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
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
In this paper, the minimal dissipation local discontinuous Galerkin method is studied to solve the parabolic interface problems in two-dimensional convex polygonal domains. The interface may be arbitrary smooth curves...In this paper, the minimal dissipation local discontinuous Galerkin method is studied to solve the parabolic interface problems in two-dimensional convex polygonal domains. The interface may be arbitrary smooth curves. The proposed method is proved to be L2 stable and the order of error estimates in the given norm is O（h｜logh｜^1/2）. Numerical experiments show the efficiency and accuracy of the method.展开更多
We propose several immersed interface hybridized difference methods(IHDMs),combined with the Crank-Nicolson time-stepping scheme,for parabolic interface problems.The IHDM is the same as the hybrid difference method aw...We propose several immersed interface hybridized difference methods(IHDMs),combined with the Crank-Nicolson time-stepping scheme,for parabolic interface problems.The IHDM is the same as the hybrid difference method away from the interface cells,but the finite difference operators on the interface cells are modified tomaintain the same accuracy throughout the entire domain.For themodification process,we consider virtual extensions of two sub-solutions in the interface cells in such a way that they satisfy certain jump equations between them.We propose several different sets of jump equations and their resulting discrete methods for one-and two-dimensional problems.Some numerical results are presented to demonstrate the accuracy and robustness of the proposed methods.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.
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Supported by the National Natural Science Foundation of China(Grant No.11171038)Youth Foundation of Tianyuan Mathematics(Grant No.11126279)+1 种基金The Science Foundation of China Academy of Engineering Physics(Grant No.2013A0202011)Defense Industrial Technology Development Program(Grant No.B1520133015)
文摘In this paper, the minimal dissipation local discontinuous Galerkin method is studied to solve the parabolic interface problems in two-dimensional convex polygonal domains. The interface may be arbitrary smooth curves. The proposed method is proved to be L2 stable and the order of error estimates in the given norm is O（h｜logh｜^1/2）. Numerical experiments show the efficiency and accuracy of the method.
基金supported by the National Research Foundation of Korea under the grant NRF 2018R1D1A1A09082082.
文摘We propose several immersed interface hybridized difference methods(IHDMs),combined with the Crank-Nicolson time-stepping scheme,for parabolic interface problems.The IHDM is the same as the hybrid difference method away from the interface cells,but the finite difference operators on the interface cells are modified tomaintain the same accuracy throughout the entire domain.For themodification process,we consider virtual extensions of two sub-solutions in the interface cells in such a way that they satisfy certain jump equations between them.We propose several different sets of jump equations and their resulting discrete methods for one-and two-dimensional problems.Some numerical results are presented to demonstrate the accuracy and robustness of the proposed methods.
