We approximate the solution of the stream function formulation of the Oseen equations on general domains by designing a nonconforming Morley-type virtual element method. Under a suitable assumption on the continuous problem’s coefficients, the discrete scheme is well-posed. By introducing an enriching operator , we derive an a priori estimate of the error in a discrete H 2 norm. By post-processing the discrete stream function, we compute the discrete velocity and vorticity fields. Furthermore, we recover an approximate pressure field by solving a Stokes-like problem in a nonconforming Crouzeix-Raviart -type virtual element space that is in a Stokes-complex relation with the Morley-type space of the virtual element approximation. Finally, we confirm our theoretical estimates by solving benchmark problems that include a convex and a nonconvex domain.
{"title":"The nonconforming virtual element method for Oseen’s equation using a stream-function formulation","authors":"D. Adak, G. Manzini","doi":"10.1051/m2an/2023075","DOIUrl":"https://doi.org/10.1051/m2an/2023075","url":null,"abstract":"We approximate the solution of the stream function formulation of the Oseen equations on general domains by designing a nonconforming Morley-type virtual element method. Under a suitable assumption on the continuous problem’s coefficients, the discrete scheme is well-posed. By introducing an enriching operator , we derive an a priori estimate of the error in a discrete H 2 norm. By post-processing the discrete stream function, we compute the discrete velocity and vorticity fields. Furthermore, we recover an approximate pressure field by solving a Stokes-like problem in a nonconforming Crouzeix-Raviart -type virtual element space that is in a Stokes-complex relation with the Morley-type space of the virtual element approximation. Finally, we confirm our theoretical estimates by solving benchmark problems that include a convex and a nonconvex domain.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73584185","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 present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier-Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.
{"title":"On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system","authors":"Yukun Yue, Franziska Weber","doi":"10.1051/m2an/2023071","DOIUrl":"https://doi.org/10.1051/m2an/2023071","url":null,"abstract":"We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier-Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87056651","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 aim of this paper is to develop numerical analysis for the two-dimensional peridynamics which depicts nonlocal phenomena with artificial boundary conditions (ABCs). To this end, the artificial boundary conditions for the fully discretized peridynamcis are proposed. Then, the numerical analysis of the fully discretized scheme is developed such that the artificial boundary conditions solve the corner reflection problem with second-order accuracy. Finally numerical examples are given to verify theoretical results.
{"title":"A second-order absorbing boundary condition for two-dimensional peridynamics","authors":"Gang Pang, Songsong Ji, Leiyu Chao","doi":"10.1051/m2an/2023072","DOIUrl":"https://doi.org/10.1051/m2an/2023072","url":null,"abstract":"The aim of this paper is to develop numerical analysis for the two-dimensional peridynamics which depicts nonlocal phenomena with artificial boundary conditions (ABCs). To this end, the artificial boundary conditions for the fully discretized peridynamcis are proposed. Then, the numerical analysis of the fully discretized scheme is developed such that the artificial boundary conditions solve the corner reflection problem with second-order accuracy. Finally numerical examples are given to verify theoretical results.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80818798","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 novel algorithm for the three-field formulation of Biot's consolidation model based on mixed and divergence-free nonconforming virtual element methods is developed and analyzed. By establishing a discrete counterpart of Korn's inequality, we ensure the well-posedness of this algorithm without special constraints in the context of nonconforming methods. In addition, we also derive a unified error estimate for this fully discrete algorithm no matter whether the specific storage coefficient vanishes or not. Moreover, this algorithm has several features, including supporting general polygonal meshes and arbitrary space approximation orders, and without Poisson's locking and pressure oscillations. Numerical experiments are presented to validate the performance of this algorithm.
