a multi level fast multipole bem for 3 d elastodynamics in the frequency domain to reduce computational complexity and memory requirement for 3 d elastodynamics using the boundary element method bem a multi level fast multipole bem fm bem is proposed the diagonal form for the expansion of the elastodynamic fundamental solution is used with a truncation parameter adjusted to the subdivision level a feature necessary for achieving optimal computational efficiency both the single level and mul ti level forms of the elastodynamic fm bem are considered with emphasis on the latter crucial implementation issues including the truncation of the multipole expansion the optimal number of levels the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments a complexity analysis for both the single level and multi level versions is conducted the correctness and computational performances of the proposed elastodynamic fmm are demonstrated on numerical examples featuring up to o 10 6 dofs run on a single processor pc and including the diffraction of an incident p plane wave by a semi spherical or semi ellipsoidal canyon representative of topographic site effects

numerical analysis of the cahn hilliard equation with a logarithmic free energy a fully discrete finite element method for the cahn hilliard equation with a logarithmic free energy based on the backward euler method is analysed existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented

study of a 3d ginzburg landau functional with a discontinuous pinning term abstract in a convex domain r 3 we consider the minimization of a 3 d ginzburg landau type energy e e u 1 2 u 2 1 2 e 2 a 2 u 2 2 with a discontinuous pinning term a among h 1 c maps subject to a dirichlet boundary condition g h 1 2 s 1 the pinning term a r 3 r takes a constant value b 0 1 in an inner strictly convex subdomain of and 1 outside we prove energy estimates with various error terms depending on assumptions on and g in some special cases we identify the vorticity defects via the concentration of the energy under hypotheses on the singularities of g the singularities are polarized and quantified by their degrees which are 1 vorticity defects are geodesics computed w r t a geodesic metric d a 2 depending only on a joining two paired singularities of g p i n i where is a minimal connection computed w r t a metric d a 2 of the singularities of g and p 1 p k are the positive resp n 1 n k are the negative singularities

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

a fully discrete evolving surface finite element method in this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface in earlier papers using a suitable variational formulation in time dependent sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces ima j numer anal 25 2007 pp 385 407 math comp to appear optimal order l 2 gamma t and h 1 gamma t error bounds were proved for linear finite elements in this work we prove optimal order error bounds for a backward euler time discretization

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

a quantization property for static ginzburg landau vortices for any critical point of the complex ginzburg landau functional in dimension 3 we prove that for large coupling constants if the energy of this critical point on a ball of a given radius r is relatively small compared to r log then the ball of half radius contains no vortex the modulus of the solution is larger than we then show how this property can be applied to describe limiting vortices as e 0 2001 john wiley sons inc

a practical determination strategy of optimal threshold parameter for matrix compression in wavelet bem a practical strategy is developed to determine the optimal threshold parameter for wavelet based boundary element be analysis the optimal parameter is determined so that the amount of storage and computational work is minimized without reducing the accuracy of the be solution in the present study the beylkin type truncation scheme is used in the matrix assembly to avoid unnecessary integration concerning the truncated entries of a coefficient matrix a priori estimation of the matrix entries is introduced and thus the truncated entries are determined twice before and after matrix assembly the optimal threshold parameter is set based on the equilibrium of the truncation and discretization errors these errors are estimated in the residual sense for laplace problems the discretization error is in particular indicated with the potential s contribution c to the residual norm r used in error estimation for mesh adaptation since the normalized residual norm c u u the potential components of be solution cannot be computed without main be analysis the discretization error is estimated by the approximate expression constructed through subsidiary be calculation with smaller degree of freedom dof the matrix compression using the proposed optimal threshold parameter enables us to generate a sparse matrix with o n 1 7 0 1 non zero entries although the quasi optimal memory requirements and complexity are not attained the compression rate of a few per cent can be achieved for n 1000

boundary element and finite element coupling for aeroacoustics simulations we consider the scattering of acoustic perturbations in the presence of a flow we suppose that the space can be split into a zone where the flow is uniform and a zone where the flow is potential in the first zone we apply a prandtl glauert transformation to recover the helmholtz equation the well known setting of boundary element method for the helmholtz equation is available in the second zone the flow quantities are space dependent we have to consider a local resolution namely the finite element method herein we carry out the coupling of these two methods and present various applications and validation test cases the source term is given through the decomposition of an incident acoustic field on a section of the computational domain s boundary validations against analytic another numerical method and measurements on different test cases are presented

a nystrom algorithm for electrostatics of an anisotropic composite the construction of efficient numerical algorithms for the electrostatics of locally anisotropic composites seems to be a poorly explored area in this paper coupled first and second kind fredholm integral equations for the electrostatics of anisotropic inclusions in an anisotropic matrix are derived for a square array of circular cylinders the integral equations are solved to high accuracy through pointwise discretization in the process a recent renormalization method is extended for the evaluation of lattice sums j math phys 35 6036 1994 to encompass the calculation of sums on stretched lattices conditionally convergent stretched lattice sums are computed from physical considerations

