In this paper,we propose a numerical method for solving parabolic interface problems with nonhomogeneous flux jump condition and nonlinear jump condition.The main idea is to use traditional finite element method on se...In this paper,we propose a numerical method for solving parabolic interface problems with nonhomogeneous flux jump condition and nonlinear jump condition.The main idea is to use traditional finite element method on semi-Cartesian mesh coupled with Newton’s method to handle nonlinearity.It is easy to implement even though variable coefficients are used in the jump condition instead of constant in previous work for elliptic interface problem.Numerical experiments show that our method is about second order accurate in the L1 norm.展开更多
Elliptic interface problems with multi-domains and triple junction points have wide applications in engineering and science. However, the corner singularity makes it a chal- lenging problem for most existing methods. ...Elliptic interface problems with multi-domains and triple junction points have wide applications in engineering and science. However, the corner singularity makes it a chal- lenging problem for most existing methods. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve the elliptic interface problems with multi-domains and triple junctions. The resulting linear system of equations is positive definite if the matrix coefficients for the elliptic equations in the domains are positive definite. Numerical experiments show that this method is about second order accurate in the L~ norm for piecewise smooth solutions. Corner singularity can be handled in a way such that the accuracy does not degenerate. The triple junction is carefully resolved and it does not need to be placed on the grid, giving our method the potential to treat moving interface problems without regenerating mesh.展开更多
In this paper,a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using nonbody-fitted grid.Different cases the interface cut the c...In this paper,a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using nonbody-fitted grid.Different cases the interface cut the cell are discussed.The condition number of the large sparse linear system is studied.Numerical results demonstrate that the method is nearly second order accurate in the L^(∞)norm and L^(2) norm,and is first order accurate in the H^(1) norm.展开更多
We propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with sharp-edged interfaces. All possible situations that the interface cuts the grid a...We propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with sharp-edged interfaces. All possible situations that the interface cuts the grid are considered. Both Diriehlet and Neumann boundary conditions are discussed. The coefficient matrix data can be given only on the grids, rather than an analytical function. Extensive numerical experiments show that this method is second order accurate in the L∞ norm.展开更多
Solving elasticity equationswith interfaces is a challenging problemformost existing methods.Nonetheless,it has wide applications in engineering and science.An accurate and efficient method is desired.In this paper,an...Solving elasticity equationswith interfaces is a challenging problemformost existing methods.Nonetheless,it has wide applications in engineering and science.An accurate and efficient method is desired.In this paper,an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces.The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface.The resulting linear system of equations is shown to be positive definite under certain assumptions.Numerical experiments show that thismethod is second order accurate in the L¥norm for piecewise smooth solutions.More than 1.5th order accuracy is observed for solution with singularity(second derivative blows up)on the sharp-edged interface corner.展开更多
基金L.Shi’s research is supported by National Natural Science Foundation of China(No.11701569)S.Hou’s research is supported by Dr.Walter Koss Professorship made available through Louisiana Board of RegentsL.Wang’s research is supported by Science Foundations of China University of Petroleum-Beijing(No.2462015BJB05).
文摘In this paper,we propose a numerical method for solving parabolic interface problems with nonhomogeneous flux jump condition and nonlinear jump condition.The main idea is to use traditional finite element method on semi-Cartesian mesh coupled with Newton’s method to handle nonlinearity.It is easy to implement even though variable coefficients are used in the jump condition instead of constant in previous work for elliptic interface problem.Numerical experiments show that our method is about second order accurate in the L1 norm.
文摘Elliptic interface problems with multi-domains and triple junction points have wide applications in engineering and science. However, the corner singularity makes it a chal- lenging problem for most existing methods. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve the elliptic interface problems with multi-domains and triple junctions. The resulting linear system of equations is positive definite if the matrix coefficients for the elliptic equations in the domains are positive definite. Numerical experiments show that this method is about second order accurate in the L~ norm for piecewise smooth solutions. Corner singularity can be handled in a way such that the accuracy does not degenerate. The triple junction is carefully resolved and it does not need to be placed on the grid, giving our method the potential to treat moving interface problems without regenerating mesh.
基金The author would like to thank the referees for the helpful suggestions.L.Shi’s research is supported by National Natural Science Foundation of China(No.11701569)L.Wang’s research is supported by Science Foundation of China University of Petroleum-Beijing(No.2462015BJB05).S.Hou’s research is supported by Dr.Walter Koss Endowed Professorship.This professorship is made available through the State of Louisiana Board of Regents Support Funds.
文摘In this paper,a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using nonbody-fitted grid.Different cases the interface cut the cell are discussed.The condition number of the large sparse linear system is studied.Numerical results demonstrate that the method is nearly second order accurate in the L^(∞)norm and L^(2) norm,and is first order accurate in the H^(1) norm.
文摘We propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with sharp-edged interfaces. All possible situations that the interface cuts the grid are considered. Both Diriehlet and Neumann boundary conditions are discussed. The coefficient matrix data can be given only on the grids, rather than an analytical function. Extensive numerical experiments show that this method is second order accurate in the L∞ norm.
基金supported by Louisiana Board of Regents RCS Grant No.LEQSF(2008-11)-RD-A-18The second author is partially supported by theAROgrants 56349MA MA,the AFSOR grant FA9550-09-1-0520the NSF grant DMS-0911434,the NIH grant 096195-01 and CNFS 11071123.
文摘Solving elasticity equationswith interfaces is a challenging problemformost existing methods.Nonetheless,it has wide applications in engineering and science.An accurate and efficient method is desired.In this paper,an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces.The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface.The resulting linear system of equations is shown to be positive definite under certain assumptions.Numerical experiments show that thismethod is second order accurate in the L¥norm for piecewise smooth solutions.More than 1.5th order accuracy is observed for solution with singularity(second derivative blows up)on the sharp-edged interface corner.
