization, it works on partitions of arbitrary polygon or polyhedron. In this article, the weak Galerkin method is applied to discretize the Ciarlet–Raviart mixed formulation for the biharmonic equation. In particular, an a priori error estimation is given for the corresponding finite element approximations. The error analysis essentially follows the framework of Babus̆ka, Osborn, and Pitkäranta (Math. Comp. 35:1039–1062, 1980) and uses specially designed mesh-dependent norms. The proof is technically tedious due to the discontinuous nature of the weak Galerkin finite element functions. Some computational results are presented to demonstrate the efficiency of the method.


Weak Galerkin finite element methods Discrete gradient Biharmonic equations Mixed finite element methods 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LancingUSA
  2. 2.Division of Mathematical SciencesNational Science FoundationArlingtonUSA
  3. 3.Department of MathematicsOklahoma State UniversityStillwaterUSA
  4. 4.Department of MathematicsUniversity of Arkansas at Little RockLittle RockUSA

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