PROC
A discontinuous Galerkin least-squares finite element method for solving Fisher's equation
Runchang Lin Huiqing Zhu
Conference Publications 2013, 2013(special): 489-497 doi: 10.3934/proc.2013.2013.489
In the present study, a discontinuous Galerkin least-squares finite element algorithm is developed to solve Fisher's equation. The present method is effective and can be successfully applied to problems with strong reaction, to which obtaining stable and accurate numerical traveling wave solutions is challenging. Numerical results are given to demonstrate the convergence rates of the method and the performance of the algorithm in long-time integrations.
keywords: Fisher's equation discontinuous Galerkin method. least-squares finite element method
PROC
A robust finite element method for singularly perturbed convection-diffusion problems
Runchang Lin
Conference Publications 2009, 2009(Special): 496-505 doi: 10.3934/proc.2009.2009.496
In this paper, we consider a convection-diffusion boundary value problem with singular perturbation. A finite element method (FEM) is proposed based on discontinuous Galerkin (DG) discretization of least-squares variational formulation. Numerical tests on representative problems reveal that the method is robust and efficient.
keywords: Finite element methods singular perturbation problems convection-diffusion problems discontinuous Galerkin methods least-squares methods
DCDS-B
$L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems
Huiqing Zhu Runchang Lin
Discrete & Continuous Dynamical Systems - B 2013, 18(5): 1493-1505 doi: 10.3934/dcdsb.2013.18.1493
Pointwise error estimates of the local discontinuous Galerkin (LDG) method for a one-dimensional singularly perturbed problem are studied. Several uniform $L^\infty$ error bounds for the LDG approximation to the solution and its derivative are established on a Shishkin-type mesh. Numerical experiments are presented.
keywords: Shishkin mesh. singular perturbation Local discontinuous Galerkin method

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