Locally Conservative Fluxes for the Continuous Galerkin Method

Abstract

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 interfaces and is motivated by the structure of the numerical traces of discontinuous Galerkin methods. This computation is nonlocal 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 asymptotically 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 continuous 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. Introduction.In this paper, we revisit the classical finite element method [13,20], otherwise known as the continuous Galerkin (CG) method, for second order elliptic problems, with the intention of showing how to overcome what is perhaps its main disadvantage, namely, the discontinuity of the normal component of the approximate flux across element interfaces. We show how to achieve this by means of an efficient postprocessing of the approximate solution provided by the CG method. We also show that the postprocessed flux is competitive with the flux provided by the Raviart-Thomas mixed method of corresponding degree.We illustrate our technique in the framework of the model second order elliptic boundary value problem

Superconvergent discontinuous Galerkin methods for second-order elliptic problems

“…Instead, we rely on an approach based on trying to rewrite the general formulation (1.2) as that of the Raviart-Thomas method. This approach was introduced in the study of the local conservativity properties of the continuous Galerkin method; see [11]. In our setting, this approach allows us to conclude that the convergence and superconvergence properties under consideration hold for any method satisfying the formulation (1.2) when, roughly speaking, the normal component of q h − q h is small enough, or, when the approximate flux q h is close to being in H(div, Ω).…”

To CG or to HDG: A Comparative Study

“…(This is unlike all the DG methods considered in the unified analysis of DG methods [4], which provide approximations converging with the same orders than the CG method.) A typically cited disadvantage of the CG method, namely, that it does not provide an approximation of the gradient whose normal component is single valued on the boundaries of the elements, can now be overcome [24] by using an L 2 -like global projection. It is hence not considered here as a competing issue.…”

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

“…A considerable amount of effort has been devoted to finding ways for postprocessing the approximate pressure of the CGFEM to produce a numerical velocity that is locally conservative and has continuous normal components across element interfaces [5,8,15]. Implementing the CG postprocessing procedure investigated in [8] takes also additional effort.…”

“…Implementing the CG postprocessing procedure investigated in [8] takes also additional effort. There are difficulties in formulating the discrete system for the flux traces on the mesh skeleton, especially choosing a good base for the involved space of jumps.…”

“…The CGFEM usually has a small number of pressure unknowns and the resulting discrete system is symmetric positive definite, but its numerical flux is not locally conservative and the normal flux is not continuous across element interfaces. Nontrivial postprocessing procedures need to be established [8] to obtain a numerical flux having the two desired properties. The DGFEM uses discontinuous shape functions on elements and hence has great flexibility in handling complicated geometry, even though interior penalty terms have to be introduced to enforce weak continuity across element interfaces.…”