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

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

the method of polarized traces for the 2d helmholtz equation we present a solver for the 2d high frequency helmholtz equation in heterogeneous acoustic media with online parallel complexity that scales optimally as o n l where n is the number of volume unknowns and l is the number of processors as long as l grows at most like a small fractional power of n the solver decomposes the domain into layers and uses transmission conditions in boundary integral form to explicitly define polarized traces i e up and down going waves sampled at interfaces local direct solvers are used in each layer to precompute traces of local green s functions in an embarrassingly parallel way the offline part and incomplete green s formulas are used to propagate interface data in a sweeping fashion as a preconditioner inside a gmres loop the online part adaptive low rank partitioning of the integral kernels is used to speed up their application to interface data the method uses second order finite differences the complexity scalings are empirical but motivated by an analysis of ranks of off diagonal blocks of oscillatory integrals they continue to hold in the context of standard geophysical community models such as bp and marmousi 2 where convergence occurs in 5 to 10 gmres iterations while the parallelism in this paper stems from decomposing the domain we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

an efficient solution for cauchy likesystems of linear equations using the displacement approach and transformation of vectors the complexity ofsolving strongly nonsingular cauchy like systems of linear equations email protected email protected will be improved off a factor of log n from the known algorithms of the recursive triangular factorization where f is a field and email protected f n x 1 is a vector and c f n x n is a strongly nonsingular cauchy like matrix the efficient algorithms were presented by morf bitmead and anderson to solve strongly nonsingular toeplitz like equations of linear systems by using the recursive triangular factorization in 1980 recently this approach of the recursive triangular factorization was extended to solve cauchy like systems with the complexity of o n log 3n operations by pan et al this is the best known complexity bound by using the direct approach of recursive triangular factorization in cauchy like cases however these algorithms are still slower than the well known algorithms with the asymptotic bound of o n log 2n operations which have been proposed by the means of reducing cauchy like matrices into toeplitz like matrices in our present paper the algorithms of the recursive triangular factorization are modified so that the complexity bound of the direct recursive approach can be decreased to o n log 2n operations this matches the asymptotic bound without transforming to toeplitz like matrices our improvement of the direct recursive approach is by a factor of log n due to changing the original vectors expressed in the given cauchy like matrix into the special vectors where the entries are unit roots the applications of structured matrices include nevanlinna pick tangential interpolation problems

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

robust exponential convergence of hp fem in balanced norms for singularly perturbed reaction diffusion problems corner domains abstract the h p version of the finite element method is applied to singularly perturbed reaction diffusion type equations on polygonal domains the solution exhibits boundary layers as well as corner layers on a class of meshes that are suitably refined near the boundary and the corners robust exponential convergence in the polynomial degree is shown in both a balanced norm and the maximum norm

short note moore s law and the saffman taylor instability ten years ago hou lowengrub and shelley 4 published a state of the art boundary integral simulation of a classical viscous fingering problem the saffman taylor instability 12 in terms of complexity and level of detail those computations 4 are still among the most ramified and accurately computed interfacial instability patterns that have appeared in the literature since 1994 the computational power of a standard workstation has increased a hundredfold as predicted by moore s law 7 the purpose of this note is to consider moore s law and its consequences in computational science and in particular its impact on studying the saffman taylor instability we illustrate moore s law and fast algorithms in action by presenting the worlds largest viscous fingering simulation to date viscous fingering is one of the fundamental interfacial instabilities in fluid dynamics perturbations to an expanding circular air bubble displacing a viscous fluid in a thin gap flow device become unstable resulting in intricate densely branched interfacial patterns the viscous fingering problem is governed by the three dimensional incompressible navier stokes equations with moving free boundaries at the fluid air interface without simplification this problem is computationally intractable and will remain so for the foreseeable future the simulations in this note build upon decades of advances in mathematical modeling and numerical methods i the fluid dynamics in a thin gap is reduced from the threedimensional navier stokes equations by asymptotic analysis to a darcy s law 12

data sparse approximation of non local operators by h2 matrices many of today s most efficient numerical methods are based on multilevel decompositions the multigrid algorithm is based on a hierarchy of grids wavelet techniques use a hierarchy of basis functions while fast panel clustering and multipole methods employ a hierarchy of clusters the high efficiency of these methods is due to the fact that the hierarchies are nested i e that the information present on a coarser level is also present on finer levels thus allowing efficient recursive algorithms h2 matrices employ nested local expansion systems in order to approximate matrices in optimal or for some problem classes at least optimal up to logarithmic factors order of complexity this paper presents a criterion for the approximability of general matrices in the h2 matrix format and an algorithm for finding good nested expansion systems and constructing the approximation efficiently

