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.
Cockburn, Bernardo; Gopalakrishnan, Jay; and Wang, Haiying, "Locally Conservative Fluxes for the Continuous Galerkin Method" (2007). Mathematics and Statistics Faculty Publications and Presentations. 64.