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Locally Conservative Fluxes for the Continuous Galerkin Method
Mathematics and Statistics Faculty Publications and Presentations
  • Bernardo Cockburn, University of Minnesota - Twin Cities
  • Jay Gopalakrishnan, Portland State University
  • Haiying Wang, University of Minnesota - Twin Cities
Document Type
Publication Date
  • Galerkin methods,
  • Finite element method,
  • Elliptic differential equations,
  • Numerical analysis,
  • Conservation laws (Mathematics)
The standard continuous Galerkin (CG) finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations, a much needed quantity in many applications. We show how to overcome this shortcoming by using a two step postprocessing. The first step is the computation of a numerical flux trace defined on element inter- faces and is motivated by the structure of the numerical traces of discontinuous Galerkin methods. This computation is non-local in that it requires the solution of a symmetric positive definite system, but the system is well conditioned independently of mesh size, so it can be solved at asymptoti- cally optimal cost. The second step is a local element by element postprocessing of the CG solution incorporating the result of the first step. This leads to a conservative flux approximation with con- tinuous normal components. This postprocessing applies for the CG method in its standard form or for a hybridized version of it. We present the hybridized version since it allows easy handling of variable-degree polynomials and hanging nodes. Furthermore, we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the Raviart-Thomas mixed method of corresponding degree.

This is the author’s version of a work that was accepted for publication in SIAM Journal on Numerical Analysis. A definitive version was subsequently published in SIAM Journal on Numerical Analysis, 2007. Vol. 45 Issue 4, p. 1742-1776.

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Citation Information
Bernardo Cockburn, Jay Gopalakrishnan and Haiying Wang. "Locally Conservative Fluxes for the Continuous Galerkin Method" (2007)
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