a fast 3d dual boundary element method based on hierarchical matrices in this paper a fast solver for three dimensional bem and dbem is developed the technique is based on the use of hierarchical matrices for the representation of the collocation matrix and uses a preconditioned gmres for the solution of the algebraic system of equations the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format special algorithms are developed to deal with crack problems within the context of dbem the structure of dbem matrices has been efficiently exploited and it has been demonstrated that since the cracks form only small parts of the whole structure the use of hierarchical matrices can be particularly advantageous test examples presented show that with the proposed technique substantial increase in number of elements over the crack surfaces leads only to moderate increases in memory storage and solution time

a mixed form thin stratified medium fast multipole algorithm for both low and mid frequency problems a mixed form thin stratified medium fast multipole algorithm is proposed for fast simulation of general microstrip structures at both low and mid frequencies the newly developed matrix friendly formula of layered medium green s function is applied in this algorithm for well separated interactions the path deformation technique is implemented to achieve a smoother and exponentially convergent integral the two dimensional addition theorem is then incorporated into the integrand to expedite the matrix vector product in our approach multipole expansion low frequency fast multiple algorithm as well as the plane wave expansion mid frequency fast multipole algorithm of the translational addition theorem are combined into a single multilevel tree to capture quasi static physics and wave physics simultaneously the outgoing wave is represented first in terms of multipole expansion at leafy levels and then switched to plane wave expansion automatically at higher levels this seamless connection makes the algorithm applicable in simulations where subwavelength interaction and wave physics both exist

analysis of arbitrarily shaped dielectric radomes using adaptive integral method based on volume integral equation the adaptive integral method aim is employed to solve the volume integral equation vie for analyzing the radiation of the antenna with an arbitrarily shaped radome small dipoles are used as exciting sources modeling the radomes by tetrahedron cells the induced volume current is determined by the aim based on vie the application of aim significantly reduces cpu time and computer memory requirement hence the method presented in the paper can be applied to simulate electrically large sized radomes finally the radiation patterns of small dipole arrays in the presence of spherical and conical radomes are calculated

an efficient surface integral algorithm applied to unsteady gravity waves a computationally fast method for calculating the unsteady motion of a surface on a two dimensional fluid is described cauchy s integral theorem is used iteratively to solve laplace s equation for successive time derivatives of the surface motion and time stepping is performed using truncated taylor series this allows fairly large time steps to be made for a given accuracy while the required number of spatial points is minimised by using high order differencing formulae this reduces the overall number of required calculations the numerical implementation of the method is found to be accurate and efficient a fairly thorough examination of this implementation is carried out revealing that high accuracies are often achievable using surprisingly few numerical surface points extensive calculations are also performed using modest computing resources some numerical instabilities are identified although these would not usually be significant in practical calculations a model analysis reveals that two of these instabilities can be eliminated by using suitable methods of time stepping should the third steep wave instability become significant it is shown that it can be completely controlled by using high order smoothing techniques at little cost to accuracy using a routine to ensure asymptotic conservation of energy this is confirmed by time stepping a very steep but stable wave over thousands of wave periods 0 1992 academx press inc since all of the interior properties of a body of fluid undergoing inviscid incompressible irrotational motion are fully determined by properties at its boundary it is possible to reduce the calculation of the motion of such a fluid to the evaluation of the motion of its surface alone in a numerical scheme the entire motion can thus be modelled using only a point discretisation of the surface such an approach was first suggested by svendsen in 1971 as a means of calculating the motion of two dimensional gravity waves on water 13 since then the idea has been implemented using a variety of boundary integral or conformal mapping techniques by longuet higgins and cokelet 2 vinje and brevig 3 baker meiron and orszag 4 roberts s fornberg 6 and based on his original formulation by svendsen and co workers 7 although this paper presents yet another numerical method the approach is aimed specifically at producing a computationally fast and efficient numerical scheme in order to be able to study problems of increased complexity in numerically representing complicated surface motions ranging from the breaking of waves to the evolution of instabilities in a train of travelling waves it typically becomes necessary to use a large number of surface points if running times tend to increase too rapidly with the number of points then this can destroy any apparent advantage in being able to treat the whole fluid only in terms of its moving surface the penalty is particularly severe for algorithms that require matrix inversion or factorisation techniques yielding running times per time step that increase like n3 the cube of the number of surface points by using iterative techniques the penalty can be reduced to running times per time step that increase in proportion to n2 while being less severe this still has the effect that increased numbers of surface points become increasingly expensive to calculate the use of a conformal mapping technique 6 can lead to running times per timestep that vary like n log n however this method is severely limited in its ability to resolve steep wave surfaces as are found in breaking waves requiring disproportionately many surface points to achieve a given accuracy it is therefore vital to ensure computational elliciency if it is to be possible accurately to describe wave processes for which large numbers of surface points may be required the most direct way of achieving this is to employ the lowest number of points that can adequately describe a fluid surface in this paper a further improvement is introduced by showing that higher time derivatives of the motion can be calculated directly using exactly the same technique that is used to determine the velocity of the surface at each timestep this yields significantly more information about the motion for relatively little extra computation in turn this allows larger time steps to be performed for a given accuracy so that the overall number of calculations can be reduced the method presented here combines these two ideas into producing a practical numerical algorithm the properties of which are examined in the following pages in appendix a the formulae are derived that make it

