Analysis of optimal superconvergence of the local discontinuous Galerkin method for nonlinear fourth-order boundary value problems

Baccouch, Mahboub

doi:10.1007/s11075-020-00947-0

Cited by 4 publications

(1 citation statement)

References 23 publications

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“…The LDG method is a form of finite-element method; it was first proposed as a generalisation of the discontinuous Galerkin (DG) method for a convection-diffusion Symmetry 2021, 13, 2291 2 of 26 problem [12]. Later, it was applied to solve the purely elliptic problem [13], convectiondiffusion problems [14], Stokes system [15], KdV-type equations [16], Hamilton-Jacobi equations [17], time-dependent fourth-order problems [18] and other higher-order partial differential equations [19]. Because the LDG method shares many advantages of the DG methods and can effectively simulate the acute change of a singular solution, it is particularly suited to solving singularly perturbed problems.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Mei

Wang

et al. 2021

Symmetry

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We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

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Section: Introductionmentioning

confidence: 99%

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Mei

Wang

et al. 2021

Symmetry

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Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

Popov

2024

J Sci Comput

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A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

Baccouch

2022

Numer Algor

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Analysis of optimal superconvergence of the local discontinuous Galerkin method for nonlinear fourth-order boundary value problems

Cited by 4 publications

References 23 publications

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations

A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

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