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Alternative Methods Of The NIPG DGFEM Superconvergence

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\subsection{NIPG DGFEM Superconvergence 2011}

Now that the continuous standard Galerkin solution over a quasi-uniform mesh may oscillate as $\epsilon \to 0$. An alternative tool will be a discontinuous Galerkin (DG) method where the oscillation can be avoided provided that an appropriate mesh refinement is applied, to capture the boundary layer behavior. The origins of the DG methods can be traced back to the seventies where they had been proposed as variational methods for numerically solving initial-value problems and transport problems. It is well known that the DG methods, in particular the local DG (LDG) method \cite{210}, are highly stable and effective for convection diffusion problems \cite{211}. Whereas, the main feature of the …show more content…

\cite{212,213} and Zhang et al. \cite{214} adopted the local discontinuous Galerkin (LDG) method to solve convection diffusion equations and analyzed the corresponding superconvergence properties. On the other hand, non-symmetric discontinuous Galerkin method with interior penalty (the NIPG method), originally designed for elliptic equations, is analyzed by Zarin and Roos \cite{87} for convection–diffusion problems with parabolic layers.
A disadvantage of DG method is that the method produces more degrees of freedom than the continuous finite element method (CFEM). With this motivation, this work derives and analyzes a coupled approach of LDG and CFEM on a layer adapted Shishkin mesh for singularly perturbed convection–diffusion problems. By splitting the domain into the coarse and the fine part, we adopt the CFEM with linear elements in the fine part where the mesh size is comparable with $\epsilon$, and use LDG method in the coarse part for its stabilization.

The idea of combining DG and CFEM to obtain the advantages of both methods is not new. A coupled LDG-CFEM approach has also been studied by Perugia and Sch{\"o}tzau \cite{215} for the modeling of elliptic problems arising in electromagnetics. Roos and Zarin \cite{79}, Zarin \cite{82} analyzed the NIPG-CFEM coupled method on Shishkin mesh for two-dimensional convection diffusion problems with exponentially layers or characteristic layers. In \cite{209}, Zhu et

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