on spectral multigrid methods for the time dependent navier stokes equations a new splitting scheme is proposed for the numerical solution of the time dependent incompressible navier stokes equations by spectral methods a staggered grid is used for the pressure improved intermediate boundary conditions are employed in the split step for the velocity and spectral multigrid techniques are used for the solution of the implicit equations

finite element preconditioning for spectral multigrid methods abstract the good performance of finite element preconditioning for the algebraic spectral systems is demonstrated different types of multigrid strategies are compared the most efficient strategy consists of a pure richardson relaxation preconditioned by one v cycle of a finite element multigrid method the good smoothing properties of line relaxation become obvious the computational work for the various methods is estimated

handling solid fluid interfaces for viscous flows explicit jump approximation vs ghost cell approaches the ghost cell approaches gca for handling stationary solid boundaries regular or irregular are first investigated theoretically and numerically for the diffusion equation with dirichlet boundary conditions the main conclusion of this part of investigation is that the approximation for the diffusion term has to be second order accurate everywhere in order for the numerical solution to be rigorously second order accurate violating this principle the linear and quadratic gcas have the following shortcomings 1 restrictive constraints on grid size when the viscosity is small 2 susceptibleness to instability of a time explicit formulation for strongly transient flows 3 convergence deterioration to zeroth or first order for solutions with high frequency modes therefore the widely used linear extrapolation for enforcing no slip boundary conditions should be avoided even for regular solid boundaries as a remedy a simple method based on explicit jump approximation eja is proposed eja hinges on the idea that a velocity derivative jump at the boundary reduces to the value of the velocity derivative at the fluid side because the velocity of the stationary boundary is zero although the time marching linear system of eja is not symmetric it is strictly diagonal dominant with positive diagonal entries numerical results show that over a large range of viscosity and grid sizes eja performs much better than gcas in terms of stability and accuracy furthermore the second order convergence of eja does not depend on viscosity and the spectrum of the solution as those of gcas do this paper is written with enough details so that one can reproduce the numerical results

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

efficient pseudospectral flow simulations in moderately complex geometries a computationally efficient pseudospectral method is developed for incompressible flow simulations in two dimensional geometries involving periodicity in one direction and significant surface deformations a pseudoconformal mapping is used to map the flow domain into a rectangle thereby establishing an orthogonal curvilinear coordinate system within which the governing equations are formulated the time integration of the spectrally discretized two dimensional momentum equations is performed by a second order mixed explicit implicit time integration scheme the satisfaction of the continuity equation is obtained through the solution of a poisson equation for the pressure and the use of the influence matrix technique a highly efficient iterative solver has been developed for the solution of a generalized stokes problem at each time step based on a spectrally preconditioned biconjugate gradient algorithm which exhibits almost linear scalability requiring an ordernlog2nnumber of operations wherenis the number of unknowns numerical results are presented for two dimensional steady oscillatory and peristaltic flows within an undulating channel which agree well with previous results that have appeared in the literature

improved condition number for spectral methods introduction d une methode spectrale avec un nombre de conditionnement o n 2 n etant le degre maximum des polynomes

superconvergence of the shortley weller approximation for dirichlet problems abstract this paper presents a superconvergence property of the shortley weller s w approximation applied to the poisson type dirichlet problem in a bounded domain r 2 with the boundary this means that if the exact solution belongs to c 3 1 then the approximate solution obtained by the s w formula gives o h 3 accuracy at every grid point whose distance to is o h and o h 2 accuracy at other grid points where h denotes the equal mesh size in x and y directions the similar property holds for the case u c l 2 where l 0 or 1 and 0 1 stands for the holder exponent numerical examples are also given which illustrate our results

computational analysis of elastic field of deep curved beams rings using displacement function equilibrium method a new computational approach is proposed for determining the elastic field of deep curved beams rings which is based on a displacement function equilibrium method the displacement function is defined in terms of the radial and circumferential components of displacements which is governed by a single partial differential equation of equilibrium solution of the elastic field is obtained in terms of the displacement function in which both the uniform and the mixed mode of boundary conditions are treated with equal sophistication the application of the single function computational method is demonstrated through the numerical solutions of a number of problems of curved beams and rings with mixed boundary conditions the soundness and accuracy of the investigation are verified through the comparison of results with those obtained by conventional computational and analytical approaches available

a general fictitious domain method with immersed jumps and multilevel nested structured meshes this study addresses a new fictitious domain method for elliptic problems in order to handle general and possibly mixed embedded boundary conditions e b c robin neumann and dirichlet conditions on an immersed interface the main interest of this fictitious domain method is to use simple structured meshes possibly uniform cartesian nested grids which do not generally fit the interface but define an approximate one a cell centered finite volume scheme with a non conforming structured mesh is derived to solve the set of equations with additional algebraic transmission conditions linking both flux and solution jumps through the immersed approximate interface hence a local correction is devised to take account of the relative surface ratios in each control volume for the robin or neumann boundary condition then the numerical scheme conserves the first order accuracy with respect to the mesh step this opens the way to combine the e b c method with a multilevel mesh refinement solver to increase the precision in the vicinity of the interface such a fictitious domain method is very efficient the l 2 and l norm errors vary like o h l where h l is the grid step of the finest refinement level around the interface until the residual first order discretization error of the non refined zone is reached the numerical results reported here for convection diffusion problems with dirichlet robin and mixed dirichlet and robin boundary conditions confirm the expected accuracy as well as the performances of the present method

