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Search Results (4)

Search Parameters:
Keywords = Raviart–Thomas

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25 pages, 1953 KiB  
Article
Numerical Algorithms for Divergence-Free Velocity Applications
by Giacomo Barbi, Antonio Cervone and Sandro Manservisi
Mathematics 2024, 12(16), 2514; https://doi.org/10.3390/math12162514 - 14 Aug 2024
Viewed by 1177
Abstract
This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas [...] Read more.
This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart–Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier–Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
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20 pages, 372 KiB  
Article
A Mixed Finite Element Method for the Multi-Term Time-Fractional Reaction–Diffusion Equations
by Jie Zhao, Shubin Dong and Zhichao Fang
Fractal Fract. 2024, 8(1), 51; https://doi.org/10.3390/fractalfract8010051 - 12 Jan 2024
Cited by 1 | Viewed by 1633
Abstract
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution [...] Read more.
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete L∞(L2(Ω)) norm and for the auxiliary variable λ in the discrete L∞((L2(Ω))2) and L∞(H(div,Ω)) norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
20 pages, 8457 KiB  
Article
Mixed Isogeometric Analysis of the Brinkman Equation
by Lahcen El Ouadefli, Omar El Moutea, Abdeslam El Akkad, Ahmed Elkhalfi, Sorin Vlase and Maria Luminița Scutaru
Mathematics 2023, 11(12), 2750; https://doi.org/10.3390/math11122750 - 17 Jun 2023
Cited by 2 | Viewed by 1517
Abstract
This study focuses on numerical solution to the Brinkman equation with mixed Dirichlet–Neumann boundary conditions utilizing isogeometric analysis (IGA) based on non-uniform rational B-splines (NURBS) within the Galerkin method framework. The authors suggest using different choices of compatible NURBS spaces, which may be [...] Read more.
This study focuses on numerical solution to the Brinkman equation with mixed Dirichlet–Neumann boundary conditions utilizing isogeometric analysis (IGA) based on non-uniform rational B-splines (NURBS) within the Galerkin method framework. The authors suggest using different choices of compatible NURBS spaces, which may be considered a generalization of traditional finite element spaces for velocity and pressure approximation. In order to investigate the numerical properties of the suggested elements, two numerical experiments based on a square and a quarter of an annulus are discussed. The preliminary results for the Stokes problem are presented in References. Full article
(This article belongs to the Section C2: Dynamical Systems)
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13 pages, 3157 KiB  
Article
Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model
by Denis Spiridonov, Jian Huang, Maria Vasilyeva, Yunqing Huang and Eric T. Chung
Mathematics 2019, 7(12), 1212; https://doi.org/10.3390/math7121212 - 10 Dec 2019
Cited by 10 | Viewed by 3073
Abstract
In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; [...] Read more.
In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method. Full article
(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)
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