analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

accurate computation of stokes flow driven by an open immersed interface we present numerical methods for computing two dimensional stokes flow driven by forces singularly supported along an open immersed interface two second order accurate methods are developed one for accurately evaluating boundary integral solutions at a point and another for computing stokes solution values on a rectangular mesh we first describe a method for computing singular or nearly singular integrals such as a double layer potential due to sources on a curve in the plane evaluated at a point on or near the curve to improve accuracy of the numerical quadrature we add corrections for the errors arising from discretization which are found by asymptotic analysis when used to solve the stokes equations with sources on an open immersed interface the method generates second order approximations for both the pressure and the velocity and preserves the jumps in the solutions and their derivatives across the boundary we then combine the method with a mesh based solver to yield a hybrid method for computing stokes solutions at n 2 grid points on a rectangular grid numerical results are presented which exhibit second order accuracy to demonstrate the applicability of the method we use the method to simulate fluid dynamics induced by the beating motion of a cilium the method preserves the sharp jumps in the stokes solution and their derivatives across the immersed boundary model results illustrate the distinct hydrodynamic effects generated by the effective stroke and by the recovery stroke of the ciliary beat cycle

the treatment of lubrication forces in boundary integral equations despite their many advantages over other numerical methods boundary integral formulations still fail to provide accurate predictions of mesoscale motion in dense suspensions of rigid particles because the nearly singular flow between surfaces in close proximity cannot be resolved accurately a procedure for incorporating analytical solutions for the lubrication flow within a large scale boundary integral equation method is shown although the method is applied to the case of spherical particles in conjunction with the completed double layer boundary integral equation it can be developed further to treat more complex geometries and can be adapted to other numerical techniques in contrast to other apparently similar approaches the present method does not resort to effective medium approximations and in principle retains all the advantages typical of boundary integral approaches the framework also allows for forces other than those due to hydrodynamic lubrication between particles provided that they are a linear function of the relative velocity or at least can be linearized for example forces due to sub continuum fluid behaviour or forces resulting from surface chemistry it is shown using several benchmarks that the relative motion between two particles in various flows is captured accurately both statically and dynamically in situations where uncorrected simulations fail moreover the computational effort is reduced substantially by the application of the method

accelerated boundary integral method for multiphase flow in non periodic geometries an accelerated boundary integral method for stokes flow of a suspension of deformable particles is presented for an arbitrary domain and implemented for the important case of a planar slit geometry the computational complexity of the algorithm scales as o n or o n log n where n is proportional to the product of number of particles and the number of elements employed to discretize the particle this technique is enabled by the use of an alternative boundary integral formulation in which the velocity field is expressed in terms of a single layer integral alone even in problems with non matched viscosities the density of the single layer integral is obtained from a fredholm integral equation of the second kind involving the double layer integral acceleration in this implementation is provided by the use of general geometry ewald like method ggem for computing the velocity and stress fields driven by a set of point forces in the geometry of interest for the particular case of the slit geometry a fourier chebyshev spectral discretization of ggem is developed efficient implementations employing the ggem methodology are presented for the resulting single and the double layer integrals the implementation is validated with test problems on the velocity of rigid particles and drops between parallel walls in pressure driven flow the taylor deformation parameter of capsules in simple shear flow and the particle trajectories in pair collisions of capsules in simple shear flow the computational complexity of the algorithm is verified with results from several large scale multiparticle simulations

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

integral equation methods for particle simulations in creeping flows abstract integral equation methods for computing the hydrodynamic interactions among solid particles suspended in a creeping flow are presented the particles may have arbitrary shape and they may be suspended in an unbounded or wall bounded fluid the analytic formulation of the integral equation is based on complex variables and the fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary thus large scale problems with complex geometry can be solve using modest computational resources from the hydrodynamic interactions the particle motions are determined either by computing a sequence of steady state stokes flow problems or by coupling the particles equation of motion thereby including the weak effects of the particles solid inertia examples will include the sedimentation of particles in a quiescent fluid towards or parallel to a plane wall and the motion of neutrally buoyant particles in a shear flow

