Fourier Type Super Convergence Study on DDGIC and Symmetric DDG Methods

In this paper, using Fourier analysis technique, we study the super convergence property of the DDGIC (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and the symmetric DDG (Vidden and Yan in J Comput Math 31(6):638–662, 2013) methods for diffusion equation. With $$k\hbox {th}$$kth degree piecewise polynomials applied, the convergence to the solution’s spatial derivative is $$k\hbox {th}$$kth order measured under regular norms. On the other hand when measuring the error in the weak sense or in its moment format, the error is super convergent with $$(k+2)\hbox {th}$$(k+2)th and $$(k+3)\hbox {th}$$(k+3)th orders for its first two moments with even order degree polynomial approximations. We carry out Fourier type (Von Neumann) error analysis and obtain the desired super convergent orders for the case of $$P^2$$P2 quadratic polynomial approximations. The theoretical predicted errors agree well with the numerical results.

[1]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model , 2017, Journal of Scientific Computing.

[2]  Chi-Wang Shu,et al.  Superconvergence and time evolution of discontinuous Galerkin finite element solutions , 2008, J. Comput. Phys..

[3]  Chi-Wang Shu,et al.  L2 Stability Analysis of the Central Discontinuous Galerkin Method and a Comparison between the Central and Regular Discontinuous Galerkin Methods , 2008 .

[4]  Hailiang Liu,et al.  The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems , 2008, SIAM J. Numer. Anal..

[5]  Zheng Chen,et al.  Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes , 2015, J. Comput. Phys..

[6]  Wei Guo,et al.  Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: Eigen-structure analysis based on Fourier approach , 2013, J. Comput. Phys..

[7]  Chad Vidden,et al.  A NEW DIRECT DISCONTINUOUS GALERKIN METHOD WITH SYMMETRIC STRUCTURE FOR NONLINEAR DIFFUSION EQUATIONS , 2013 .

[8]  Alexander Kurganov,et al.  New Interior Penalty Discontinuous Galerkin Methods for the Keller-Segel Chemotaxis Model , 2008, SIAM J. Numer. Anal..

[9]  Jue Yan,et al.  THE DIRECT DISCONTINUOUS GALERKIN (DDG) METHOD FOR DIFFUSION WITH INTERFACE CORRECTIONS , 2010 .

[10]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[11]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[12]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[13]  Jue Yan,et al.  A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations , 2013, J. Sci. Comput..

[14]  Mary F. Wheeler,et al.  A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[15]  Waixiang Cao,et al.  Superconvergence of the direct discontinuous Galerkin method for convection‐diffusion equations , 2017 .

[16]  Mengping Zhang,et al.  AN ANALYSIS OF THREE DIFFERENT FORMULATIONS OF THE DISCONTINUOUS GALERKIN METHOD FOR DIFFUSION EQUATIONS , 2003 .

[17]  Zheng Chen,et al.  Direct Discontinuous Galerkin Method and Its Variations for Second Order Elliptic Equations , 2017, J. Sci. Comput..

[18]  Yekaterina Epshteyn,et al.  Discontinuous Galerkin methods for the chemotaxis and haptotaxis models , 2009 .

[19]  Chi-Wang Shu,et al.  Central local discontinuous galerkin methods on overlapping cells for diffusion equations , 2011 .

[20]  Yingda Cheng,et al.  Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-diffusion equations , 2009 .

[21]  Chi-Wang Shu,et al.  Numerical resolution of discontinuous Galerkin methods for time dependent wave equations , 2011 .

[22]  Mengping Zhang,et al.  An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods , 2005 .

[23]  Mengping Zhang,et al.  Fourier Type Error Analysis of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations , 2012, J. Sci. Comput..