A new analysis of isoparametric bilinear finite volume element scheme based on trapezoidal rule for anisotropic diffusion problems

Shengying Mu; Yanhui Zhou; Shengying Mu; Yanhui Zhou

doi:10.3934/math.2026138

AIMS Mathematics

2026, Volume 11, Issue 2: 3394-3424. doi: 10.3934/math.2026138

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Research article

A new analysis of isoparametric bilinear finite volume element scheme based on trapezoidal rule for anisotropic diffusion problems

Shengying Mu ¹,
Yanhui Zhou ^{2
,
,}

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China
2.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, China

Received: 22 November 2025 Revised: 16 January 2026 Accepted: 26 January 2026 Published: 04 February 2026
MSC : 65N08, 65N12

We studied the coercivity and error estimate of a modified isoparametric bilinear finite volume element scheme for anisotropic diffusion problems on quadrilateral meshes, where the scheme is obtained by employing the trapezoidal rule to approximate the line integrals in classical $ Q_1 $-finite volume element method. By an element analysis approach, we propose a new sufficient condition to ensure the coercivity result of the scheme, which is better than the existing results in [Q. Hong and J. Wu, Adv. Comput. Math., 44 (2018), 897-922]. Under $ h^2 $-uniform quadrilateral mesh assumption, we prove the superconvergence $ |u_I-u_{h}|_{1} = \mathcal{O}(h^2) $, where $ u_I $ is the isoparametric bilinear interpolation of exact solution $ u $, and $ u_h $ is the finite volume element solution. As a result, an optimal $ L^2 $ error estimate of $ u_h $ is obtained. Some numerical experiments were carried out to verify the theoretical findings.
- isoparametric bilinear finite volume element scheme,
- coercivity,
- superconvergence,
- $ L^2 $ error estimate,
- anisotropic diffusion problem
Citation: Shengying Mu, Yanhui Zhou. A new analysis of isoparametric bilinear finite volume element scheme based on trapezoidal rule for anisotropic diffusion problems[J]. AIMS Mathematics, 2026, 11(2): 3394-3424. doi: 10.3934/math.2026138

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Abstract

We studied the coercivity and error estimate of a modified isoparametric bilinear finite volume element scheme for anisotropic diffusion problems on quadrilateral meshes, where the scheme is obtained by employing the trapezoidal rule to approximate the line integrals in classical $ Q_1 $-finite volume element method. By an element analysis approach, we propose a new sufficient condition to ensure the coercivity result of the scheme, which is better than the existing results in [Q. Hong and J. Wu, Adv. Comput. Math., 44 (2018), 897-922]. Under $ h^2 $-uniform quadrilateral mesh assumption, we prove the superconvergence $ |u_I-u_{h}|_{1} = \mathcal{O}(h^2) $, where $ u_I $ is the isoparametric bilinear interpolation of exact solution $ u $, and $ u_h $ is the finite volume element solution. As a result, an optimal $ L^2 $ error estimate of $ u_h $ is obtained. Some numerical experiments were carried out to verify the theoretical findings.

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