application of the inverse fast multipole method as a preconditioner in a 3d helmholtz boundary element method abstract we investigate an efficient preconditioning of iterative methods such as gmres for solving dense linear systems a x b that follow from a boundary element method bem for the 3d helmholtz equation focusing on the low frequency regime while matrix vector products in gmres can be accelerated through the low frequency fast multipole method lffmm the bem often remains computationally expensive due to the large number of gmres iterations we propose the application of the inverse fast multipole method ifmm as a preconditioner to accelerate the convergence of gmres the ifmm is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low rank approximations the proposed ifmm based preconditioning has a tunable parameter e that balances the cost to construct a preconditioner m which is an approximation of a 1 and the cost to perform the iterative process by means of m namely using a small respectively large value of e takes a long respectively short time to construct m while the number of iterations can be small respectively large a comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method we show that the ifmm preconditioner with a nearly optimal e of 10 2 clearly outperforms some common preconditioners for the bem achieving 1 2 10 8 times speed up of the computations in particular when the scale of the underlying scatterer is about five wavelengths or more in addition the ifmm preconditioner is capable of solving complicated problems in a reasonable amount of time that bd preconditioner can not

bem with linear complexity for the classical boundary integral operators alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as e g galerkin boundary elements including numerical quadrature and panel clustering these representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations low order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered we will present fully discrete galerkin boundary element methods where the storage amount and the cpu time grow only linearly with respect to the number of unknowns

accelerating boundary integral equation method using a special purpose computer we present a new solution to accelerate the boundary integral equation method biem the calculation time of the biem is dominated by the evaluation of the layer potential in the boundary integral equation we performed this task using mdgrape 2 a special purpose computer designed for molecular dynamics simulations mdgrape 2 calculates pairwise interactions among particles e g atoms and ions using hardwired pipeline processors we combined this hardware with an iterative solver during the iteration process mdgrape 2 evaluates the layer potential the rest of the calculation is performed on a conventional pc connected to mdgrape 2 we applied this solution to the laplace and helmholtz equations in three dimensions numerical tests showed that biem is accelerated by a factor of 10 100 our rather naive solution has a calculation cost of o n2 niter where n is the number of unknowns and niter is the number of iterations copyright 2005 john wiley sons ltd

on applications of a complex variable method in compressible flows in this paper we develop a complex variable formulation for the potential equations of compressible fluid flow and discuss the possibility of its application to the solution of compressible fluid flow problems a numerical method to solve inverse problems using this formulation is discussed 6 1990 academic press iiic in this paper we discuss the complex variable formulation of compressible fluid flow equations and its application to inverse problems this paper shoud be viewed as an attempt to extend the complex variable approach cva to compressible fluid flow computation as we know the success of cva with incompressible flow lies in the cauchy riemann formulation of the fluid flow equations this suggests that a similar formulation for compressible fluid flow equations may prove useful for many applications as it turns out the compressible fluid flow equations admit an almost similar representation in an appropriate plane 1 2 the equations in this plane are the nonlinear cauchy riemann equations cre as we will see the nonlinearity in this plane is very mild in the sense that these equations are linear cre with an o m4 nonlinear contribution where m is the local mach number this formulation has a number of significant applications and is likely to be useful in fluid flow applications a similar notion using the partial differential equation in the real plane has been first used by the author 2 and has proven useful in aerodynamic application neglecting the nonlinear contribution provides an almost exact solution by a simple fast fourier transform the principle idea has been to compute this almost exact solution and then to use this as a driver to get the exact solution this approach is naturally favorable provided the nonlinear contribution can be computed economically in 2 the appropriate partial differential equations were solved using this approach and proved quite economical there a clever approach is taken to remove some of the singularities such singularities are encountered at the stagnation points as differential operators are not suitable to deal with such singularities we feel that it is worthwhile to attempt an alternate approach that has inherent

a fast high order solver for problems of scattering by heterogeneous bodies a new high order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented here a scatterer is described by a continuously or discontinuously varying refractive index n x within a two dimensional 2d bounded region solutions of the associated helmholtz equation under given incident fields are then obtained by high order inversion of the lippmann schwinger integral equation the algorithm runs in o nlog n operations where n is the number of discretization points a wide variety of numerical examples provided include applications to highly singular geometries high contrast configurations as well as acoustically electrically large problems for which supercomputing resources have been used recently our method provides highly accurate solutions for such problems on small desktop computers in cpu times of the order of seconds

