Article
Mathematics, Applied
Maria Gabriela Armentano, Maria Lorena Stockdale
Summary: In this work, we solved a Stokes-Darcy coupled problem in a plane curved domain using curved elements. The velocity-pressure pair was approximated using the MINI-element method for the entire coupled problem. It was demonstrated that, under appropriate assumptions about the curved domain, the proposed method achieved optimal accuracy in terms of solution regularity and had a simple implementation. Numerical tests also showed the good performance of the proposed method.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2021)
Article
Mathematics, Applied
Jiaping Yu, Yuhong Zhang
Summary: This paper introduces a fully mixed finite element scheme for the Navier-Stokes/Darcy problem based on Nitsche's type interface stabilizations, ensuring stability and optimal convergence in the coupling of fluid region and porous media domain. The dependence and requirement of the stabilization parameter for optimal error estimates are explicitly derived, and numerical tests confirm the stability and efficiency of this stabilized mixed method.
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Hui Peng, Qilong Zhai, Ran Zhang, Shangyou Zhang
Summary: This paper proposes a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition, and validates the theoretical analysis through numerical experiments.
SCIENCE CHINA-MATHEMATICS
(2021)
Article
Engineering, Multidisciplinary
Yizhong Sun, Weiwei Sun, Haibiao Zheng
Summary: In this paper, a parallel domain decomposition method is proposed for solving the fully-mixed Stokes-Darcy coupled problem with the Beavers-Joseph-Saffman interface conditions. The method decouples the original problem into two independent subproblems with newly constructed Robin-type boundary conditions and modified weak formulation. Convergence analysis and numerical examples demonstrate the effectiveness of the proposed method.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Zerihun Kinfe Birhanu, Tadele Mengesha, Abner J. Salgado
Summary: The study focused on the flow of a viscous incompressible fluid through a porous medium where the permeability of the medium depends exponentially on the pressure. A splitting formulation was used to define the permeability through a convection diffusion problem, which was then incorporated into a linear Darcy equation. The discretization of the problem and error analysis were also conducted in this research.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Xinhui Wang, Guangzhi Du, Liyun Zuo
Summary: A new local and parallel finite element method based on two-grid discretization is proposed and investigated for the mixed Navier-Stokes-Darcy problem. The method considers partition of unity method to generate local subproblems and achieves global continuous approximations. Theoretical analysis and numerical tests are conducted to support the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Guangzhi Du, Liyun Zuo
Summary: In this paper, a local and parallel partition of unity scheme is proposed for the mixed Navier-Stokes-Darcy problem. The method combines the two-grid method with a partition of unity and exhibits several advantageous features, including parallel computing and optimal error bounds. The convergence of the method is also proven and numerical experiments are conducted to validate the theoretical results.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Jessika Camano, Carlos Garcia, Ricardo Oyarzua
Summary: In this paper, a new momentum conservative mixed finite element method for the Navier-Stokes problem in nonstandard Banach spaces is proposed and analyzed. The method conserves momentum by introducing a pseudostress tensor and using Raviart-Thomas elements for the pseudostress tensor and discontinuous piece-wise polynomial elements for the velocity. The unique solvability is proven using Banach-Necas-Babuska and Banach's fixed-point theorems, and the method exhibits optimal rate of convergence for the error decay.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Ingeborg G. Gjerde, Kundan Kumar, Jan M. Nordbotten
Summary: In this work, we analyze the Poisson equation with a line source using a dual-mixed variational formulation. By making assumptions on the problem parameters, we split the solution into higher- and lower-regularity terms, and propose a singularity removal-based mixed finite element method to approximate the higher-regularity terms, which significantly improves the convergence rate compared to approximating the full solution.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
V. Kosin, S. Beuchler, T. Wick