{"title":"A virtual element method for overcoming locking phenomena in Biot's consolidation model","authors":"Xin Liu, Zhangxin Chen","doi":"10.1051/m2an/2023073","DOIUrl":"https://doi.org/10.1051/m2an/2023073","url":null,"abstract":"A novel algorithm for the three-field formulation of Biot's consolidation model based on mixed and divergence-free nonconforming virtual element methods is developed and analyzed. By establishing a discrete counterpart of Korn's inequality, we ensure the well-posedness of this algorithm without special constraints in the context of nonconforming methods. In addition, we also derive a unified error estimate for this fully discrete algorithm no matter whether the specific storage coefficient vanishes or not. Moreover, this algorithm has several features, including supporting general polygonal meshes and arbitrary space approximation orders, and without Poisson's locking and pressure oscillations. Numerical experiments are presented to validate the performance of this algorithm.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78377754","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}
Abstract. This work is concerned with entropy functions of the reactive Euler equations describing inviscid compressible flow with chemical reactions. In our recent work [40] we point out that for these equations as a hyperbolic system, the classical entropy function associated with the thermodynamic entropy is no longer strictly convex under the equation of state (EoS) for the ideal gas. In this work, we propose two strategies to address this issue. The first one is to correct the entropy function. Namely, we present a class of strictly convex entropy functions by adding an extra term to the classical one. Such strictly entropy functions contain that constructed in [40] as a special case. The second strategy is to modify the EoS. We show that there exists a family of EoS (for the nonideal gas) such that the classical entropy function is strictly convex. Under these new EoS, the reactive Euler equations are proved to satisfy the conservation dissipation conditions for general hyperbolic relaxation systems, which guarantee the existence of zero relaxation limit. Additionally, an elegant eigen-system of the Jacobian matrix is derived for the reactive Euler equations under the proposed EoS. Numerical experiments demonstrate that the proposed EoS can also generate ZND detonations. Extension of the present results to high dimensions is direct.
{"title":"On strictly convex entropy functions for the reactive Euler equations","authors":"Weifeng Zhao","doi":"10.1051/m2an/2023067","DOIUrl":"https://doi.org/10.1051/m2an/2023067","url":null,"abstract":"Abstract. This work is concerned with entropy functions of the reactive Euler equations describing inviscid compressible flow with chemical reactions. In our recent work [40] we point out that for these equations as a hyperbolic system, the classical entropy function associated with the thermodynamic entropy is no longer strictly convex under the equation of state (EoS) for the ideal gas. In this work, we propose two strategies to address this issue. The first one is to correct the entropy function. Namely, we present a class of strictly convex entropy functions by adding an extra term to the classical one. Such strictly entropy functions contain that constructed in [40] as a special case. The second strategy is to modify the EoS. We show that there exists a family of EoS (for the nonideal gas) such that the classical entropy function is strictly convex. Under these new EoS, the reactive Euler equations are proved to satisfy the conservation dissipation conditions for general hyperbolic relaxation systems, which guarantee the existence of zero relaxation limit. Additionally, an elegant eigen-system of the Jacobian matrix is derived for the reactive Euler equations under the proposed EoS. Numerical experiments demonstrate that the proposed EoS can also generate ZND detonations. Extension of the present results to high dimensions is direct.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76923047","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 and analyze a fully discrete finite element method for a constrained transport (CT) model of the incompressible magnetohydrodynamic (MHD) equations. The spatial discretization is based on mixed finite elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, the magnetic field and magnetic vector potential are discretized by H(curl)-conforming edge element. The time marching is combining a backward Euler scheme and some subtle implicit-explicit treatments for nonlinear and coupling terms. With these treatments, the fully discrete scheme is linear and the computation of the magnetic vector potential is decoupled from the whole coupled system. The most attractive feature of this scheme that it can yield the exactly divergence-free magnetic field and current density on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic field and magnetic vector potential are further demonstrated under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.