why finite element discretizations can be factored by triangular hierarchical matrices although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second order elliptic partial differential operators is far from being optimal these methods are often preferred over modern iterative methods this is mainly due to their robustness in this article it is shown that an approximate lu decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical lu decomposition low precision approximations may be used for preconditioning iterative solvers as a byproduct we prove that schur complements of stiffness matrices can be approximated with almost linear complexity

a fast algorithm to solve the beltrami equation with applications to quasiconformal mappings two algorithms are provided for the fast and accurate computation of the solution of beltrami equations in the complex plane in the interior of a unit disk there are two integral operators which are fundamental in the construction of this solution a fast algorithm to evaluate one of these integrals is given by daripa siam j sci stat comput nov 1992 an algorithm for fast evaluation of the second integral is provided here these algorithms are based on representation of the solution in terms of a double integral some recursive relations in fourier space and fast fourier transforms the theoretical computational complexity of our algorithm iso mn inn wherem andn are the number of nodes in the circular and radial directions we show the application of these algorithms in constructing numerical methods for quasiconformal mappings a numerical method is provided and explored numerically for the construction of quasiconformal mappings using the beltrami equation in particular we show the effect of dilatations on the jacobian of the mappings which controls what is known as the crowding phenomenon in the literature

fast algorithms for spectral collocation with non periodic boundary conditions we present a method for the numerical solution of partial differential equations using spectral collocation by employing a structured representation of linear operators we are able to use fast algorithms without being restricted to periodic boundary conditions the underlying ideas are introduced and developed in the context of linearly implicit methods for stiff equations we show how different boundary conditions may be applied and illustrate the technique on the allen cahn equation and the diffusion equation

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

adaptive spatial decomposition in fast multipole method this work presents a new adaptive node cluster algorithm for fast multipole method in the algorithm we use rectangular boxes instead of cubes subdivide a box based on its shape and tighten the child boxes at each subdivision step more importantly we determine the number of expansion terms in multipole to local translations according to the distance between the two interaction boxes our method is tested using benchmark examples for three dimensional potential problems the results obtained show that the new algorithm can solve a problem with 100 thousands nodes in about 20 min and runs nearly three times faster than the standard algorithm the proposed algorithm is especially suitable for treating slender and shell like structures

a fast algorithm for the multiplication of generalized hilbert matrices with vectors on presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de hilbert generalisees par des vecteurs

rcs computation of large inhomogeneous objects using a fast integral equation solver in this paper we apply a fast fourier transform fft accelerated volume integral equation solver to compute the radar cross section of large scale inhomogeneous objects this method is related to bojarski s k space method and the subsequent conjugate cg and biconjugate gradient bcg fft algorithms the method developed here combines the weak form discretization with the bcg and stabilized bcg bcgs solvers we show that this method has marked improvements over related algorithms the numerical method has been validated by mie series for multilayer spheres and applied to some practical problems with this method we are currently able to solve a three dimensional problem with a volume of size 3648 spl lambda sup 3 21 23 million unknowns on a single workstation

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

accelerating t sne using tree based algorithms the paper investigates the acceleration of t sne an embedding technique that is commonly used for the visualization of high dimensional data in scatter plots using two tree based algorithms in particular the paper develops variants of the barnes hut algorithm and of the dual tree algorithm that approximate the gradient used for learning t sne embeddings in o n log n our experiments show that the resulting algorithms substantially accelerate t sne and that they make it possible to learn embeddings of data sets with millions of objects somewhat counterintuitively the barnes hut variant of t sne appears to outperform the dual tree variant

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

pinning with a variable magnetic field of the two dimensional ginzburg landau model abstract we study the ginzburg landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded smooth two dimensional domain supposing that the ginzburg landau parameter and the intensity of the magnetic field are large and of the same order we determine an accurate asymptotic formula for the minimizing energy this asymptotic formula displays the influence of the pinning term also we discuss the existence of non trivial solutions and prove some asymptotics of the third critical field

a fast solver for the helmholtz equation on long thin structures we present a fast solver for the helmholtz equation on long thin structures it operates on an integral equation formulation of the problem in which the solution is represented as a superposition of fields generated by sources on the structure usually on the boundary or boundaries of the structure it uses a standard iterative solver for linear equations in conjunction with a novel method for applying the forward matrix whose computational complexity is o n where n is the number of points on which the integral equation is solved the algorithm is suitable for structures in either two dimensions 2 d or three dimensions it does not depend in any great detail on the specifics of the helmholtz equation and thus is also suitable for similar equations we demonstrate the algorithm by using it to simulate scattering in 2 d from dielectric structures using an integral equation formulation constructed using a combination of single layer and double layer potentials yielding a second kind integral equation numerical results show the algorithm to be efficient and accurate

