Abstract
A new local discontinuous Galerkin method for convection–diffusion equations on overlapping mesh was introduced in Du et al. (BIT Numer Math 1–24, 2019). In the new method, the primary variable u and auxiliary variable \(p=u_x\) are solved on different meshes. The stability and suboptimal error estimates for problems with periodic boundary conditions were derived. Numerical experiments demonstrated that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. Several alternatives to gain optimal convergence rates were demonstrated in Du et al. (2019). However, the reason for accuracy degeneration is still unclear. In this paper, we will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. We explicitly write out the error between the numerical and exact solutions, and investigate the reason for the accuracy degeneration. Moreover, we also find out some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh. Since the current work is based on Fourier analysis, we only consider uniform meshes. Numerical experiments will be given to verify the theoretical analysis.
Similar content being viewed by others
References
Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279 (1997)
Cao, W., Zhang, Z.: Superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. Math. Comput. 85, 63–84 (2016)
Cheng, Y., Shu, C.-W.: Superconvergence of local discontinuous Galerkin methods for convection–diffusion equations. Comput. Struct. 87, 630–641 (2009)
Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection–diffusion equations in one space dimension. SIAM J. Numer. Anal. 47, 4044–4072 (2010)
Chuenjarern, N., Xu, Z., Yang, Y.: High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes. J. Comput. Phys. 378, 110–128 (2019)
Chung, E., Lee, C.S.: A staggered discontinuous Galerkin method for convection–diffusion equations. J. Numer. Math. 20, 1–31 (2012)
Cockburn, B., Hou, S., Shu, C.W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)
Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: one-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)
Cockburn, B., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: general framework. Math. Comput. 52, 411–435 (1989)
Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin method for conservation laws. V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)
Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
Douglas Jr., J., Ewing, R.E., Wheeler, M.F.: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, R.A.I.R.O. Analyse numérique 17, 249–256 (1983)
Douglas Jr., J., Ewing, R.E., Wheeler, M.F.: The approximation of the pressure by a mixed method in the simulation of miscible displacement, R.A.I.R.O. Analyse numérique 17, 17–33 (1983)
Du, J., Yang, Y.: Maximum-principle-preserving third-order local discontinuous Galerkin methods on overlapping meshes. J. Comput. Phys. 377, 117–141 (2019)
Du, J., Yang, Y., Chung, E.: Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes. BIT Numer. Math. (2019). https://doi.org/10.1007/s10543-019-00757-4
Gelfand, I.M.: Some questions of analysis and differential equations. Am. Math. Soc. Transl. 26, 201–219 (1963)
Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
Guo, H., Yu, F., Yang, Y.: Local discontinuous Galerkin method for incompressible miscible displacement problem in porous media. J. Sci. Comput. 71, 615–633 (2017)
Guo, H., Yang, Y.: Bound-preserving discontinuous Galerkin method for compressible miscible displacement problem in porous media. SIAM J. Sci. Comput. 39, A1969–A1990 (2017)
Hurd, A.E., Sattinger, D.H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Am. Math. Soc. 132, 159–174 (1968)
Keller, E.F., Segel, L.A.: Initiation on slime mold aggregation viewed as instability. J. Theor. Biol. 26, 399–415 (1970)
Li, X., Shu, C.-W., Yang, Y.: Local discontinuous Galerkin method for the Keller–Segel chemotaxis model. J. Sci. Comput. 73, 943–967 (2017)
Liu, Y., Shu, C.-W., Tadmor, E., Zhang, M.: Central local discontinuous Galerkin method on overlapping cells for diffusion equations. ESAIM Math. Model. Numer. Anal. (M2AN) 45, 1009–1032 (2011)
Patlak, C.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311338 (1953)
Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Scientific Laboratory Report LA-UR-73-479, Los Alamos (1973)
Wang, H., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for advection–diffusion problems. SIAM J. Numer. Anal. 53, 206–227 (2015)
Wang, H., Shu, C.-W., Zhang, Q.: Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems. Appl. Math. Comput. 272, 237–258 (2016)
Wang, H., Wang, S., Zhang, Q., Shu, C.-W.: Local discontinuous Galerkin methods with implicit–explicit time marching for multi-dimensional convection–diffusion problems. ESAIM M2AN 50, 1083–1105 (2016)
Xu, Z., Yang, Y., Guo, H.: High-order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes. J. Comput. Phys. 390, 323–341 (2019)
Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. J. Comput. Math. 33, 323–340 (2015)
Yu, F., Guo, H., Chuenjarern, N., Yang, Y.: Conservative local discontinuous Galerkin method for compressible miscible displacements in porous media. J. Sci. Comput. 73, 1249–1275 (2017)
Zhang, M., Yan, J.: Fourier type error analysis of the direct discontinuous Galerkin method and its variations for diffusion equations. J. Sci. Comput. 52, 638–655 (2012)
Zhang, M., Shu, C.W.: An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13, 395–413 (2003)
Zhong, X., Shu, C.-W.: Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Eng. 200, 2814–2827 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by National Science Foundation DMS-1818467.
Rights and permissions
About this article
Cite this article
Chuenjarern, N., Yang, Y. Fourier Analysis of Local Discontinuous Galerkin Methods for Linear Parabolic Equations on Overlapping Meshes. J Sci Comput 81, 671–688 (2019). https://doi.org/10.1007/s10915-019-01030-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-019-01030-0