An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem

Article Mathematics, Applied

Optimal pointwise convergence of the LDG method for singularly perturbed convection-diffusion problem

Yao Cheng et al.

Summary: This paper considers a local discontinuous Galerkin (LDG) method for a one-dimensional singularly perturbed convection-diffusion problem with an exponential boundary layer. Based on the technique of discrete Green's function, the optimal pointwise convergence (up to a logarithmic factor) of the LDG method is established on three typical families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type, and Bakhvalov-type. Numerical experiments are also provided.

APPLIED MATHEMATICS LETTERS (2023)

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Article Mathematics, Applied

SUPERCLOSENESS OF THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM

Y. A. O. Cheng et al.

Summary: This paper numerically solves a convection-diffusion problem on the unit square in Double-struck capital R2 with exponential boundary layers using the local discontinuous Galerkin (LDG) method. They establish the superconvergence property of the LDG solution on three types of layer-adapted meshes, which leads to an optimal bound for the L2 error. Numerical experiments confirm their theoretical results.

MATHEMATICS OF COMPUTATION (2023)

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Article Mathematics, Applied

Balanced-norm error estimate of the local discontinuous Galerkin method on layer-adapted meshes for reaction-diffusion problems

Yao Cheng et al.

Summary: This paper studies the application of the local discontinuous Galerkin method on layer-adapted meshes for singularly perturbed problems. A novel projection method is proposed to improve the balanced-error estimate, and numerical experiments are conducted to validate the theoretical results.

NUMERICAL ALGORITHMS (2022)

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Article Mathematics, Applied

The local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems

Yao Cheng et al.

Summary: In this paper, we analyze the error of both the semi-discretization and the full discretization of a time-dependent convection-diffusion problem. By using the local discontinuous Galerkin method and the implicit theta-scheme, we obtain error estimates with respect to space and time and discuss improved estimates and the use of a discontinuous Galerkin discretization in time.

COMPUTERS & MATHEMATICS WITH APPLICATIONS (2022)

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Article Mathematics, Applied

Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems

Yao Cheng et al.

Summary: In this study, the local discontinuous Galerkin method is applied to time-dependent singularly perturbed semi-linear problems with boundary layers. The method is equipped with a general numerical flux and global projections, and an optimal error estimate is established in a local region near the domain boundary. Numerical experiments support the theoretical results for both semi-discrete and fully discrete LDG methods.

COMPUTATIONAL METHODS IN APPLIED MATHEMATICS (2021)

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Article Mathematics, Interdisciplinary Applications