a wideband fast multipole method for the helmholtz kernel this work presents a new fast multipole method fmm based on plane wave expansions pwfmm combining the advantages of the low and high frequency formulations we revisit the method of greengard etal 1998 devoted to the low frequency regime and based on the splitting of the greens function into a propagative and an evanescent part more precisely we give an explicit formula of the filtered translation function for the propagative part we derive a new formula for the evanescent part and we provide a new interpolation algorithm at all steps we check the accuracy of the method by providing error estimates these theoretical developments are used to propose a wideband fmm based entirely on plane wave expansions the numerical efficiency and accuracy of this broadband pwfmm are illustrated with a numerical example

an inverse method for subcritical flows the inverse problem in the tangent gas approximation is considered an exact method for designing airfoils is presented constraints on the speed distribution are easily implemented a simple numerical algorithm which is fast and accurate is presented comparison of designed airfoils using the tangent gas method with exact euler results is found to be excellent for subcritical flows

improving error bounds for multipole based treecodes rapid evaluation of potentials in particle systems is an important time consuming step in many physical simulations over the past decade the development of treecodes such as the fast multipole method and barnes hut method has enabled large scale simulations in astrophysics molecular dynamics material science etc these methods use fixed degree polynomial approximations of the potential at a point due to a set of particles in a hierarchical representation of the particle system in this paper we present analysis and experiments to illustrate that fixed degree multipole approximations can lead to large aggregate errors we describe an alternate strategy based on careful selection of the multipole degree that leads to asymptotically lower errors while incurring minimal computation overhead first we estimate the error associated with each particle cluster interaction and the aggregate error for each particle then we describe a technique for computing the multipole degree of each interaction with a view to reducing aggregate error and establish the computational complexity of the new method following this we discuss numerical experiments that demonstrate favorable error properties of the new method at the expense of marginal increase in computation finally we report a parallel implementation on the sgi origin 2000 that achieves excellent speedup for problems from domains such as astrophysics and boundary element solvers

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

a broadband volume integral formulation based on edge elements for full wave analysis of lossy interconnects a new numerical fully three dimensional 3 d volume integral formulation for the electromagnetic analysis from static to microwave frequencies of penetrable materials dielectric eventually lossy and conductors with finite conductivity is here discussed its key feature is the introduction of a volumetric loop star decomposition for treating piecewise homogeneous materials the associated shape functions have been determined to decompose the volume current density in a solenoidal and a nonsolenoidal part in analogy to the surface loop star shape functions used for modeling surface current densities on perfect electric conductors the possibility of modeling volumetric ohmic and polarization current densities allows to compute in an accurate way the electromagnetic field in complex 3 d geometries such as high speed interconnects on a broad range of frequencies

phase ordering kinetics on curved surfaces abstract we report results of our study of phase ordering kinetics in curved two dimensional systems using a time dependent ginzburg landau equation specifically we generalize the model a and b equations to curved surfaces derive an interface equation of motion for model a dynamics and propose a new method of calculation of the structure factor valid for curved surfaces

pdes on moving surfaces via the closest point method and a modified grid based particle method partial differential equations pdes on surfaces arise in a wide range of applications the closest point method ruuth and merriman 2008 20 is a recent embedding method that has been used to solve a variety of pdes on smooth surfaces using a closest point representation of the surface and standard cartesian grid methods in the embedding space the original closest point method cpm was designed for problems posed on static surfaces however the solution of pdes on moving surfaces is of considerable interest as well here we propose solving pdes on moving surfaces using a combination of the cpm and a modification of the grid based particle method leung and zhao 2009 12 the grid based particle method gbpm represents and tracks surfaces using meshless particles and an eulerian reference grid our modification of the gbpm introduces a reconstruction step into the original method to ensure that all the grid points within a computational tube surrounding the surface are active we present a number of examples to illustrate the numerical convergence properties of our combined method experiments for advection diffusion equations that are strongly coupled to the velocity of the surface are also presented

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

fast adaptive summation of point forces in the two dimensional poisson equation direct summation of the velocity field introduced by point vortices tends to be time consuming since the velocity of each vortex is found as a sum over all other vortices the resulting number of numerical operations is proportional to the square of the number of vortices here a relatively simple procedure is outlined which significantly reduces the number of operations by replacing selected partial sums by asymptotic series the resulting number of operations appears to vary roughly in proportion to the number of unknowns corresponding to a fast solver

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

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

evolving surface finite element method for the cahn hilliard equation we use the evolving surface finite element method to solve a cahn hilliard equation on an evolving surface with prescribed velocity we start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting the finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity we go on to show a rigorous well posedness result for the continuous equations by showing convergence along a subsequence of the finite element scheme we conclude the paper by deriving error estimates and present various numerical examples

