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273
Finite Element Methods and Their Convergence for Elliptic and Parabolic Interface Problems
 Numer. Math
, 1996
"... In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of ar ..."
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Cited by 79 (12 self)
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In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Mathematics Subject Classification (1991): 65N30, 65F10. A running title: Finite element methods for interface problems. Correspondence to: Dr. Jun Zou Email: zou@math.cuhk.edu.hk Fax: (852) 2603 5154 1 Institute of Mathematics, Academia Sinica, Beijing 100080, P.R. China. Email: zmchen@math03.math.ac.cn. The work of this author was partially supported by China National Natural Science Foundation. 2 Department of Mathematics, the Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: zou@math.cuhk....
A secondorderaccurate symmetric discretization of the Poisson equation on irregular domain
 J. Comput. Phys
"... In this paper, we consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather qui ..."
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Cited by 75 (17 self)
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In this paper, we consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem. Multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method. In addition, we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains. Then, we briefly consider Stefan problems.
A Fast Iterative Algorithm For Elliptic Interface Problems
 SIAM J. Numer. Anal
, 1995
"... . A fast, second order accurate iterative method is proposed for the elliptic equation r \Delta (fi(x; y)ru) = f(x; y) in a rectangular region\Omega in 2 space dimensions. We assume that there is an irregular interface across which the coefficient fi, the solution u and its derivatives, and/or the ..."
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Cited by 71 (20 self)
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. A fast, second order accurate iterative method is proposed for the elliptic equation r \Delta (fi(x; y)ru) = f(x; y) in a rectangular region\Omega in 2 space dimensions. We assume that there is an irregular interface across which the coefficient fi, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the jump in fi is large. The interface may or may not align with a underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [SINUM, 4 (1994), pp. 10191044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second order difference scheme for a corresponding Poisson equation in the region, and a second order discretization for a Ne...
Evolution, implementation, and application of level set and fast marching methods for advancing fronts
 J. Comput. Phys
, 2001
"... A variety of numerical techniques are available for tracking moving interfaces. In this review, we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbol ..."
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Cited by 67 (2 self)
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A variety of numerical techniques are available for tracking moving interfaces. In this review, we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws. This link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws. We discuss the evolution of these techniques, the fundamental numerical approximations, involved, implementation details, and applications. In particular, we review some work on three aspects of materials sciences: semiconductor process simulations, seismic processing, and optimal structural topology design. c ○ 2001 Academic Press 1.
A sharp interface Cartesian grid method for simulating flows with complex moving boundaries
 J. Comput. Phys
, 2001
"... A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian– Lagrangian framework is employed, which allows us to treat the imm ..."
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Cited by 66 (4 self)
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A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian– Lagrangian framework is employed, which allows us to treat the immersed moving boundary as a sharp interface. The incompressible Navier–Stokes equations are discretized using a secondorderaccurate finitevolume technique, and a secondorderaccurate fractionalstep scheme is employed for time advancement. The fractionalstep method and associated boundary conditions are formulated in a manner that properly accounts for the boundary motion. A unique problem with sharp interface methods is the temporal discretization of what are termed “freshly cleared ” cells, i.e., cells that are inside the solid at one time step and emerge into the fluid at the next time step. A simple and consistent remedy for this problem is also presented. The solution of the pressure Poisson equation is usually the most timeconsuming step in a fractional step scheme and this is even more so for moving boundary problems where the flow domain changes constantly. A multigrid method is presented and is shown to
S.: The numerical solution of diffusion problems in strongly heterogeneous nonisotropic materials
 J .Comput. Phys
, 1997
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A Fourth Order Accurate Discretization for the Laplace and Heat Equations on Arbitrary Domains, with Applications to the Stefan Problem
 J. Comput. Phys
, 2004
"... In this paper, we first describe a fourth order accurate finite di#erence discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent tim ..."
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Cited by 52 (5 self)
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In this paper, we first describe a fourth order accurate finite di#erence discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent time step restrictions associated with explicit schemes. We then turn our focus to the Stefan problem and construct a third order accurate method that also includes an implicit time discretization.
The Explicit Jump Immersed Interface Method and Interface Problems for Differential Equations
, 1998
"... The Explicit Jump Immersed Interface Method and Interface Problems for Differential Equations by Andreas Wiegmann Chairperson of Supervisory Committee: Professor Kenneth P. Bube Department of Mathematics We study and numerically solve elliptic differential equations in the presence of interfaces wh ..."
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Cited by 51 (7 self)
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The Explicit Jump Immersed Interface Method and Interface Problems for Differential Equations by Andreas Wiegmann Chairperson of Supervisory Committee: Professor Kenneth P. Bube Department of Mathematics We study and numerically solve elliptic differential equations in the presence of interfaces where the solution is not smooth. We use uniform Cartesian grids and do not require the interfaces to be aligned with the grid. We develop a onedimensional theory for the new Explicit Jump Immersed Interface Method (EJIIM), which culminates in a proof of secondorder convergence for piecewiseconstant coefficients for singlepoint interfaces. The proof is interesting in not requiring the numerical scheme to satisfy a discrete maximum principle, the usual means by which such results are proved, and in providing error bounds that are independent of the geometry and the contrast in the coefficients. EJIIM works by focusing on the jumps in the solutions and their derivatives, rather than on findin...
The Blob Projection Method for Immersed Boundary Problems
, 1999
"... this paper is based on the projection formulation of the NavierStokes equations. The following standard decomposition theorem is needed in order to state the formulation. ..."
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Cited by 47 (4 self)
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this paper is based on the projection formulation of the NavierStokes equations. The following standard decomposition theorem is needed in order to state the formulation.