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Nodal superconvergence of the local discontinuous Galerkin method for singularly perturbed problems. (English) Zbl 1376.65110

Summary: In this paper, a superconvergence of order \((\ln N/N)^{2k+1}\) for the numerical traces of the local discontinuous Galerkin (LDG) approximation to a one dimensional singularly perturbed convection-diffusion-reaction problem is proved. The LDG method is applied on a Shishkin mesh with \(2N\) elements, and we use polynomials of degree at most \(k\) on each element. This result puts the numerical finding reported in Xie and Zhang (2007), Xie et al. (2009) on firm mathematical ground.

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
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