Summary: In this paper, a new mixed method proposed by Rafetseder and Zulehner is investigated for Kirchhoff plates and applied to fourth order eigenvalue problems. This new mixed method uses two auxiliary variables to require only H(1) regularity for the displacement and the auxiliary variables, without the demand of a convex domain. A direct comparison is provided to the C-0-IPG method and Ciarlet-Raviart's mixed method, specifically in view of convergence orders, for vibration problems with clamped and simply supported plates. Numerical experiments are conducted using the open-source finite element library deal.II and incorporating non-trivial boundary conditions with the coupling of finite elements with elements on the boundary.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Wietse M. Boon, Timo Koch, Miroslav Kuchta, Kent-Andre Mardal
Summary: This paper presents mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem, considering three different formulations and their discretizations in various finite element and finite volume methods. Robust preconditioners are derived using a unified theoretical framework, utilizing operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Jing Wen, Zhangxing Chen, Yinnian He
Summary: This paper investigates a method of gradient-divergence stabilization in discontinuous Galerkin methods for a coupled Stokes and Darcy problem. The method penalizes the jumps of normal velocities over facets of the triangulation to improve the accuracy of velocity and mass conservation. The paper proves the existence, uniqueness, and stability of the solution, and demonstrates the convergence of the method with an analogue of gradient-divergence stabilization.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Sergio Caucao, Marco Discacciati
Summary: This paper presents an a priori analysis of a mixed finite element method for the filtration of incompressible fluid through non-deformable porous media with heterogeneous permeability. It considers the Brinkman-Forchheimer and Darcy equations for flow in different permeable regions, with appropriate transmission conditions. The finite element discretization involves various elements for different variables, and stability, convergence, and error estimates are obtained. Numerical tests confirm the theoretical findings. (c) 2023 IMACS. Published by Elsevier B.V. All rights reserved.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Jessika Camano, Sergio Caucao, Ricardo Oyarzua, Segundo Villa-Fuentes
Summary: In this paper, a new momentum conservative mixed finite element method is developed for solving the steady-state Navier-Stokes problem in two and three dimensions, and an a posteriori error analysis is conducted. By extending standard techniques from Hilbert spaces to Banach spaces, a reliable and efficient residual-based a posteriori error estimator is derived for the mixed finite element scheme on arbitrary polygonal and polyhedral regions. The efficiency of the proposed error indicator is proven using inverse inequalities and the localization technique based on bubble functions.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Haoran Liu, Michael Neilan, M. Baris Otus
Summary: This paper proposes and analyzes a boundary correction finite element method for solving the Stokes problem. The method is based on the Scott-Vogelius pair on Clough-Tocher splits. It utilizes continuous piecewise polynomials for the velocity space and piecewise polynomials without continuity constraints for the pressure space. A Lagrange multiplier space is introduced to enforce boundary conditions and address the lack of pressure-robustness. The paper proves several inf-sup conditions, ensuring the well-posedness of the method, and demonstrates its convergence with optimal order and divergence-free velocity approximation.
JOURNAL OF NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Carlos Garcia, Gabriel N. Gatica, Salim Meddahi
IMA JOURNAL OF NUMERICAL ANALYSIS
(2017)
Article
Mathematics, Applied
Ana Alonso Rodriguez, Antonio Marquez, Salim Meddahi, Alberto Valli
Article
Mathematics, Applied
Felipe Lepe, Salim Meddahi, David Mora, Rodolfo Rodriguez
MATHEMATICS OF COMPUTATION
(2019)
Article
Mathematics, Applied
Antonio Marquez, Salim Meddahi, Thanh Tran
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2018)
Article
Mathematics, Applied
Felipe Lepe, Salim Meddahi, David Mora, Rodolfo Rodriguez
NUMERISCHE MATHEMATIK
(2019)
Article
Mathematics, Applied
Gabriel N. Gatica, Salim Meddahi
JOURNAL OF NUMERICAL MATHEMATICS
(2020)
Article
Mathematics, Applied
Gabriel N. Gatica, Antonio Marquez, Salim Meddahi
Summary: The study introduces and analyzes a new mixed finite element method for the standard linear model in viscoelasticity, showing that the resulting variational formulation is well-posed and proving the convergence of a class of H(div)-conforming semi-discrete schemes. Additionally, the use of the Newmark trapezoidal rule results in a fully discrete scheme with established convergence results. Numerical examples are provided to demonstrate the method's performance.