{"title":"A fully discrete finite element method for a constrained transport model of the incompressible MHD equations","authors":"Xiaodi Zhang, Haiyan Su, Xianzhu Li","doi":"10.1051/m2an/2023061","DOIUrl":"https://doi.org/10.1051/m2an/2023061","url":null,"abstract":"In this paper, we propose and analyze a fully discrete finite element method for a constrained transport (CT) model of the incompressible magnetohydrodynamic (MHD) equations. The spatial discretization is based on mixed finite elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, the magnetic field and magnetic vector potential are discretized by H(curl)-conforming edge element. The time marching is combining a backward Euler scheme and some subtle implicit-explicit treatments for nonlinear and coupling terms. With these treatments, the fully discrete scheme is linear and the computation of the magnetic vector potential is decoupled from the whole coupled system. The most attractive feature of this scheme that it can yield the exactly divergence-free magnetic field and current density on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic field and magnetic vector potential are further demonstrated under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84283053","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 devise a fully explicit scheme for the nonlinear acoustic wave equation in its second-order formulation in time, using the HHO method for space discretization and the leapfrog scheme for time integration. The key idea for the explicitation is an iterative procedure to approximate at each time step the static nonlinear coupling between the cell and face unknowns. This procedure is based on a splitting of the HHO stabilization operator and, in the linear case, is proved to converge for a large enough weight of the stabilization uniformly in the mesh size. Increasing the stabilization weight turns out to have a moderate impact on the CFL condition. The numerical experiments demonstrate the computational efficiency of the splitting procedure compared to a semi-implicit scheme for the static coupling between cell and face unknowns.
{"title":"Time-explicit Hybrid High-Order method for the nonlinear acoustic wave equation","authors":"Morgane Steins, A. Ern, O. Jamond, Florence Drui","doi":"10.1051/m2an/2023066","DOIUrl":"https://doi.org/10.1051/m2an/2023066","url":null,"abstract":"We devise a fully explicit scheme for the nonlinear acoustic wave equation in its second-order formulation in time, using the HHO method for space discretization and the leapfrog scheme for time integration. The key idea for the explicitation is an iterative procedure to approximate at each time step the static nonlinear coupling between the cell and face unknowns. This procedure is based on a splitting of the HHO stabilization operator and, in the linear case, is proved to converge for a large enough weight of the stabilization uniformly in the mesh size. Increasing the stabilization weight turns out to have a moderate impact on the CFL condition. The numerical experiments demonstrate the computational efficiency of the splitting procedure compared to a semi-implicit scheme for the static coupling between cell and face unknowns.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75085572","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 work, we present a family of layer-averaged models for the Navier-Stokes equations. For its derivation, we consider a layerwise linear vertical profile for the horizontal velocity component. As a particular case, we also obtain layer-averaged models with the common layerwise constant approximation of the horizontal velocity. The approximation of the derivatives of the velocity components is set by following the theory of distributions to account for the discontinuities at the internal interfaces. Several models has been proposed, depending on the order of approximation of an asymptotic analysis respect to the shallowness parameter. Then, we obtain a hydrostatic model with vertical viscous effects, a hydrostatic model where the pressure depends on the stress tensor, and fully non-hydrostatic models, with a complex rheology. It is remarkable that the proposed models generalize plenty of previous models in the literature. Furthermore, all of them satisfy an exact dissipative energy balance. We also propose a model that is second-order accurate in the vertical direction thanks to a correction of the shear stress approximation. Finally, we show how effective the layerwise linear approach is to notably improve, with respect to the layerwise constant method, the approximation of the velocity profile for some geophysical flows. Namely, a Newtonian fluid and some complex viscoplastic (dry granular and Herschel-Bulkley) materials are considered.
