In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in $L^2$-norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in $L^2$-norm and $W^{1,p}$-seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.
{"title":"A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source","authors":"Haitao Leng, Yanping Chen","doi":"10.1051/m2an/2022005","DOIUrl":"https://doi.org/10.1051/m2an/2022005","url":null,"abstract":"In this paper,\u0000\u0000we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures.\u0000\u0000Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in $L^2$-norm\u0000\u0000is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in $L^2$-norm and $W^{1,p}$-seminorm\u0000\u0000are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80580287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The modified Maxwell's Steklov eigenvalue problem is a new problem arising from the study of inverse electromagnetic scattering problems. In this paper, we investigate two finite element methods for this problem and perform the convergence analysis. Moreover, the monotonic convergence of the discrete eigenvalues computed by one of the methods is analyzed.
{"title":"Convergence analysis of two finite element methods for the modified Maxwell's Steklov eigenvalue problem","authors":"Bo Gong","doi":"10.1051/m2an/2022001","DOIUrl":"https://doi.org/10.1051/m2an/2022001","url":null,"abstract":"The modified Maxwell's Steklov eigenvalue problem is a new problem arising from the study of inverse electromagnetic scattering problems. In this paper, we investigate two finite element methods for this problem and perform the convergence analysis. Moreover, the monotonic convergence of the discrete eigenvalues computed by one of the methods is analyzed.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84142841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the ``{zero-energy-contribution}" feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.
{"title":"Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme \u0000\u0000of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cell","authors":"Xiaofeng Yang","doi":"10.1051/m2an/2022003","DOIUrl":"https://doi.org/10.1051/m2an/2022003","url":null,"abstract":"We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the ``{zero-energy-contribution}\" feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90491267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Type-I model of a multicomponent system of fluids with non-constant temperature is derived as the high-friction limit of a Type-II model via a Chapman-Enskog expansion. The asymptotic model is shown to fit into the general theory of hyperbolic-parabolic systems, by exploiting the entropy structure inherited through the asymptotic procedure. Finally, by deriving the relative entropy identity for the Type-I model, two convergence results for smooth solutions are presented, from the system with mass-diffusion and heat conduction to the corresponding system without mass-diffusion but including heat conduction and to its hyperbolic counterpart.
{"title":"Asymptotic derivation of multicomponent compressible flows with heat conduction and mass diffusion","authors":"S. Georgiadis, A. Tzavaras","doi":"10.1051/m2an/2022065","DOIUrl":"https://doi.org/10.1051/m2an/2022065","url":null,"abstract":"A Type-I model of a multicomponent system of fluids with non-constant temperature is derived as the high-friction limit of a Type-II model via a Chapman-Enskog expansion. The asymptotic model is shown to fit into the general theory of hyperbolic-parabolic systems, by exploiting the entropy structure inherited through the asymptotic procedure. Finally, by deriving the relative entropy identity for the Type-I model, two convergence results for smooth solutions are presented, from the system with mass-diffusion and heat conduction to the corresponding system without mass-diffusion but including heat conduction and to its hyperbolic counterpart.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78996823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width O(ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.
{"title":"Analysis of a diffuse interface method for the Stokes-Darcy coupled problem","authors":"Martina Bukavc, B. Muha, A. Salgado","doi":"10.1051/m2an/2023062","DOIUrl":"https://doi.org/10.1051/m2an/2023062","url":null,"abstract":"We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width O(ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88994770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and R, and v is valued in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D and 3D cases where the solution does not belong to H 1(Ω), show that this method can provide accurate results. We then construct a numerical scheme which presents a convergence property to the entropy weak solution of the problem in the case where the right-hand side belongs to L1 . This is achieved owing to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation, and adding an upstream weighting evaluation and a nonlinear p−Laplace vanishing stabilisation term.