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

a fast bie iteration method for an arbitrary body in a flow of incompressible inviscid fluid the paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics the stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid with a discrete numerical treatment we approximate the exact matrix by a certain toeplitz one and then apply a fast algorithm for this matrix on each iteration step we illustrate the convergence of this iteration scheme by a number of numerical examples both for hard and soft boundary conditions it appears that the method is highly efficient for hard boundaries being much less efficient for soft boundaries

an adaptive multigrid technique for evaluating long range forces in biomolecular simulations we propose an adaptive multigrid method for evaluating the nonlocal long range interactions in biomolecular models the method provides o h 3 accuracy for the forces where h is the mesh size of the finest cubic grid even though the stencil size is minimal two new algorithmic components are developed an adaptive technique for spreading a three dimensional charge distribution onto a space mesh and a procedure for performing coarse to fine grid interpolation of the potential force components and their derivatives grid to particle interpolation at the finest level in a consistent fashion the adaptive mesh used in the spreading procedure contains two types of nodes regular positioned at nodes cubic grid and flexible one per cubic cell thus the charges located in each grid cell are spread to these nine nodes the method conserves charge dipole and quadrupole moments of cells the spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils implementation is also straightforward our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique

a fast dual boundary element method for 3d anisotropic crack problems in the present paper a fast solver for dual boundary element analysis of 3d anisotropic crack problems is formulated implemented and tested the fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix the admissible low rank blocks are computed by adaptive cross approximation aca the performance of aca against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated focusing on the balance between the kernel representation accuracy and the accuracy required for aca the system solution is computed by a preconditioned gmres and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format the effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated highlighting the accuracy as well as the significant reduction in memory storage and analysis time in particular it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom obtaining up to solution speedups of order 10 for systems of order 104 moreover the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed copyright 2009 john wiley sons ltd

adaptive multiscale moment method ammm for analysis of scattering from three dimensional perfectly conducting structures the adaptive multiscale moment method ammm is presented for the analysis of scattering from three dimensional 3d perfectly conducting bodies this algorithm employs the conventional moment method mm using the subsectional triangular patch basis functions and a special matrix transformation which is derived from solving the fredholm equation of the first kind by the multiscale technique this methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time the widely used triangular patch vector basis functions developed by rao et al 1982 are used for expansion and testing functions in the conventional mm the objective here is to compress the unknowns in existing mm codes which solves for the currents crossing the edges of the triangular patch basis functions by use of a matrix transformation the currents source terms and impedance matrix can be arranged in the form of different scales from one scale to another scale the initial guess for the solution can be predicted according to the properties of the multiscale technique ammm can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional mm the basic difference between this methodology and the other wavelet based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown two numerical results are presented which demonstrate that the ammm is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3d perfectly conducting bodies

accelerated cartesian expansion ace based framework for the rapid evaluation of diffusion lossy wave and klein gordon potentials diffusion lossy wave and klein gordon equations find numerous applications in practical problems across a range of diverse disciplines the temporal dependence of all three green s functions are characterized by an infinite tail this implies that the cost complexity of the spatio temporal convolutions associated with evaluating the potentials scales as on s 2n t 2 where n s and n t are the number of spatial and temporal degrees of freedom respectively in this paper we discuss two new methods to rapidly evaluate these spatio temporal convolutions by exploiting their block toeplitz nature within the framework of accelerated cartesian expansions ace the first scheme identifies a convolution relation in time amongst ace harmonics and the fast fourier transform fft is used for efficient evaluation of these convolutions the second method exploits the rank deficiency of the ace translation operators with respect to time and develops a recursive numerical compression scheme for the efficient representation and evaluation of temporal convolutions it is shown that the cost of both methods scales as o n sn tlog 2n t several numerical results are presented for the diffusion equation to validate the accuracy and efficacy of the fast algorithms developed here

a new fast multipole accelerated poisson solver in two dimensions we present an adaptive fast multipole method for solving the poisson equation in two dimensions the algorithm is direct assumes that the source distribution is discretized using an adaptive quad tree and allows for dirichlet neumann periodic and free space conditions to be imposed on the boundary of a square the amount of work per grid point is comparable to that of classical fast solvers even for highly nonuniform grids