an efficient numerical method for studying interfacial motion in two dimensional creeping flows we present new methods for computing the motion of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal spacing in arclength of the marker points a small scale decomposition is performed to extract the dominant term in the evolution of the interface and we show that this dominant term leads to a cfl type stability constraint when in an equal arclength frame this term is linear and we show that implicit time integration schemes that are explicit in fourier space can be formulated we verify this analysis through several numerical examples

laplace s equation and the dirichlet neumann map a new mode for mikhlin s method mikhlin s method for solving laplace s equation in domains exterior to a number of closed contours is discussed with particular emphasis on the dirichlet neumann map in the literature there already exist two computational modes for mikhlin s method here a new mode is presented the new mode is at least as stable as the previous modes furthermore its computational complexity in the number of closed contours is better as a result highly accurate solutions in domains exterior to tens of thousands of closed contours can be obtained on a simple workstation

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

integral equation methods for elliptic problems with boundary conditions of mixed type laplace s equation with mixed boundary conditions that is dirichlet conditions on parts of the boundary and neumann conditions on the remaining contiguous parts is solved on an interior planar domain using an integral equation method rapid execution and high accuracy is obtained by combining equations which are of fredholm s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results this opens up for the rapid and accurate solution of several elliptic problems of mixed type

fast integral equation methods for rothe s method applied to the isotropic heat equation we present an efficient integral equation approach to solve the forced heat equation u t x email protected x f x u t in a two dimensional multiply connected domain with dirichlet boundary conditions instead of using an integral equation formulation based on the heat kernel we discretize in time first this approach known as rothe s method leads to a non homogeneous modified helmholtz equation that is solved at each time step we formulate the solution to this equation as a volume potential plus a double layer potential and both of these potentials are calculated with available tools accelerated by the fast multipole method for a total of n points in the discretization of the boundary and the domain the total computational cost per time step is o n we demonstrate our approach on the heat equation and the allen cahn equation

convolution theorems with weights analogues of young s inequality and the convolution theorem are shown to hold when the lp and l p q spaces have underlying measure defined in terms of power weights

a complex valued first integral of navier stokes equations unsteady couette flow in a corrugated channel system for a two dimensional incompressible viscous flow a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady navier stokes equations in terms of complex variables and the subsequent introduction of a complex potential field complementary solid and free surface boundary conditions are formulated the methodology is used to solve the challenging problem of unsteady couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral in the stokes flow limit the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature similarly for unsteady flow the results are in accord with related investigations exploring material transfer between trapped eddies and the associated bulk flow and vice versa it is shown how the work relates to the classical complex variable method for solving the

fast multipole method for the biharmonic equation in three dimensions the evaluation of sums matrix vector products of the solutions of the three dimensional biharmonic equation can be accelerated using the fast multipole method while memory requirements can also be significantly reduced we develop a complete translation theory for these equations it is shown that translations of elementary solutions of the biharmonic equation can be achieved by considering the translation of a pair of elementary solutions of the laplace equations the extension of the theory to the case of polyharmonic equations in r 3 is also discussed an efficient way of performing the fmm for biharmonic equations using the solution of a complex valued fmm for the laplace equation is presented compared to previous methods presented for the biharmonic equation our method appears more efficient the theory is implemented and numerical tests presented that demonstrate the performance of the method for varying problem sizes and accuracy requirements in our implementation the fmm for the biharmonic equation is faster than direct matrix vector product for a matrix size of 550 for a relative l 2 accuracy e 2 10 4 and n 3550 for e 2 10 1 2

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

time frequency analysis of the dirac equation abstract the purpose of this paper is to investigate several issues concerning the dirac equation from a time frequency analysis perspective more precisely we provide estimates in weighted modulation and wiener amalgam spaces for the solutions of the dirac equation with rough potentials we focus in particular on bounded perturbations arising as the weyl quantization of suitable time dependent symbols as well as on quadratic and sub quadratic non smooth functions hence generalizing the results in 40 we then prove local well posedness on the same function spaces for the nonlinear dirac equation with a general nonlinearity including power type terms and the thirring model for this study we adopt the unifying framework of vector valued time frequency analysis 56 most of the preliminary results are stated under general assumptions and hence they may be of independent interest

