This paper analyzes the local discontinuous Galerkin (LDG) method using generalized alternating numerical fluxes for solving a singularly perturbed reaction-diffusion problem defined on the unit square <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varOmega \subset {\mathbb {R}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. Generalized alternating numerical fluxes offer several key advantages: they are easier to formulate, reduce numerical errors more effectively than traditional alternating fluxes, and provide flexibility in modeling layer behavior through adjustable weights. We prove <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-independent uniform convergence of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {O}}\left( (M^{-1}\ln M)^{k+1}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>ln</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for the energy-norm errors on Shishkin-type meshes, with k being the polynomial degree and M the mesh elements per spatial direction. The analysis employs generalized Gauss–Radau projections as well as their approximation and superapproximation properties. Numerical experiments confirm our theoretical bounds.

Uniform convergence analysis of the LDG method using generalized alternating numerical fluxes for 2D singularly perturbed reaction-diffusion problems

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发表日期：

2025年12月15日

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Donghui Shi,
Yao Cheng

摘要

This paper analyzes the local discontinuous Galerkin (LDG) method using generalized alternating numerical fluxes for solving a singularly perturbed reaction-diffusion problem defined on the unit square $\varOmega \subset {\mathbb {R}}^2$ $Ω \subset R^{2}$ . Generalized alternating numerical fluxes offer several key advantages: they are easier to formulate, reduce numerical errors more effectively than traditional alternating fluxes, and provide flexibility in modeling layer behavior through adjustable weights. We prove $\varepsilon $ $ε$ -independent uniform convergence of order ${\mathcal {O}}\left( (M^{-1}\ln M)^{k+1}\right) $ $O ({(M^{- 1} ln M)}^{k + 1})$ for the energy-norm errors on Shishkin-type meshes, with k being the polynomial degree and M the mesh elements per spatial direction. The analysis employs generalized Gauss–Radau projections as well as their approximation and superapproximation properties. Numerical experiments confirm our theoretical bounds.