Numerische Mathematik

, Volume 139, Issue 2, pp 447–478 | Cite as

Interior energy error estimates for the weak Galerkin finite element method

  • Hengguang LiEmail author
  • Lin Mu
  • Xiu Ye


Consider the Poisson equation in a polytopal domain \(\Omega \subset {\mathbb {R}}^d\) (\(d=2, 3\)) as the model problem. We study interior energy error estimates for the weak Galerkin finite element approximation to elliptic boundary value problems. In particular, we show that the interior error in the energy norm is bounded by three components: the best local approximation error, the error in negative norms, and the trace error on the element boundaries. This implies that the interior convergence rate can be polluted by solution singularities on the domain boundary, even when the solution is smooth in the interior region. Numerical results are reported to support the theoretical findings. To the best of our knowledge, this is the first local energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes.


Weak Galerkin Finite element methods Interior estimates Second-order elliptic problems 

Mathematics Subject Classification

Primary 65N15 65N30 Secondary 35J50 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  3. 3.Department of MathematicsUniversity of Arkansas at Little RockLittle RockUSA

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