stokes darcy boundary integral solutions using preconditioners in a system where a free fluid flow is coupled to flow in a porous medium different pdes are solved simultaneously in two subdomains we consider steady stokes equations in the free region coupled across a fixed interface to darcy equations in the porous substrate recently the numerical solution of this system was obtained using the boundary integral formulation combined with a regularization correction procedure the correction process also results in the improvement of the condition number of the linear system in this paper an appropriate preconditioner based on the singular part of corrections is introduced to improve the convergence of a krylov subspace method applied to solve the integral formulation

gevrey regularity for a class of dissipative equations with applications to decay abstract in this paper following the techniques of foias and temam we establish gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional laplacian the initial data is taken to be in besov type spaces defined via caloric extension we apply our result to the navier stokes equations the surface quasi geostrophic equations the kuramoto sivashinsky equation and the barotropic quasi geostrophic equation consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the navier stokes equations in the 3d case we extend the class of initial data where such decay holds while in 2d we provide a new class for such decay similar decay result and uniform analyticity band on the attractor is also proven for the sub critical 2d surface quasi geostrophic equation

a parallel fast multipole accelerated integral equation scheme for 3d stokes equations in this paper we discuss a numerical scheme for the stokes equations in three dimensions it uses an integral equation formulation and is accelerated by the new version of fast multipole method first introduced by greengard and rokhlin in 1997 acta numerica 1997 6 229 269 the code is parallelized to solve problems of extremely large size the resulting numerical solver can be applied to stokes flows in complex geometry and also serves as a building block for solving the navier stokes equations of low to moderate reynold s numbers copyright 2006 john wiley sons ltd

integral equation methods for unsteady stokes flow in two dimensions we present an integral equation formulation for the unsteady stokes equations in two dimensions this problem is of interest in its own right as a model for slow viscous flow but perhaps more importantly as an ingredient in the solution of the full incompressible navier stokes equations using the unsteady green s function the velocity evolves analytically as a divergence free vector field this avoids the need for either the solution of coupled field equations as in fully implicit pde based marching schemes or the projection of the velocity field onto a divergence free field at each time step as in operator splitting methods in addition to discussing the analytic properties of the operators that arise in the integral formulation we describe a family of high order accurate numerical schemes and illustrate their performance with several examples

integral equation methods for stokes flow and isotropic elasticity in the plane we present a class of integral equation methods for the solution of biharmonic boundary value problems with applications to two dimensional stokes flow and isotropic elasticity the domains may be multiply connected and finite infinite or semi infinite in extent our analytic formulation is based on complex variables and our fast multipole based iterative solution procedure requireso n operations wherenis the number of nodes in the discretization of the boundary the performance of the methods is illustrated with several large scale numerical examples

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

on the numerical solution of the biharmonic equation in the plane the biharmonic equation arises in a variety of problems in applied mathematics most notably in plane elasticity and in viscous incompressible flow integral equation methods are natural candidates for the numerical solution of such problems since they discritize the boundary alone are easy to apply in the case of free or moving boundaries and achieve superalgebraic convergence rates on sufficiently smooth domains regardless of shape in this paper we follow the work of mayo and greenbaum and make use of the sherman lauricella integral equation which is a fredholm equation with bounded kernel we describe a fast algorithm for the evaluation of the integral operators appearing in that equation when combined with a conjugate gradient like algorithm we are able to solve the discretized integral equation in an amount of time proportional to n where n is the number of nodes in the discretization of the boundary

fast integral equation methods for the modified helmholtz equation we present integral equation methods for the solution to the two dimensional modified helmholtz equation u x email protected email protected x 0 in bounded or unbounded multiply connected domains we consider both dirichlet and neumann problems we derive well conditioned fredholm integral equations of the second kind which are discretized using high order hybrid gauss trapezoid rules our fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary we demonstrate the performance of our methods on several numerical examples and we show that they have both the ability to handle highly complex geometry and the potential to solve large scale problems

stress calculations on multiply connected domains the outstanding problem of finding a simple muskhelishvili type integral equation for stress problems on multiply connected domains is solved complex potentials are represented in a way which allows for the incorporation of cracks and inclusions several numerical examples demonstrate the generality and extreme stability of the approach the stress field is resolved with a relative error of less than 10 10 on a large yet simply reproducible setup with a loaded square plate containing 4096 holes and cracks comparison with previous results in the literature indicates that general purpose finite element software may perform better than many special purpose codes

