In this paper,a new finite element and finite difference(FE-FD)method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes.In the spatial discretization,...In this paper,a new finite element and finite difference(FE-FD)method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes.In the spatial discretization,the standard P,FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite,while near the interface,the maximum principle preserving immersed interface discretization is applied.In the time discretization,a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate.Correction terms are needed when the interface crosses grid lines.The moving interface is represented by the zero level set of a Lipschitz continuous function.Numerical experiments presented in this paper confirm second orderconvergence.展开更多
Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interface...Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed.Note that,while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature,there is a gap in dealing with Navier-Stokes equations,where the non-linear term includes an additional component involving the coordinateφ,even if the geometry and force term are axis-symmetric.Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity,in addition to the pole singularity at r=0.In the newly developed algorithm,we have derived the jump conditions under the cylindrical coordinates.The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates.The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface.As expected,the method is shown to be second-order accurate for the velocity.The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.展开更多
In this paper two dimensional elliptic interface problem with imperfect contact is considered,which is featured by the implicit jump condition imposed on the imperfect contact interface,and the jumping quantity of the...In this paper two dimensional elliptic interface problem with imperfect contact is considered,which is featured by the implicit jump condition imposed on the imperfect contact interface,and the jumping quantity of the unknown is related to the flux across the interface.A finite difference method is constructed for the 2D elliptic interface problems with straight and curve interface shapes.Then,the stability and convergence analysis are given for the constructed scheme.Further,in particular case,it is proved to be monotone.Numerical examples for elliptic interface problems with straight and curve interface shapes are tested to verify the performance of the scheme.The numerical results demonstrate that it obtains approximately second-order accuracy for elliptic interface equations with implicit jump condition.展开更多
This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitt...This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.展开更多
This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explic...This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions.The method reformulates the governing equation with interface conditions into boundary integral equations(BIEs)and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region.Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver.The structure of the coefficient matrix is preserved even with the existence of the interface.In the whole calculation process,analytical expressions of Green’s functions are never determined,formulated or computed.This is the novelty of the proposed kernel-free boundary integral(KFBI)method.Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.展开更多
基金partially supported by the National Natural Science Foundation of China(Grant No.12261070)the Ningxia Key Research and Development Project of China(Grant No.2022BSB03048)+2 种基金partially supported by the Simons(Grant No.633724)and by Fundacion Seneca grant 21760/IV/22partially supported by the Spanish national research project PID2019-108336GB-I00by Fundacion Séneca grant 21728/EE/22.Este trabajo es resultado de las estancias(21760/IV/22)y(21728/EE/22)financiadas por la Fundacion Séneca-Agencia de Ciencia y Tecnologia de la Region de Murcia con cargo al Programa Regional de Movilidad,Colaboracion Internacional e Intercambio de Conocimiento"Jimenez de la Espada".(Plan de Actuacion 2022).
文摘In this paper,a new finite element and finite difference(FE-FD)method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes.In the spatial discretization,the standard P,FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite,while near the interface,the maximum principle preserving immersed interface discretization is applied.In the time discretization,a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate.Correction terms are needed when the interface crosses grid lines.The moving interface is represented by the zero level set of a Lipschitz continuous function.Numerical experiments presented in this paper confirm second orderconvergence.
基金supported by Fundación Sáneca Grant 21728/EE/22(Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad,Colaboración Internacional e Intercambio de Conocimiento“Jiménez de la Espada”)B.Dong is partially supported by the National Natural Science Foundation of China Grant No.12261070+1 种基金Ningxia Key Research and Development Project of China Grant No.2022BSB03048Z.Li is partially supported by Simons Grant 633724 and by Fundación Séneca Grant 21760/IV/22(Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad,Colaboración Internacional Intercambio de Conocimiento“Jiménez de la Espada”).
文摘Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed.Note that,while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature,there is a gap in dealing with Navier-Stokes equations,where the non-linear term includes an additional component involving the coordinateφ,even if the geometry and force term are axis-symmetric.Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity,in addition to the pole singularity at r=0.In the newly developed algorithm,we have derived the jump conditions under the cylindrical coordinates.The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates.The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface.As expected,the method is shown to be second-order accurate for the velocity.The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.
基金supported by the National Natural Science Foundation of China(Grants 12261067,12161067,12361088,62201298,12001015,51961031)the Inner Mongolia Autonomous Region"Youth Science and Technology Talents"support program(Grant NJYT20B15)+1 种基金the Inner Mongolia Scientific Fund Project(Grants 2020MS06010,2021LHMS01006,2022MS01008)by the Innovation fund project of Inner Mongolia University of science and technology-Excellent Youth Science Fund Project(Grant 2019YQL02).
文摘In this paper two dimensional elliptic interface problem with imperfect contact is considered,which is featured by the implicit jump condition imposed on the imperfect contact interface,and the jumping quantity of the unknown is related to the flux across the interface.A finite difference method is constructed for the 2D elliptic interface problems with straight and curve interface shapes.Then,the stability and convergence analysis are given for the constructed scheme.Further,in particular case,it is proved to be monotone.Numerical examples for elliptic interface problems with straight and curve interface shapes are tested to verify the performance of the scheme.The numerical results demonstrate that it obtains approximately second-order accuracy for elliptic interface equations with implicit jump condition.
文摘This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.
基金the National Natural Science Foundation of China(Grant No.DMS-12101553,Grant No.DMS-11771290)the Natural Science Foundation of Zhejiang Province(Grant No.LQ22A010017)+4 种基金the National Key Research and Development Program of China(Project No.2020YFA0712000)the Science Challenge Project of China(Grant No.TZ2016002)the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25000400)the National Science Foundation of America(Grant No.ECCS-1927432)also partially supported by the National Science Foundation of America(Grant No.DMS-1720420).
文摘This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions.The method reformulates the governing equation with interface conditions into boundary integral equations(BIEs)and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region.Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver.The structure of the coefficient matrix is preserved even with the existence of the interface.In the whole calculation process,analytical expressions of Green’s functions are never determined,formulated or computed.This is the novelty of the proposed kernel-free boundary integral(KFBI)method.Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.
