Higher-order NIPG methods on Bakhvalov-type meshes for a singularly perturbed Volterra integro-differential equation

Weining Chen; Ke Liu; Lei Xu; Ying Liang; Weining Chen; Ke Liu; Lei Xu; Ying Liang

doi:10.3934/nhm.2026081

Networks and Heterogeneous Media

2026, Volume 21, Issue 6: 2180-2195. doi: 10.3934/nhm.2026081

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Research article

Higher-order NIPG methods on Bakhvalov-type meshes for a singularly perturbed Volterra integro-differential equation

1.
Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, China
2.
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
3.
School of Primary Education, Nanning Normal University, Nanning 530299, China

Received: 24 January 2026 Revised: 26 September 2026 Accepted: 20 October 2026 Published: 23 October 2026

In the present article, a higher-order NIPG method on a Bakhvalov-type mesh is developed to solve a second-order singularly perturbed Volterra integro-differential equation. Drawing on the distinctive properties of Bakhvalov-type meshes, we construct a composite interpolation operator, which enables the derivation of parameter-uniform error estimates for the proposed NIPG method. It is shown that the accuracy of this method attains an order of $ k $ in the energy norm, where k denotes the degree of the piecewise polynomials employed in the finite element space. Numerical experiments are presented to confirm our theoretical results.
- singularly perturbed,
- Volterra integro-differential equation,
- NIPG method,
- Bakhvalov-type mesh,
- uniform convergence
Citation: Weining Chen, Ke Liu, Lei Xu, Ying Liang. Higher-order NIPG methods on Bakhvalov-type meshes for a singularly perturbed Volterra integro-differential equation[J]. Networks and Heterogeneous Media, 2026, 21(6): 2180-2195. doi: 10.3934/nhm.2026081

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Abstract

In the present article, a higher-order NIPG method on a Bakhvalov-type mesh is developed to solve a second-order singularly perturbed Volterra integro-differential equation. Drawing on the distinctive properties of Bakhvalov-type meshes, we construct a composite interpolation operator, which enables the derivation of parameter-uniform error estimates for the proposed NIPG method. It is shown that the accuracy of this method attains an order of $ k $ in the energy norm, where k denotes the degree of the piecewise polynomials employed in the finite element space. Numerical experiments are presented to confirm our theoretical results.

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