solving partial differential equations on irregular domains with moving interfaces with applications to superconformal electrodeposition in semiconductor manufacturing we present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces instead of the typical approach of solving in a larger rectangular domain our approach performs most calculations only in the desired domain to do so efficiently we have developed a one sided multigrid method to solve the corresponding large sparse linear systems our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling we begin by motivating the set of equations which model the electrodeposition process building on existing models for superconformal electrodeposition we develop a model which naturally arises from a conservation law form of surface additive evolution we then introduce several numerical algorithms including a conservative material transport level set method and our multigrid method for one sided diffusion equations we then analyze the accuracy of our numerical methods finally we compare our result with experiment over a wide range of physical parameters

the method of regularized stokeslets a numerical method for computing stokes flows in the presence of immersed boundaries and obstacles is presented the method is based on the smoothing of the forces leading to regularized stokeslets the resulting expressions provide the pressure and velocity field as functions of the forcing the latter expression can also be inverted to find the forces that impose a given velocity boundary condition the numerical examples presented demonstrate the wide applicability of the method and its properties solutions converge with second order accuracy when forces are exerted along smooth boundaries examples of segmented boundaries and forcing at random points are also presented

patchmatch filter edge aware filtering meets randomized search for visual correspondence though many tasks in computer vision can be formulated elegantly as pixel labeling problems a typical challenge discouraging such a discrete formulation is often due to computational efficiency recent studies on fast cost volume filtering based on efficient edge aware filters provide a fast alternative to solve discrete labeling problems with the complexity independent of the support window size however these methods still have to step through the entire cost volume exhaustively which makes the solution speed scale linearly with the label space size when the label space is huge or even infinite which is often the case for subpixel accurate stereo and optical flow estimation their computational complexity becomes quickly unacceptable developed to search approximate nearest neighbors rapidly the patchmatch method can significantly reduce the complexity dependency on the search space size but its pixel wise randomized search and fragmented data access within the 3d cost volume seriously hinder the application of efficient cost slice filtering this paper presents a generic and fast computational framework for general multi labeling problems called patchmatch filter pmf we explore effective and efficient strategies to weave together these two fundamental techniques developed in isolation i e patchmatch based randomized search and efficient edge aware image filtering by decompositing an image into compact superpixels we also propose superpixel based novel search strategies that generalize and improve the original patchmatch method further motivated to improve the regularization strength we propose a simple yet effective cross scale consistency constraint which handles labeling estimation for large low textured regions more reliably than a single scale pmf algorithm focusing on dense correspondence field estimation in this paper we demonstrate pmf s applications in stereo and optical flow our pmf methods achieve top tier correspondence accuracy but run much faster than other related competing methods often giving over 10 100 times speedup

effective preconditioning of uzawa type schemes for a generalized stokes problem the schur complement of a model problem is considered as a preconditioner for the uzawa type schemes for the generalized stokes problem the stokes problem with the additional term alpha boldmath u in the motion equation the implementation of the preconditioned method requires for each iteration only one extra solution of the poisson equation with neumann boundary conditions for a wide class of 2d and 3d domains a theorem on its convergence is proved in particular it is established that the method converges with a rate that is bounded by some constant independent of alpha some finite difference and finite element methods are discussed numerical results for finite difference mac scheme are provided

a robust and efficient hybrid cut cell ghost cell method with adaptive mesh refinement for moving boundaries on irregular domains abstract a new robust and efficient cartesian grid method is developed for the moving boundary problem on an arbitrary complex domain a difficult problem in the application of cut cell methods is the presence of degenerate cut cells which are defined as cut cells that are intersected by a boundary curve at more than two points a new strategy is proposed for handling the problem of degenerate cells within the cut cell methodology in the proposed approach gradient fluxes through all sections of the boundary curve intersecting a cut cell are computed using surface integrals with the consequence that all cut cells in the solution domain whether they are regular or degenerate are handled using the same procedure marker points on the boundary curve are used to define cubic splines which provide a high fidelity representation for the boundary curve from which various geometrical properties e g volume face areas centroid etc of cut cells can be determined with high accuracy in order to enhance the robustness of the present approach the concept of a ghost point for degenerate cut cells is introduced which facilitates the evaluation of gradient fluxes in the fluid domain this is advantageous as there is no longer any need to deal individually with each usually small split cut cell in consequence the method proposed here can be interpreted as a hybrid cut cell ghost cell method in which the conventional cut cell method is used to deal with regular cut cells and the ghost cell method is used to handle degenerate cut cells the accuracy and efficiency of this method can be improved further by application of an adaptive mesh refinement amr method based on a fully threaded tree data structure to the cut cells a multigrid acceleration technique is used to solve the resulting algebraic system of discretized equations six test cases for which exact solutions are available are used to demonstrate the accuracy and efficiency of the proposed algorithm for both fixed and moving boundary problems using a triangulated surface mesh to represent an internal boundary surface our proposed approach is generalized to deal with three dimensional fluid flow problems a test case of a three dimensional free surface problem is used to validate the generalization of our cartesian grid approach for complex boundaries in three dimensions