{"title":"Convergence of nonlinear numerical approximations for\u0000\u0000an elliptic linear problem with irregular data","authors":"R. Eymard, David Maltese","doi":"10.1051/m2an/2021079","DOIUrl":"https://doi.org/10.1051/m2an/2021079","url":null,"abstract":"This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and R, and v is valued in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the\u0000\u0000existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D and 3D cases where the solution does not belong to H 1(Ω), show that this method can provide accurate results. We then construct a numerical scheme which presents a convergence property to the entropy weak solution of the problem in the case where the right-hand side belongs to L1 . This is achieved owing to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation, and adding an upstream weighting evaluation and a nonlinear p−Laplace vanishing stabilisation term.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80628061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering $S$-matrix and corresponding ``truly 2D well-balanced'' numerical schemes. A first scheme is obtained by directly implementing truncated Fourier-Bessel series, whereas another proceeds by applying an exponential modulation to a former, conservative, one. Consistency with the asymptotic damped parabolic approximation is checked for both algorithms. These findings are confirmed by means of practical benchmarks carried out on coarse Cartesian computational grids.
{"title":"Diffusive limits of 2D well-balanced schemesfor kinetic models of neutron transport","authors":"G. Bretti, L. Gosse, N. Vauchelet","doi":"10.1051/m2an/2021077","DOIUrl":"https://doi.org/10.1051/m2an/2021077","url":null,"abstract":"Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering $S$-matrix and corresponding ``truly 2D well-balanced'' numerical schemes. A first scheme is obtained by directly implementing truncated Fourier-Bessel series, whereas another proceeds by applying an exponential modulation to a former, conservative, one. Consistency with the asymptotic damped parabolic approximation is checked for both algorithms. These findings are confirmed by means of practical benchmarks carried out on coarse Cartesian computational grids.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89866496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.
{"title":"Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem","authors":"Fei Xu, Liu Chen, Qiumei Huang","doi":"10.1051/m2an/2021076","DOIUrl":"https://doi.org/10.1051/m2an/2021076","url":null,"abstract":"In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81240991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ibtihel Ben Gharbia, M. Haddou, Quang Huy Tran, Duc Thach Son Vu
In this paper , we conduct a thorough mathematical analysis of the unified formulation advocated by Lauser et al . [ Adv . Water Res . 34 (2011), 957-966] for compositional multiphase flows in porous media . The interest of this formulation lies in its potential to automatically handle the appearance and disappearance of phases. However , its practical implementation turned out to be not always robust for realistic fugacity laws associated with cubic equations of state , as shown by Ben Gharbia and Flauraud [ Oil & Gas Sci . Technol . 74 (2019), 43]. By focusing on the subproblem of phase equilibrium , we derive sufficient conditions for the existence of the corresponding system of equations. We trace back the difficulty of cubic laws to a deficiency of the Gibbs functions that comes into play due to the `` unifying '' feature of the new formulation. We propose a partial remedy for this problem by extending the domain of definition of these functions in a natural way . Besides , we highlight the crucial but seemingly unknown fact that the unified formulation encapsulates all the properties known to physicists on phase equilibrium , such as the tangent plane criterion and the minimization of the Gibbs energy of the mixture.
{"title":"An analysis of the unified formulation for the equilibrium problem of compositional multiphase mixtures","authors":"Ibtihel Ben Gharbia, M. Haddou, Quang Huy Tran, Duc Thach Son Vu","doi":"10.1051/m2an/2021075","DOIUrl":"https://doi.org/10.1051/m2an/2021075","url":null,"abstract":"In this paper , we conduct a thorough mathematical analysis of the unified formulation advocated by Lauser et al . [ Adv . Water Res . 34 (2011), 957-966] for compositional multiphase flows in porous media . The interest of this formulation lies in its potential to automatically handle the appearance and disappearance of phases. However , its practical implementation turned out to be not always robust for realistic fugacity laws associated with cubic equations of state , as shown by Ben Gharbia and Flauraud [ Oil & Gas Sci . Technol . 74 (2019), 43]. By focusing on the subproblem of phase equilibrium , we derive sufficient conditions for the existence of the corresponding system of equations. We trace back the difficulty of cubic laws to a deficiency of the Gibbs functions that comes into play due to the `` unifying '' feature of the new formulation. We propose a partial remedy for this problem by extending the domain of definition of these functions in a natural way . Besides , we highlight the crucial but seemingly unknown fact that the unified formulation encapsulates all the properties known to physicists on phase equilibrium , such as the tangent plane criterion and the minimization of the Gibbs energy of the mixture.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84324686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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