A Galerkin-characteristic unified finite element method for moving thermal fronts in porous media

Salhi, Loubna; El‐Amrani, Mofdi; Seaı̈d, Mohammed

doi:10.1016/j.cam.2020.113159

Cited by 8 publications

(8 citation statements)

References 33 publications

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“…Notice that it is evident to prove that any triplet (u, p, Θ) in H 0 (div, Ω) × L 2 0 (Ω) × H 1 0 (Ω) solving the problem (1) in the sense of distributions in Ω is a solution of the variational problem (17), see [14,42] for more details. It is also worth noting that the main advantage of the proposed method is that it does not require the use of mixed formulations such as those widely employed in the literature, see for instance [2,41].…”

Section: Unified Galerkin-characteristic Finite Element Approximationmentioning

confidence: 99%

“…Next, to solve the convection-diffusion equation (17b), we use the modified method of characteristics for the transport term. This class of methods has been used for solving many convection-dominated flow problems, see for example [6,22,19,38,36,37,42]. The main idea is to treat the transport part DΘ Dt of equation (17b) separately in the finite element discretization.…”

Section: Unified Galerkin-characteristic Finite Element Approximationmentioning

confidence: 99%

“…Next, to establish a priori error estimates for the stabilized solutions of (34), we shall use the following Theorem, , the proof for which can be found in [42]: THEOREM 3. Let (S h , Q h ) be the pair of spaces defined in (3), let (u, p) ∈ H 0 (div, Ω) × L 2 0 (Ω) be the solution of the Darcy problem (17a) and let (u h , p h ) ∈ S h × Q h be the solution of the stabilized mixed problem (21), where the operator Π k−1 defined in (7) satisfies (6).…”

Section: Error Estimates For Velocity and Pressurementioning

confidence: 99%

“…Moreover, it offers the possibility of using time steps that exceed those allowed by the stability CFL condition for the conventional Eulerian methods. The Galerkin-characteristic methods have been subject of investigations in many references including [6,19,20,21,36,37,38,42,43,44,46]. In [19], the semi-Lagrangian method combined with a finite difference discretization has been studied and applied to convection-diffusion problems.…”

Section: Introductionmentioning

confidence: 99%

“…In [23,25], the method has proved to be stable and accurate when used to study the thermal incompressible Navier-Stokes equations. Semi-Lagrangian methods have also been investigated for the simulation of natural and mixed convection flows in [21,43], for tidal flows in [24], and for moving thermal fronts in porous media in [42]. A first-order Galerkin-characteristic method combined with the finite element discretization has been analyzed for the Navier-Stokes equations in [38].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a Galerkin-characteristic finite element method for convection-diffusion problems in porous media

Salhi¹,

El‐Amrani²,

Seaı̈d³

2021

APAM

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L'objectif de ce travail est l'étude d'une méthode de résolution numérique par éléments finis semilagrangiennes afin de résoudre les problémes évolutifs de convection-diffusion issus des milieux poreux. La méthode proposée permet d'utiliser une approximation par éléments finis d'ordre égal pour toutes les solutions du probléme. En outre, la condition standard de Courant-Friedrichs-Lewy est assouplie avec le traitement lagrangien des termes de convection, et les erreurs de troncature sont réduites dans la partie diffusion-réaction du probléme. Dans cette étude, une analyse de la convergence et de la stabilité de la méthode proposée est aussi présentée, ainsi que les estimations des erreurs dans la norme L 2 dérivées pour toutes les solutions numériques. Les tests numériques sont illustrées par quelques exemples afin de vérifier les estimations théoriques et de démontrer la grande précision et l'efficacité de la méthode proposée. ABSTRACT. We present a Galerkin-characteristic finite element method for the numerical solution of time-dependent convection-diffusion problems in porous media. The proposed method allows the use of equal-order finite element approximations for all solutions in the problem. In addition, the standard Courant-Friedrichs-Lewy condition is relaxed with the Lagrangian treatment of convection terms, and the time truncation errors are reduced in the diffusion-reaction part. Analysis of convergence and stability of the proposed method is also investigated in this study and error estimates in the L 2 -norm are established for the numerical solutions. Numerical performance of the method is examined using two examples to verify the theoretical estimates and to demonstrate the high accuracy and efficiency of the proposed Galerkin-characteristic finite element method. MOTS-CLÉS. Problémes de convection-diffusion, Équation de Darcy, Milieux poreux, Méthode semi-lagrangienne, Éléments finis, Estimation des erreurs a priori.

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Section: Unified Galerkin-characteristic Finite Element Approximationmentioning

confidence: 99%

Section: Unified Galerkin-characteristic Finite Element Approximationmentioning

confidence: 99%

Section: Error Estimates For Velocity and Pressurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis of a Galerkin-characteristic finite element method for convection-diffusion problems in porous media

Salhi¹,

El‐Amrani²,

Seaı̈d³

2021

APAM

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A priori error estimates for a semi‐Lagrangian unified finite element method for coupled Darcy‐transport problems

Salhi

El‐Amrani

Seaı̈d

2023

Numerical Methods Partial

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A semi‐Lagrangian unified finite element method is investigated for solving time‐dependent coupled Darcy‐transport problems. In this method, the modified method of characteristics is combined with a Galerkin finite element discretization allowing the same finite element space to be used for all solutions of the problem including the pressure, velocity and concentration. Convergence and stability of the method are also analyzed in this study and error estimates in the ‐norm are established for the numerical solutions. For the time integration we consider a second‐order implicit method and discrete error estimates are provided. Due to the Lagrangian treatment of convection terms in the considered problems, the conventional Courant‐Friedrichs‐Lewy condition is relaxed and the time truncation errors are reduced in the diffusion‐reaction part. Numerical results are presented for several test examples to demonstrate the reliability and efficiency of the proposed method. The computed results support our expectations for an accurate and highly efficient semi‐Lagrangian unified finite element method for time‐dependent coupled Darcy‐transport problems.

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An Enhanced Finite Element Algorithm for Thermal Darcy Flows with Variable Viscosity

Salhi

El‐Amrani

Seaı̈d

2021

Computational Science – ICCS 2021

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A Galerkin-characteristic unified finite element method for moving thermal fronts in porous media

Cited by 8 publications

References 33 publications

Analysis of a Galerkin-characteristic finite element method for convection-diffusion problems in porous media

Analysis of a Galerkin-characteristic finite element method for convection-diffusion problems in porous media

A priori error estimates for a semi‐Lagrangian unified finite element method for coupled Darcy‐transport problems

An Enhanced Finite Element Algorithm for Thermal Darcy Flows with Variable Viscosity

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