Article
Mathematics, Applied
Gabriel N. Gatica, Salim Meddahi, Ricardo Ruiz-Baier
Summary: In this work, a new fully mixed finite element method is presented and analyzed for the nonlinear problem of coupling the Darcy and heat equations. The introduction of the pseudoheat flux as an additional unknown leads to saddle point-type schemes in Banach spaces. Convenient choices of Lebesgue and H(div)-type spaces are made based on the solvability of a related Neumann problem. The unique solvability of the continuous formulation is established using fixed-point operator, classical Banach theorem, Babuska-Brezzi theory, Banach-Necas-Babuska theorem, suitable operators mapping Lebesgue spaces into themselves, regularity assumptions, and the aforementioned Neumann problem. The existence of a solution is proved using the Brouwer theorem for the associated Galerkin scheme. The a priori error analysis and numerical examples are also provided.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Antonio Marquez, Salim Meddahi
Summary: The stress-based formulation for Zener's model in linear viscoelasticity is introduced and analyzed in this study. The method is designed to effectively handle heterogeneous materials with purely elastic and viscoelastic components. The mixed variational formulation of the problem is presented in terms of a class of tensorial wave equations, and an energy estimate is obtained to ensure the well-posedness of the problem. Mixed continuous and discontinuous Galerkin space discretizations are proposed and analyzed, with optimal error bounds derived for each semidiscrete solution in the corresponding energy norm. Full discretization strategies for both Galerkin methods are discussed in the conclusion.
JOURNAL OF NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Gabriel N. Gatica, George C. Hsiao, Salim Meddahi
Summary: This paper presents new insights on the application of the boundary-field equation approach for solving nonlinear exterior transmission problems. It extends classical coupling procedures and introduces a new modification method. Well-posedness of continuous and discrete schemes is established by analyzing primal and dual-mixed variational formulations, with a priori error estimates.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Editorial Material
Mathematics, Applied
Gabriel N. Gatica, Norbert Heuer, Salim Meddahi
Summary: This article is the preface of a special issue dedicated to the memory of Francisco Javier Sayas. The articles discuss Sayas' main research interests in the numerical analysis of partial differential equations, including contributions on the scattering and propagation of acoustic and electromagnetic waves, and the analysis of discontinuous Galerkin schemes, boundary element methods, and coupled schemes. An overview of the results covered by this special issue is also provided.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Salim Meddahi
Summary: We present a pure-stress formulation for the eigenvalue problem in elasticity with mixed boundary conditions. A discontinuous Galerkin method based on H(div) is proposed to enforce the stress symmetry strongly in the discretization. Under suitable assumptions on the mesh and polynomial approximation, we prove the spectral correctness of the discrete scheme and derive optimal convergence rates for eigenvalues and eigenfunctions. Numerical examples in two and three dimensions are provided.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Interdisciplinary Applications
Salim Meddahi, Ricardo Ruiz-Baier
Summary: This paper proposes a strongly symmetric stress approximation for the Brinkman equations with mixed boundary conditions. The Cauchy stress is solved using a symmetric interior penalty discontinuous Galerkin method, and pressure and velocity can be readily post-processed from the stress. A second post-process is shown to produce exactly divergence-free discrete velocities. The stability of the method and explicit error estimates are demonstrated, and optimal rates of convergence for the stress and for the post-processed variables are derived. Furthermore, optimal L-2 error estimates for the stress are proven under appropriate assumptions on the mesh. Numerical examples in 2D and 3D are provided.
NETWORKS AND HETEROGENEOUS MEDIA
(2022)
Article
Mathematics, Applied
Gabriel N. Gatica, Salim Meddahi
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2019)
Article
Mathematics, Applied
Carlos Garcia, Gabriel N. Gatica, Antonio Marquez, Salim Meddahi