Discontinuous Galerkin method for hybrid–dimensional fracture models of the advection–diffusion equation

Shuangshuang Chen; Longchao Jin; Shuangshuang Chen; Longchao Jin

doi:10.3934/era.2026192

Electronic Research Archive

2026, Volume 34, Issue 7: 4325-4358. doi: 10.3934/era.2026192

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Discontinuous Galerkin method for hybrid–dimensional fracture models of the advection–diffusion equation

Shuangshuang Chen ^,,
Longchao Jin

School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China

Received: 19 January 2026 Revised: 29 April 2026 Accepted: 07 May 2026 Published: 20 May 2026

This paper presents a numerical method for the advection-diffusion process in fractured porous media described by a kind of hybrid–dimensional fracture models, in which fractures are represented as $ (n-1) $–dimensional interfaces embedded within the $ n $–dimensional porous matrix. We develop a numerical approach that combines the discontinuous Galerkin (DG) method for spatial discretization with the backward Euler method for temporal discretization. Leveraging the advantage of the DG method in flexibly handling boundary conditions, the DG–backward Euler method is extended from the model with a single fracture to one with intersecting fractures. Theoretical analysis shows that the optimal order error estimate for the concentration in the discrete $ H^1 $–norm is obtained. Numerical experiments with a single fracture and intersecting fractures are all carried out to verify the accuracy of the theoretical results.
- hybrid–dimensional fracture model,
- advection–diffusion equation,
- discontinuous Galerkin method,
- backward–Euler method
Citation: Shuangshuang Chen, Longchao Jin. Discontinuous Galerkin method for hybrid–dimensional fracture models of the advection–diffusion equation[J]. Electronic Research Archive, 2026, 34(7): 4325-4358. doi: 10.3934/era.2026192

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Abstract

This paper presents a numerical method for the advection-diffusion process in fractured porous media described by a kind of hybrid–dimensional fracture models, in which fractures are represented as $ (n-1) $–dimensional interfaces embedded within the $ n $–dimensional porous matrix. We develop a numerical approach that combines the discontinuous Galerkin (DG) method for spatial discretization with the backward Euler method for temporal discretization. Leveraging the advantage of the DG method in flexibly handling boundary conditions, the DG–backward Euler method is extended from the model with a single fracture to one with intersecting fractures. Theoretical analysis shows that the optimal order error estimate for the concentration in the discrete $ H^1 $–norm is obtained. Numerical experiments with a single fracture and intersecting fractures are all carried out to verify the accuracy of the theoretical results.

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