a new boundary integral formulation for stream function and vorticity in axisymmetric stokes flow a boundary integral equation approach is presented to solve the flow stream function and vorticity in the axisymmetric stokes problem without swirl the governing equations expressed in these variables require that the fundamental solutions of the involved differential operators be found the fundamental solutions in a closed form suitable for numerical treatment of integral equations are obtained and their equivalent forms in infinite integral are derived by using fourier and hankel transforms a coupled system of boundary integral equations for the stream function and the vorticity is formulated by employing new fundamental solutions solution of these equations is realized by the collocation method numerical results are included for several bounded and unbounded domain problems to validate and demonstrate the effectiveness of the proposed approach

a high order 3d boundary integral equation solver for elliptic pdes in smooth domains we present a high order boundary integral equation solver for 3d elliptic boundary value problems on domains with smooth boundaries we use nystrom s method for discretization and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals the overall asymptotic complexity of our method is o n3 2 where n is the number of discretization points on the boundary of the domain and corresponds to linear complexity in the number of uniformly sampled evaluation points a kernel independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators we describe a high order accurate method for evaluating the solution at arbitrary points inside the domain including points close to the domain boundary we demonstrate how our solver combined with a regular grid spectral solver can be applied to problems with distributed sources we present numerical results for the stokes navier and poisson problems

second kind integral equation formulation for the modified biharmonic equation and its applications a system of fredholm second kind integral equations skies is constructed for the modified biharmonic equation in two dimensions with gradient boundary conditions such boundary value problem arises naturally when the incompressible navier stokes equations are solved via a semi implicit discretization scheme and the resulting boundary value problem at each time step is then solved using the pure stream function formulation the advantages of such an approach greengard and kropinski 1998 14 are two fold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided our construction though similar to that of farkas 1989 10 for the biharmonic equation is modified such that the skie formulation has low condition numbers for large values of the parameter we illustrate the performance of our numerical scheme with several numerical examples finally the scheme can be easily coupled with standard fast algorithms such as fft fast multipole methods greengard and rokhlin 1987 15 or fast direct solvers ho and greengard 2012 martinsson and rokhlin 2005 17 25 to achieve optimal complexity bringing accurate large scale long time fluid simulations within practical reach

determinant of complexes and higher hessians abstract let x pr be a smooth algebraic curve in projective space over an algebraically closed field of characteristic zero for each m n the m flexes of x are defined as the points where the osculating hypersurface of degree m has higher contact than expected and a hypersurface h pr is called a m hessian if it cuts x along its m flexes when x is a complete intersection we give an expression for a rational m hessian as the determinant of a complex of graded free modules naturally associated to x

spline galerkin methods for the sherman lauricella equation on contours with corners spline galerkin approximation methods for the sherman lauricella integral equation on simple closed piecewise smooth contours are studied and necessary and sufficient conditions for their stability are obtained it is shown that the method under consideration is stable if and only if certain operators associated with the corner points of the contour are invertible numerical experiments demonstrate a good convergence of the spline galerkin methods and validate theoretical results moreover it is shown that if all corners of the contour have opening angles located in interval 0 1 pi 1 9 pi then the corresponding galerkin method based on splines of order 0 1 or 2 is always stable these results are in strong contrast with the behavior of the nystrom method which has a number of instability angles in the interval mentioned

second kind integral equation formulation of stokes flows past a particle of arbitary shape the problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear fredholm integral equations of the second kind for a

fourth order accurate evaluation of integrals in potential theory on exterior 3d regions we present two methods for the rapid high order accurate evaluation of integrals in potential theory on general unbounded 3d regions our methods allow for direct calculation of derivatives of the integrals as well one of the methods uses a fourth order compact stencil and the other uses a nonstandard variant of richardson extrapolation both methods involve calculation of discontinuities in high order derivatives of the integrals across the boundary of the integration region the extrapolation method in addition involves correction for the discontinuities in truncation error the number of operations required for the methods is essentially equal to twice the number of operations needed to solve poisson s equation on a regular grid both methods avoid problems associated with using quadrature methods to evaluate integrals with singular kernels numerical results are presented for experiments on a variety of geometries in free space