a hybrid ψ v hoc approach for surface tension driven flows in level set framework abstract in this study a higher order compact hoc finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of navier stokes n s equations with the level set method unlike most of the existing methods for multiphase flows that solve the incompressible navier stokes equations in the velocity pressure form or the stream function vorticity form we recast the governing equations into the biharmonic form of the n s equations which carries stream function as the only variable to be computed the coupling of this formulation with level set ls enables the simulation of two phase flows by the core computation of only the single variable free from discontinuities in the whole physical domain in the process we also propose a novel hybrid hoc algorithm for post processing the pressure on either side of the interface including the jump across it the robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves static bubble and oscillating bubble at equilibrium in a fluid at rest and deforming circular drops in shear flows we compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends our approach is proved to be very efficient in terms of its easy implementation relatively low computational time and high order of accuracy

an efficient iterative method for the generalized stokes problem the generalized stokes problem which arises frequently in the simulation of time dependent navier stokes equations for incompressible fluid flow gives rise to symmetric linear systems of equations these systems are indefinite due to a set of linear constraints on the velocity causing difficulty for most preconditioners and iterative methods this paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method this new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite numerical experiments indicating superior convergence compared to existing methods are presented a natural extension of this method to elliptic problems is also proposed along with theoretical bounds on the rate of convergence and results of experiments demonstrating robust and effective preconditioning

numerical solution of the steady stokes equations in this paper we present a fast numerical technique for finding solutions of the steady state stokes equations on both two and three dimensional dimains we implement the method on a special staggered grid for a rectangular cubic domain and obtain a solution in an order of o n log n operations for both two and three dimensional cases where n is the number of grid points in the domain the main idea is to derive from the stokes equations an equation for the pressure p ap b where the matrix a is semi positive definite and very well conditioned on the orthogonal complement of its null space

an immersed interface method for incompressible navier stokes equations the method developed in this paper is motivated by peskin s immersed boundary ib method and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed cartesian grid the ib method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points our method instead incorporates part of this force into jump conditions for the pressure avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary this has been implemented for the two dimensional incompressible navier stokes equations using a high resolution finite volume method for the advective terms and a projection method to enforce incompressibility in the projection step the correct jump in pressure is imposed in the course of solving the poisson problem this gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional ib method comparisons between this method and the ib method are presented for several test problems numerical studies of the convergence and order of accuracy are included

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

domain imbedding methods for the stokes equations we study direct and iterative domain imbedding methods for the stokes equations on certain non rectangular domains in two space dimensions we analyze a continuous analog of numerical domain imbedding for bounded smooth domains and give an example of a simple numerical algorithm suggested by the continuous analysis this algorithms is applicable for simply connected domains which can be covered by rectangular grids with uniformly spaced grid lines in at least one coordinate direction we also discuss a related fft based fast solver for stokes problems with physical boundary conditions on rectangles and present some numerical results

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

a high order immersed interface method for simulating unsteady incompressible flows on irregular domains immersed boundary methods and immersed interface methods are becoming increasingly popular for the computation of unsteady flows around complex geometries using a cartesian grid while good results both qualitative and quantitative have been obtained most of the methods rely on low order corrections to account for the immersed boundary the objective of the present work is to present as an alternative a high order modified immersed interface method for the 2d unsteady incompressible navier stokes equations in stream function vorticity formulation the method employs an explicit fourth order runge kutta time integration scheme fourth order compact finite differences for computation of spatial derivatives and a nine point fourth order compact discretization of the poisson equation for computation of the stream function corrections to the finite difference schemes are used to maintain high formal accuracy at the immersed boundary as confirmed by analytical tests to validate the method in its application to incompressible flows several physically relevant test cases are computed including uniform flow past a circular cylinder and tollmien schlichting waves in a boundary layer

an immersed interface method for simulating the interaction of a fluid with moving boundaries in the immersed interface method boundaries are represented as singular force in the navier stokes equations which enters a numerical scheme as jump conditions recently we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3d with second order spatial and temporal accuracy near the boundaries sheng xu z jane wang systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation siam j sci comput 2006 in press in this paper we implement the immersed interface method to incorporate these jump conditions in a 2d numerical scheme we study the accuracy efficiency and robustness of our method by simulating taylor couette flow flow induced by a relaxing balloon flow past single and multiple cylinders and flow around a flapping wing our results show that 1 our code has second order accuracy in the infinity norm for both the velocity and the pressure 2 the addition of an object introduces relatively insignificant computational cost 3 the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion

