Corpus ID: 8143600

LOCAL ERROR ESTIMATES OF THE LDG METHOD FOR 1-D SINGULARLY PERTURBED PROBLEMS

@article{Zhu2013LOCALEE,
title={LOCAL ERROR ESTIMATES OF THE LDG METHOD FOR 1-D SINGULARLY PERTURBED PROBLEMS},
author={Huiqing Zhu and Zhimin Zhang},
journal={International Journal of Numerical Analysis and Modeling},
year={2013},
volume={10},
pages={350-373}
}
• Published 2013
• Mathematics
• International Journal of Numerical Analysis and Modeling
In this paper local discontinuous Galerkin method (LDG) was analyzed for solving 1-D convection-diffusion equations with a boundary layer near the outflow boundary. Local error estimates are established on quasi-uniform meshes with maximum mesh size h. On a subdomain with O(hln(1/h)) distance away from the outflow boundary, the L 2 error of the approximations to the solution and its derivative converges at the optimal rate O(h k+1 ) when polynomials of degree at most k are used. Numerical… Expand

Figures and Tables from this paper

Local Analysis of Local Discontinuous Galerkin Method for the Time-Dependent Singularly Perturbed Problem
• Mathematics, Computer Science
• J. Sci. Comput.
• 2015
The local stability analysis and local error estimate for the local discontinuous Galerkin (LDG) method is presented, when solving the time-dependent singularly perturbed problems in one dimensional space with a stationary outflow boundary layer. Expand
Local Analysis of the Local Discontinuous Galerkin Method with Generalized Alternating Numerical Flux for One-Dimensional Singularly Perturbed Problem
• Mathematics, Computer Science
• J. Sci. Comput.
• 2017
The local analysis of the local discontinuous Galerkin method based on the generalized alternating numerical flux for the one-dimensional time-dependent singularly perturbed problem with a stationary boundary layer can be obtained by virtue of the generalized Gauss–Radau projection and energy technique with suitable weight function. Expand
Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems
• Computer Science, Mathematics
• Comput. Methods Appl. Math.
• 2021
Local discontinuous Galerkin method is considered for time-dependent singularly perturbed semilinear problems with boundary layer using weighted estimates and suitably designed global projections to establish optimal ( k + 1) -th error estimate in a local region at a distance of 𝒪 ⁢ ( h⁢ log ⁡ ( 1 h ) ) {\mathcal{O}(h\log(\frac{1}{h}))} from domain boundary. Expand
GALERKIN METHODS WITH GRADED MESHES FOR TWO-DIMENSIONAL REACTION-DIFFUSION PROBLEMS
• 2016
We develop high-order Galerkin methods with graded meshes for solving the twodimensional reaction-diffusion problem on a rectangle. With the help of the comparison principle, we establish upperExpand
Pressure Recovery for Weakly Over-Penalized Discontinuous Galerkin Methods for the Stokes Problem
• Mathematics, Computer Science
• J. Sci. Comput.
• 2015
It is proved that these recovery procedures have optimal order of convergence rates in the $$L^2$$L2-norm for the numerical pressure. Expand

References

SHOWING 1-10 OF 26 REFERENCES
Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems
• Computer Science, Mathematics
• Math. Comput.
• 2002
Theoretical results are confirmed in a series of numerical examples and upper bounds for the energy norm of the error which are explicit in the mesh-width h, in the polynomial degree p, and in the regularity of the exact solution are obtained. Expand
Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems
• Mathematics
• 2011
Abstract. In this paper the local discontinuous Galerkin method (LDG) is considered for solving one-dimensional singularly perturbed two-point boundary value problems of convection-diffusion type andExpand
Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D
• Computer Science, Mathematics
• Math. Comput.
• 2010
The theoretical aspect of the Local Discontinuous Galerkin (LDG) method converges uniformly under the Shishkin mesh for singularly perturbed two-point boundary problems of the convection-diffusion type under a simplified ODE model is investigated. Expand
An Optimal Streamline Diffusion Finite Element Method for a Singularly Perturbed Problem
• 2005
The stability and accuracy of a streamline diffusion finite element method (SDFEM) on arbitrary grids applied to a linear 1-d singularly perturbed problem are studied in this paper. With a specialExpand
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
• Mathematics
• 1998
In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuousExpand
Nodal Superconvergence of SDFEM for Singularly Perturbed Problems
• Mathematics, Computer Science
• J. Sci. Comput.
• 2012
It is proved that for a special group of exact solutions, the nodal error converges at a superconvergence rate of order (ln ε−1/N)2k (or (ln-N/N), on a Shishkin mesh, where ε is the singular perturbation parameter and 2N denotes the number of mesh elements. Expand
Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method
• Computer Science, Mathematics
• SIAM J. Numer. Anal.
• 2000
The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of theGalerkin method and it is shown that the hp's spectral convergence gives spectral convergence in problems with smooth time dependence. Expand
Finite element superconvergence approximation for one‐dimensional singularly perturbed problems
Superconvergence approximations of singularly perturbed two-point boundary value problems of reaction-diffusion type and convection-diffusion type are studied. By applying the standard finite elementExpand
High-Order Finite Element Methods for Singularly Perturbed Elliptic and Parabolic Problems
• Mathematics, Computer Science
• SIAM J. Appl. Math.
• 1995
We develop a framework for applying high-order finite element methods to singularly-perturbed elliptic and parabolic differential systems that utilizes special quadrature rules to confine spuriousExpand
Superconvergence of the numerical traces of discontinuous Galerkin and Hybridized methods for convection-diffusion problems in one space dimension
• Computer Science, Mathematics
• Math. Comput.
• 2007
It is proved that the so-called numerical traces of both variables superconverge at all the nodes of the mesh, provided that the traces are conservative, that is, provided they are single-valued. Expand