surface tension driven stokes flow a novel numerical scheme is presented for solving the problem of two dimensional stokes flows with free boundaries whose evolution is driven by surface tension the formulation is based on a complex variable formulation of stokes flow and use of conformal mapping to track the free boundaries the method is motivated by applications to modelling the fabrication process for microstructured optical fibres mofs also known as holey fibres and is therefore tailored for the computation of multiple interacting free boundaries we give evidence of the efficacy of the method and discuss its performance

stokes flow around cylinders in a bounded two dimensional domain using multipole accelerated boundary element methods the multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from o n2 iterative solvers or o n3 gaussian elimination to o n log n or o n where n is the number of nodes in the discretized system such a reduction in the growth of the calculation time and data storage is crucial in applications where n is large such as when modelling the macroscopic behaviour of suspensions of particles in such cases a minimum of 1000 particles is needed to obtain statistically meaningful results leading to systems with n of the order of 10 000 for the smallest problems when only boundary velocities are known the indirect boundary element formulation for stokes flow results in fredholm equations of the second kind which generally produce a well posed set of equations when discretized a necessary requirement for iterative solution methods the direct boundary element formulation on the other hand results in fredholm equations of the first kind which upon discretization produce ill conditioned systems of equations the model system here is a two dimensional wide gap couette viscometer where particles are suspended in the fluid between the cylinders this is a typical system that is efficiently modelled using boundary element method simulations the multipolar technique is applied to both direct and indirect formulations it is found that the indirect approach is sufficiently well conditioned to allow the use of fast multipole methods the direct approach results in severe ill conditioning to a point where application of the multipole method leads to non convergence of the solution iteration copyright 1999 john wiley sons ltd

a multi level boundary element method for stokes flows in irregular two dimensional domains recently we have developed multi level boundary element methods mlbem for the solution of the laplace and helmholtz equations that involve asymptotically decaying non oscillatory and oscillatory singular kernels respectively the accuracy and efficiency of the fast boundary element methods for steady state heat diffusion and acoustics problems have been investigated for square domains the current work extends the mlbem methodology to the solution of stokes equation in more complex two dimensional domains the performance of the fast boundary element method for the stokes flows is first investigated for a model problem in a unit square then we consider an example problem possessing an analytical solution in a rectangular domain with 5 1 aspect ratio and finally we study the performance of the mlbem algorithm in a c shaped domain

an integral equation method for the cahn hilliard equation in the wetting problem abstract we present an integral equation approach to solving the cahn hilliard equation equipped with boundary conditions that model solid surfaces with prescribed young s angles the discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step to solve it we split the solution into a volume potential computed with free space kernels plus the solution to a second kind integral equation skie the volume potential is evaluated with the help of a box based volume fmm method for non box domains the source density is extended by solving a biharmonic dirichlet problem the near singular boundary integrals are computed using quadrature by expansion qbx with fmm acceleration our method has linear complexity in the number of surface volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension

efficient numerical methods for multiple surfactant coated bubbles in a two dimensional stokes flow we present efficient and highly accurate numerical methods to compute the deformation of surfactant coated two dimensional bubbles in a slow viscous flow surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion in this paper we restrict our attention to the case of a dilute insoluble surfactant the convection diffusion equation for the surfactant concentration on the interface is coupled with the stokes equations in the fluid domain through a boundary condition based on the laplace young condition the stokes equations are first recast as an integral equation and then solved using a fast multipole accelerated iterative procedure the computational cost per time step is only o n log n operations with n being the number of discretization points on the interface the bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points this equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient implicit treatment of the stiffest terms in the dynamics several phenomenologically different examples are presented

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

gevrey regularity for the supercritical quasi geostrophic equation abstract in this paper following the techniques of foias and temam we establish suitable gevrey class regularity of solutions to the supercritical quasi geostrophic equations in the whole space with initial data in critical sobolev spaces moreover the gevrey class that we obtain is near optimal and as a corollary we obtain temporal decay rates of higher order sobolev norms of the solutions unlike the navier stokes or the subcritical quasi geostrophic equations the low dissipation poses a difficulty in establishing gevrey regularity a new commutator estimate in gevrey classes involving the dyadic littlewood paley operators is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty

a new fast multipole boundary element method for solving large scale two dimensional elastostatic problems a new fast multipole boundary element method bem is presented in this paper for large scale analysis of two dimensional 2 d elastostatic problems based on the direct boundary integral equation bie formulation in this new formulation the fundamental solution for 2 d elasticity is written in a complex form using the two complex potential functions in 2 d elasticity in this way the multipole and local expansions for 2 d elasticity bie are directly linked to those for 2 d potential problems furthermore their translations moment to moment moment to local and local to local turn out to be exactly the same as those in the 2 d potential case this formulation is thus very compact and more efficient than other fast multipole approaches for 2 d elastostatic problems using taylor series expansions of the fundamental solution in its original form several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole bem formulation and code bem models with more than one million equations have been solved successfully on a laptop computer these results clearly demonstrate the potential of the developed fast multipole bem for solving large scale 2 d elastostatic problems copyright 2005 john wiley sons ltd

fast multipole boundary element method for the laplace equation in a locally perturbed half plane with a robin boundary condition a fast multipole boundary element method fm bem for solving large scale potential problems ruled by the laplace equation in a locally perturbed 2 d half plane with a robin boundary condition is developed in this paper these problems arise in a wide gamut of applications being the most relevant one the scattering of water waves by floating and submerged bodies in water of infinite depth the method is based on a multipole expansion of an explicit representation of the associated green s function which depends on a combination of complex valued exponential integrals and elementary functions the resulting method exhibits a computational performance and memory requirements similar to the classic fm bem for full plane potential problems numerical examples demonstrate the accuracy and efficiency of the method

on the computation of stress fields on polygonal domains with v notches the interior stress problem is solved numerically for a single edge notched specimen under uniaxial load the algorithm is based on a modification of a fredholm second kind integral equation with compact operators due to muskhelishvili several singular basis functions for each of the seven corners in the geometry enable high uniform resolution of the stress field with a modest number of discretization points as a consequence notch stress intensity factors can be computed directly from the solution this is an improvement over other procedures where the stress field is not resolved in the corners and where notch stress intensity factors are computed in a roundabout way via a path independent integral numerical examples illustrate the superior stability and economy of the new scheme less

a fast and stable solver for singular integral equations on piecewise smooth curves a scheme for the numerical solution of singular integral equations on piecewise smooth curves is presented it relies on several techniques reduction nystrom discretization composite quadrature recursive compressed inverse preconditioning and multipole acceleration the scheme is fast and stable its computational cost grows roughly logarithmically with the precision sought and linearly with overall system size when the integral equation models a boundary value problem the achievable accuracy may be close to the condition number of that problem times machine epsilon this is illustrated by application to elastostatic problems involving zigzag shaped cracks with up to twenty thousand corners and branched cracks with hundreds of triple junctions

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

an integral equation approach to the incompressible navier stokes equations in two dimensions we present a collection of methods for solving the incompressible navier stokes equations in the plane that are based on a pure stream function formulation the advantages of this approach are twofold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided the disadvantage is that the solution of a nonlinear fourth order partial differential equation is required by recasting this partial differential equation as an integral equation we avoid the ill conditioning which hampers finite difference and finite element methods in this environment by using fast algorithms for the evaluation of volume integrals we are able to solve the equations using o m or o m log m operations where m is the number of points in the discretization of the domain

numerical methods for multiple inviscid interfaces in creeping flows we present new highly accurate and efficient methods for computing the motion of a large number of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal arclength spacing of the marker points we assume that the fluid on one side of the interface is inviscid and we discuss two different physical phenomena bubble dynamics and interfacial motion driven by surface tension viscous sintering applications from buoyancy driven bubble interactions the motion of polydispersed bubbles in an extensional flow and the removal of void spaces through viscous sintering are considered and we present large scale fully resolved simulations involving o 100 closed interfaces

the rapid evaluation of volume integrals of potential theory on general regions abstract we present a new method for the rapid high order accurate evaluation of certain volume integrals in potential theory on general irregular regions the kernels of the integrals are either a fundamental solution or a linear combination of the derivatives of a fundamental solution of a second order linear elliptic differential equation instead of using a standard quadrature formula or the exact evaluation of any integral the methods rely on rapid methods of solving the differential equation which the kernel is the solution of therefore the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid and the method requires no evaluation of the kernel