metastability of solitary roll wave solutions of the st venant equations with viscosity we study by a combination of numerical and analytical evans function techniques the stability of solitary wave solutions of the st venant equations for viscous shallowwater ow down an incline and related models our main result is to exhibit examples of metastable solitary waves for the st venant equations with stable point spectrum indicating coherence of the wave prole but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake we propose a mechanism based on dynamic spectrum of the wave prole by which a wave train of solitary pulses can stabilize each other by de amplication of convective instabilities as they pass through successive waves we present numerical time evolution studies supporting these conclusions which bear also on the possibility of stable periodic solutions close to the homoclinic for the closely related viscous jin xin model by contrast for which the essential spectrum is stable we show using the stability index of gardner zumbrun that solitary wave pulses are always exponentially unstable possessing point spectra with positive real part

an indirect boundary element method for three dimensional explosion bubbles this paper suggests the implementation of indirect boundary element method ibem to simulate three dimensional underwater explosion bubbles the source is assumed distributed continuously on the boundary surface an expression for the self induced velocity of a point on the boundary surface is obtained via mathematical analysis and with it a new definition of the normal vector valid even for a non smooth surface is given elastic mesh technique emt which is a new mesh regulation technique is further applied to maintain the regularity of the triangularelement mesh used to discretize the dynamic boundary surfaces during the evolution of explosion bubble s the bi cg iterative solver is employed to solve the resulting linear system all these efforts make the present approach viable and robust and which is validated by computations of several bubble dynamics problems

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

an adaptive mesh redistribution method for the incompressible mixture flows using phase field model a phase field model which describes the motion of mixtures of two incompressible fluids is presented by liu and shen c liu j shen a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method phys d 179 2003 211 228 the model is based on an energetic variational formulation in this work we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids it is a coupled nonlinear system of navier stokes equations and allen cahn phase equation phase field equation through an extra stress term and the transport term the numerical strategy is based on the approach proposed by li et al r li t tang p w zhang moving mesh methods in multiple dimensions based on harmonic maps j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution in the pde evolution part the phase field equation is numerically solved by a conservative scheme with a lagrange multiplier and the coupled incompressible navier stokes equations are solved by the incremental pressure correction projection scheme based on the semi staggered grid method in the mesh moving part the mesh points are iteratively redistributed by solving the euler lagrange equations with a parameter free monitor function in each iteration the pressure and the phase are updated on the resulting new grid by a conservative interpolation formula while the velocity is re mapped in a non conservative approach a simple method for preserving divergence free is obtained by projecting the velocity onto the divergence free space after generating the new mesh at the last iterative step numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows

a ghost cell immersed boundary method for flow in complex geometry an efficient ghost cell immersed boundary method gcibm for simulating turbulent flows in complex geometries is presented a boundary condition is enforced through a ghost cell method the reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second order accuracy of the base solver both dirichlet and neumann boundary conditions can be treated the current ghost cell treatment is both suitable for staggered and non staggered cartesian grids the accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface numerical results are compared with experimental data and boundary fitted grid results the method is further extended to an existing ocean model mitgcm to simulate geophysical flow over a three dimensional bump the method is easily implemented as evidenced by our use of several existing codes

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

a cartesian grid method for solving the two dimensional streamfunction vorticity equations in irregular regions we describe a method for solving the two dimensional navier stokes equations in irregular physical domains our method is based on an underlying uniform cartesian grid and second order finite difference finite volume discretizations of the streamfunction vorticity equations geometry representing stationary solid obstacles in the flow domain is embedded in the cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells along the embedded boundary we determine a distribution of vorticity sources needed to impose the no slip flow conditions this distribution appears as a right hand side term in the discretized fluid equations and so we can use fast solvers to solve the linear systems that arise to handle the advective terms we use the high resolution algorithms in clawpack we show that our stokes solver is second order accurate for steady state solutions and that our full navier stokes solver is between first and second order accurate and reproduces results from well studied benchmark problems in viscous fluid flow finally we demonstrate the robustness of our code on flow in a complex domain

