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this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coefficients or/and singular sources along the interfaces. The moving interfaces then are updated using the newly developed fast level set formulation which involves computation only inside some small tubes containing the interfaces. This method combines the advantage of the two approaches and gives a second-order Eulerian discretization for interface problems. Several key steps in the implementation are addressed in detail. This newapproach is then applied to Hele-Shaw flow, an unstable flow involving two fluids with very different viscosity. 1997 Academic Press L

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this paper involves the evolution of voids under electro-migration in a conducting metal line where the driving forces for diffusion are the gradient of the curvature and the electric potential along the void boundary. The normal velocity of the void surface is given by the partial differential equation (PDE) U n = # s (C 1 # + C 2 #), (1.1) where # s is the surface Laplacian, # is the potential function associated with an applied electric field, and # is the mean curvature along the boundary; for a circle, the curvature is a positive constant. The coefficients C 1 , C 2 are related to the physical constants C 1 = eD s # 1/3 Z # k B T k , C 2 = D s # 4/3 # s k B T k , (1.2) where e is the charge of an electron, # is the atomic volume, Z # is a phenomenological constant related to the effective valence of an atom, k B is Boltzmann's constant, T k is the temperature, # s is the surface energy. The constant D s is defined as D s = D # s # s k B T k e -Q s /k B T k , (1.3) where # s is the thickness of the diffusion layer, D # s e -Q /k B T k is the surface diffusion coefficient, and Q s is the activation energy for surface diffusion. The electric potential # satisfies the Laplace equation ## = 0, with no-flux boundary condition ## #n =0 on the void boundary as well as other appropriate boundary conditions on the computational boundary. For a void bounded by a closed surface, it can be shown from the divergence theorem that the void conserves its volume (or area) during surface diffusion. In Eq. (1.1), the first term, # s #, is a nonlocal driving force which tends to drift the void along with the electric current. The second term, # s # , is a fourth order nonlinear term which only depends on the local geometry. The boundary evolution governed...

Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. We shall describe a numerical procedure for homogenization, which starts from a discretization of the multiscale differential equation. In this procedure the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert space, which also applies to the differential equation directly. The wavelet based homogenization technique is applied to discretizations of the Helmholtz equation. In one problem from electromagnetic compatibility a subgrid scale geometrical detail is represented on a coarser grid. In another a wave-guide filter is efficiently approximated in a lower dimension. The technique is also applied to the derivation of effective equations for a nonlinear problem and to the derivation of coarse grid operators in multigrid. These multigrid methods work very well for equations with highly oscillatory or discontinuous coefficients.

A level-set method for the simulation of fluid interfaces with insoluble surfactant is presented in two-dimensions. The method can be straightforwardly extended to three-dimensions and to soluble surfactants. The method couples a semi-implicit discretization for solving the surfactant transport equation recently developed by Xu and Zhao [62] with the immersed interface method originally developed by LeVeque and Li and [31] for solving the fluid flow equations and the Laplace-Young boundary conditions across the interfaces. Novel techniques are developed to accurately conserve component mass and surfactant mass during the evolution. Convergence of the method is demonstrated numerically. The method is applied to study the effects of surfactant on single drops, dropdrop interactions and interactions among multiple drops in Stokes flow under a steady applied shear. Due to Marangoni forces and to nonuniform Capillary forces, the presence of surfactant results in larger drop deformations and more complex drop-drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the level-set method has been used to simulate fluid interfaces with surfactant.

A systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations, is described. The technique is based on wavelet projections of the discrete operator followed by a sparse approximation. The resulting numerical method will accurately represent subgrid scale phenomena on a coarse grid. Applications to numerical homogenization and wave propagation in materials with subgrid inhomogeneities are presented. 1 Introduction In the numerical simulation of partial differential equations, the existence of subgrid scale phenomena poses considerable difficulties. With subgrid scale phenomena, we mean processes which should influence the solution on the computational grid but which have length scales shorter than the grid size. Fine scales in the initial values may for example interact with fine scales in the material properties and produce coarse scale contributions to the solution. There are many traditional ways to deal with this probl...

this paper, we formulate a model, based on Greenspan's lubrication theory approximation [14], that describes the coupling between the motion of a drop and the deposition of a surfactant monolayer. The model consists of a Poisson equation for the drop height on a moving domain, together with conditions for the velocity of the domain boundary that incorporate the effect of surfactant deposition. We develop a numerical scheme to compute solutions of the model, using an immersed interface method to solve the Poisson equation, and a level-set method to evolve the moving domain. The numerical 0021-9991/02 $35.00 c All rights reserved. August 27, 2002 13:50 APJ/Journal of Computational Physics JCPH7168 2 HUNTER, LI, AND ZHAO solutions include traveling drops that are qualitatively similar to those observed in experiments