{"title":"Layer-averaged approximation of Navier-Stokes system with complex rheologies","authors":"E. Fernández-Nieto, J. Garres-Díaz","doi":"10.1051/m2an/2023065","DOIUrl":"https://doi.org/10.1051/m2an/2023065","url":null,"abstract":"In this work, we present a family of layer-averaged models for the Navier-Stokes equations. For its derivation, we consider a layerwise linear vertical profile for the horizontal velocity component. As a particular case, we also obtain layer-averaged models with the common layerwise constant approximation of the horizontal velocity. The approximation of the derivatives of the velocity components is set by following the theory of distributions to account for the discontinuities at the internal interfaces. Several models has been proposed, depending on the order of approximation of an asymptotic analysis respect to the shallowness parameter. Then, we obtain a hydrostatic model with vertical viscous effects, a hydrostatic model where the pressure depends on the stress tensor, and fully non-hydrostatic models, with a complex rheology. It is remarkable that the proposed models generalize plenty of previous models in the literature. Furthermore, all of them satisfy an exact dissipative energy balance. We also propose a model that is second-order accurate in the vertical direction thanks to a correction of the shear stress approximation. Finally, we show how effective the layerwise linear approach is to notably improve, with respect to the layerwise constant method, the approximation of the velocity profile for some geophysical flows. Namely, a Newtonian fluid and some complex viscoplastic (dry granular and Herschel-Bulkley) materials are considered.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78472128","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 elliptic diffusion (steady-state heat conduction) equation with space-dependent conductivity and inhomogeneous source subject to a generalized oblique boundary condition on a part of the boundary and Dirichlet or Neumann boundary conditions on the remaining part. The oblique boundary condition represents a linear combination between the dependent variable and its normal and tangential derivatives at the boundary. We first prove the well- posedness of the continuous problems. We then develop new finite volume schemes for these problems and prove rigorously the stability and convergence of these schemes. We also address an application to the inverse corrosion problem concerning the reconstruction of the coefficients present in the generalized oblique boundary condition that is prescribed over a portion $Gamma_{0}$ of the boundary $partial Omega$ from Cauchy data on the complementary portion $Gamma_{1} = partial Omega backslash Gamma_{0}$.
{"title":"Steady-state inhomogeneous diffusion with generalized oblique boundary conditions","authors":"A. Bradji, D. Lesnic","doi":"10.1051/m2an/2023063","DOIUrl":"https://doi.org/10.1051/m2an/2023063","url":null,"abstract":"We consider the elliptic diffusion (steady-state heat conduction) equation with space-dependent conductivity and inhomogeneous source subject to a generalized oblique boundary condition on a part of the boundary and Dirichlet or Neumann boundary conditions on the remaining part. The oblique boundary condition represents a linear combination between the dependent variable and its normal and tangential derivatives at the boundary. We first prove the well- posedness of the continuous problems. We then develop new finite volume schemes for these problems and prove rigorously the stability and convergence of these schemes. We also address an application to the inverse corrosion problem concerning the reconstruction of the coefficients present in the generalized oblique boundary condition that is prescribed over a portion $Gamma_{0}$ of the boundary $partial Omega$ from Cauchy data on the complementary portion $Gamma_{1} = partial Omega backslash Gamma_{0}$.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89316923","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 dedicated to the analysis of generalized perfectly matched layers (PMLs) for 2D electromagnetic wave propagation in dispersive waveguides. Under quite general assumptions on frequency-dependent dielectric permittivity and magnetic permeability we prove convergence estimates in homogeneous waveguides and show that the PML error decreases exponentially with respect to the absorption parameter and the length of the absorbing layer. The optimality of this error estimate is studied both numerically and analytically. Finally, we demonstrate that in the case when the waveguide contains a heterogeneity supported away from the absorbing layer, instabilities may occur, even in the case of the non-dispersive media. Our findings are illustrated by numerical experiments.
{"title":"Convergence analysis of time-domain PMLs for 2D electromagnetic wave propagation in dispersive waveguides","authors":"É. Bécache, M. Kachanovska, M. Wess","doi":"10.1051/m2an/2023060","DOIUrl":"https://doi.org/10.1051/m2an/2023060","url":null,"abstract":"This work is dedicated to the analysis of generalized perfectly matched layers (PMLs) for 2D electromagnetic wave propagation in dispersive waveguides. Under quite general assumptions on frequency-dependent dielectric permittivity and magnetic permeability we prove convergence estimates in homogeneous waveguides and show that the PML error decreases exponentially with respect to the absorption parameter and the length of the absorbing layer. The optimality of this error estimate is studied both numerically and analytically. Finally, we demonstrate that in the case when the waveguide contains a heterogeneity supported away from the absorbing layer, instabilities may occur, even in the case of the non-dispersive media. Our findings are illustrated by numerical experiments.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73504144","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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