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

an efficient numerical method for the two fluid stokes equations with a moving immersed boundary we consider the immersed boundary problem in which the boundary separates two very viscous fluids with differing viscosities the moving elastic boundary may exert a force on the local fluid the model solution is obtained using the immersed interface method which computes second order accurate approximations by incorporating known jumps in the solution or its derivatives into a finite difference method these jump conditions become coupled when the fluid viscosity has a jump across the boundary and this coupling renders the application of the immersed interface method challenging we present a method that first uses boundary integral equations to reduce the two fluid stokes problem to the single fluid case and then solves the single fluid problem using the immersed interface method using this method we assess through two numerical examples how the fluid dynamics are affected by differing viscosities in the two fluid regions we also propose an implicit algorithm and a fractional step algorithm for advancing the boundary position because both algorithms make use of the integral form of the solution neither one requires the solution of a large system of coupled nonlinear equations as is traditionally the case numerical results suggest that for sufficiently stiff problems the fractional time stepping algorithm is the most efficient in the sense that it allows the largest time interval between subsequent updates of global model solutions

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

computational methods for continuum models of platelet aggregation platelet aggregation plays an important role in blood clotting robust numerical methods for simulating the behavior of fogelson s continuum models of platelet aggregation have been developed which in particular involve a hybrid finite difference and spectral method for the models link evolution equation this partial differential equation involvesfourspatial dimensions and time the new methods are used to begin investigating the influence of new chemically induced activation link formation and shear induced link breaking in determining when aggregates develop sufficient strength to remain intact and when they are broken apart by fluid stresses

the immersed boundary method a projection approach a new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion like previous methods a boundary force is applied at the immersed surface to satisfy the no slip constraint this extra constraint can be added to the incompressible navier stokes equations by introducing regularization and interpolation operators the current method gives prominence to the role of the boundary force acting as a lagrange multiplier to satisfy the no slip condition this role is analogous to the effect of pressure on the momentum equation to satisfy the divergence free constraint the current immersed boundary method removes slip and non divergence free components of the velocity field through a projection the boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger cfl numbers compared to some past methods symmetry and positive definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field examples show that the current formulation achieves second order temporal accuracy and better than first order spatial accuracy in l 2 norms for one and two dimensional test problems results from two dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies

an immersed interface method for stokes flows with fixed moving interfaces and rigid boundaries we present an immersed interface method for solving the incompressible steady stokes equations involving fixed moving interfaces and rigid boundaries irregular domains the fixed moving interfaces and rigid boundaries are represented by a number of lagrangian control points in order to enforce the prescribed velocity at the rigid boundaries singular forces are applied on the fluid at these boundaries the strength of singular forces at the rigid boundary is determined by solving a small system of equations for the deformable interfaces the forces that the interface exerts on the fluid are calculated from the configuration position of the deformed interface the jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed moving interfaces and rigid boundaries these forces are interpolated using cubic splines and applied to the fluid through the jump conditions the positions of the deformable interfaces are updated implicitly using a quasi newton method bfgs within each time step in the proposed method the stokes equations are discretized via the finite difference method on a staggered cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient uzawa type method numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible stokes flows with fixed moving interfaces on irregular domains

moving mesh finite element methods for the incompressible navier stokes equations this work presents the first effort in designing a moving mesh algorithm to solve the incompressible navier stokes equations in the primitive variables formulation the main difficulty in developing this moving mesh scheme is how to keep it divergence free for the velocity field at each time level the proposed numerical scheme extends a recent moving grid method based on harmonic mapping r li t tang and p w zhang j comput phys 170 2001 pp 562 588 which decouples the pde solver and the mesh moving algorithm this approach requires interpolating the solution on the newly generated mesh designing a divergence free preserving interpolation algorithm is the first goal of this work selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations which is the second goal of this study the performance of the moving mesh scheme is tested on the standard periodic double shear layer problem no spurious vorticity patterns appear when even fairly coarse grids are used

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

a high order hybridizable discontinuous galerkin method for elliptic interface problems we present a high order hybridizable discontinuous galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface the hybridizable discontinuous galerkin method is endowed with several distinct characteristics first they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries thereby leading to a significant reduction in the global degrees of freedom second they provide for elliptic problems with polygonal interfaces approximations of all the variables that converge with the optimal order of kc1 in the l norm where k denotes the polynomial order of the approximation spaces third they possess some superconvergence properties that allow the use of an inexpensive element by element postprocessing to compute a new approximate solution that converges with order kc2 however for elliptic problems with finite jumps in the solution across the curvilinear interface the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric the discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence to recover the optimal convergence for the approximate solution and gradient we propose to use superparametric elements at the interface copyright 2012 john wiley sons ltd

fast iterative solvers for discrete stokes equations we consider saddle point problems that result from the finite element discretization of stationary and nonstationary stokes equations three efficient iterative solvers for these problems are treated namely the preconditioned conjugate gradient method introduced by bramble and pasciak the preconditioned minres method and a method due to bank et al we give a detailed overview of algorithmic aspects and theoretical convergence results for the method of bank et al a new convergence analysis is presented a comparative study of the three methods for a three dimensional stokes problem discretized by the hood taylor p2 p1 finite element pair is given