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

combining kronecker product approximation with discrete wavelet transforms to solve dense function related linear systems a new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory this technique combines kronecker product approximation and wavelet sparsification for the kronecker product factors the user is only required to supply a procedure for computation of each entry of the given matrix the main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic skeleton method of the second author and data sparse preconditioners the incomplete lu decomposition with dynamic choice of the fill in structure with a prescribed threshold and the inverse kronecker product preconditioner constructed for the sum of kronecker products of sparsified finger like matrices computed by the discrete wavelet transform in some model but quite representative examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 gbyte operative memory

a stabilized cut finite element method for the darcy problem on surfaces we develop a cut finite element method for the darcy problem on surfaces the cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite

an implicit interface boundary integral method for poisson s equation on arbitrary domains abstract we propose a simple formulation for constructing boundary integral methods to solve poisson s equation on domains with smooth boundaries defined through their signed distance function our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain where the integrations are carried out in the level set framework using an appropriate jacobian by the coarea formula the algorithm operates in the euclidean space and does not require any explicit parameterization of the boundaries we present numerical results in two and three dimensions

boundary algebraic equations for lattice problems boundary algebraic equations corresponding to dirichlet boundary value problems on lattices are introduced these equations are based on the lattice green s function from which discrete single and double layer potentials are derived structurally the boundary algebraic equations are similar to the boundary integral equations of classical potential theory numerical experiments indicate that boundary algebraic equations possess excellent spectral properties

phase separation patterns for diblock copolymers on spherical surfaces a finite volume method we explore phase separation on spherical surfaces by solving the cahn hilliard equation modified for diblock copolymers using a finite volume method the spherical surface is discretized into almost uniform triangles by employing successive dyadic refinements of the spherical icosahedron a methodology that avoids potential mathematical and numerical problems related to the poles in spherical coordinates the finite volume method is based on averaging voronoi cells built from triangular meshes to calculate the laplace beltrami operator on the curved surface which greatly improves both the accuracy and speed of calculation as compared to the conventional finite difference method by using this method we simulate the phase separation of diblock copolymers on a spherical surface it is found that stable and intrinsic defects which would not occur in a flat space after sufficient annealing appear in the periodic arrangement of the domains on the curved surface due to the distinct euler characteristic of the surface

wavelet approximations for first kind boundary integral equations on polygons an elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation its galerkin discretization with n degrees of freedom on the boundary with spline wavelets as basis functions is analyzed a truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from o n 2 to o n log n entries the condition numbers are bounded independently of the meshwidth it is proved that the compressed scheme thus obtained yields in o n log n 2 operations approximate solutions with the same asymptotic convergence rates as the full galerkin scheme in the boundary energy norm as well as in interior points numerical examples show the asymptotic error analysis to be valid already for moderate values of n

fmm accelerated source model technique for many scatterer problems motivated by the lack of scalability of the source model technique smt to many scatterer problems a new smt based method for 3 d many scatterer problems is developed the large time complexity of the smt is reduced by deriving an implicit compact representation of the scattering problem via a fast multipole formulation which divides the interactions into far and near ones while grouping and approximating the far interactions this results in significant acceleration of classical smt which is demonstrated in a variety of 3 d many body scattering problems the improvement is notable already at the minimal number of unknowns necessary and grows with the size of the problem

numerical simulation of dealloying by surface dissolution via the evolving surface finite element method in this article we formulate a numerical method for the simulation of dealloying of a binary alloy by the selective removal of one component via electrochemical dissolution such that there is phase separation of the other component at the solid electrolyte interface the evolution of the interface is modelled by a forced mean curvature flow strongly coupled to the solution of a cahn hilliard equation modelling surface phase separation the method is based on a triangulated hypersurface whose evolution is calculated as well as the solution of the cahn hilliard equation by the evolving surface finite element method esfem the numerical experiments simulate complex morphology and concentration patterns providing evidence that the mathematical model may describe the formation of nanoporosity

a boundary element method for steady convective heat diffusion in three dimensions a higher order boundary element method recently developed by the current authors numer heat trans part b fundamentals 45 2004 109 for two dimensional steady convective heat diffusion is generalized to three dimensions in order to facilitate an accurate and efficient boundary element formulation we introduce an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence the localization of the kernels becomes more prominent as the peclet number of the flow increases this in turn leads to increasing sparsity and improved conditioning of the global matrix consequently iterative solvers for sparse matrices become the primary choice in this paper we consider an example problem with an exact solution and investigate the accuracy and efficiency of the boundary element formulation for peclet numbers in the range from 10 to 1000 the bi quartic boundary elements included in this study are shown to provide acceptable levels of resolution

the hybrid boundary node method accelerated by fast multipole expansion technique for 3d potential problems this paper presents a fast formulation of the hybrid boundary node method hybrid bnm for solving problems governed by laplace s equation in 3d the preconditioned gmres is employed for solving the resulting system of equations at each iteration step of the gmres the matrix vector multiplication is accelerated by the fast multipole method green s kernel function is expanded in terms of spherical harmonic series an oct tree data structure is used to hierarchically subdivide the computational domain into well separated cells and to invoke the multipole expansion approximation formulations for the local and multipole expansions and also conversion of multipole to local expansion are given and a binary tree data structure is applied to accelerate the moving least square approximation on surfaces all the formulations are implemented in a computer code written in c numerical examples demonstrate the accuracy and efficiency of the proposed approach copyright 2005 john wiley sons ltd

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