spectral methods for initial boundary value problems an alternative approach mea this paper presents a new approach to spectral methods for initial boundary value problems a filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated as an approximate solution the function is chosen that belongs to an appropriate finite dimensional space and minimizes a weighted average of the residuals at these points it is proved that the approximate solution converges to the actual solution at a spectral rate of accuracy in both space and time the proof is based on a priori energy estimates that have been proved for such systems although this method is restricted here to hyperbolic initial boundary value problems and chebyshev polynomials it generalizes to general initial boundary value problems boundary value problems and gegenbauer polynomials

a supra convergent finite difference scheme for the variable coefficient poisson equation on non graded grids we introduce a method for solving the variable coefficient poisson equation on non graded cartesian grids that yields second order accuracy for the solutions and their gradients we employ quadtree in 2d and octree in 3d data structures as an efficient means to represent the cartesian grid allowing for constraint free grid generation the schemes take advantage of sampling the solution at the nodes vertices of each cell in particular the discretization at one cell s node only uses nodes of two 2d or three 3d adjacent cells producing schemes that are straightforward to implement numerical results in two and three spatial dimensions demonstrate supra convergence in the l norm

a spectral collocation method for two dimensional compressible convection abstract a fourier chebyshev spectral simulation of two dimensional compressible convection is presented the fluid is a perfect gas with constant dynamic viscosity and thermal conductivity both slippery and rigid boundary conditions for the velocity are used here the temperature is maintained fixed at the upper and lower boundaries an explicit adams bashforth predictor corrector numerical scheme is used in order to overcome the courant friedriechs lewy condition the nonlinear diffusion terms are handled by an iterative method with finite differences or spectral preconditioning steady state solutions have been obtained for both types of boundary conditions critical exponents are found to be the same as in the incompressible case at least for a weak value of the stratification parameter

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

the spectrum of the chebyshev collocation operator for the heat equation it is shown that the eigenvalues of the pseudospectral chebyshev second derivative operator with separated boundary conditions are real negative and distinct an application to the full potential equation is also presented

construction of some balanced incomplete block designs with nested rows and columns par i j qac ij as j j a 1 where a a constant integer independent of the pair i j chosen satisfies the relationship a v 1 r p 1 q 1 ar ij ac ij and a i j denote respectively the numbers of rows within sets columns within sets and sets in which the pair occurs together these variance balanced binary designs introduced and discussed by singh dey 1979 with parameters v b r p q and a are abbreviated as bibrc v b r p q a note that 1 1 is an alternative version of their condition 4 3 agarwal prasad 1982 1983 and cheng 1986 gave further methods of construction of these designs cheng 1986 considered the case of k v in these designs and called them balanced complete block designs with nested rows and columns with parameters v b r p q and a and abbreviated as bcbrc v b r p q a he has equivalently used the condition

the robin hood method a novel numerical method for electrostatic problems based on a non local charge transfer we introduce a novel numerical method named the robin hood method of solving electrostatic problems the approach of the method is closest to the boundary element methods although significant conceptual differences exist with respect to this class of methods the method achieves equipotentiality of conducting surfaces by iterative non local charge transfer for each of the conducting surfaces non local charge transfers are performed between surface elements which differ the most from the targeted equipotentiality of the surface the method is tested against analytical solutions and its wide range of application is demonstrated the method has appealing technical characteristics for the problem with n surface elements the computational complexity of the method essentially scales with n a where a 2 the required computer memory scales with n while the error of the potential decreases exponentially with the number of iterations for many orders of magnitude of the error without the presence of the critical slowing down the robin hood method could prove useful in other classical or even quantum problems some future development ideas for possible applications outside electrostatics are addressed

a method for constructing balanced incomplete block designs with nested rows and columns summary a general method for constructing balanced incomplete block designs with nested rows and columns is presented many designs are obtained in which none of the treatmentblock treatment row and treatment column incidence matrices is the incidence matrix of a balanced incomplete block design if the method of construction is applied to a generally balanced design then the resulting design is also generally balanced

nested generalized cyclic row column designs summary a nested row column design has v treatments arranged in b blocks each comprising pq units grouped into p rows and q columns a class of designs based on a cyclical method of construction is defined the canonical efficiency factors of such nested generalized cyclic row column designs are examined efficient designs are tabulated for 5 v 15 p 3 q 7 and r v where r bpq v is the number of replications of each treatment row column designs are used to eliminate heterogeneity in two directions more generally an experiment may have several blocks each containing pq plots arranged in p rows and q columns where the blocks may represent a further blocking factor or provide replications of the basic row column design these designs are called nested row column designs since rows and columns are nested within blocks the lattice square designs introduced by yates 1940 are examples of such designs a general analysis and the concept of balance have been considered by singh dey 1979 several series of balanced nested row column designs have been given by singh dey 1979 street 1981 and agrawal prasad 1982a preece 1967 provides a table of nested balanced incomplete block designs and a number of these designs can also be used as balanced nested row column designs other series based on partially balanced designs have been given by street 1981 and agrawal prasad 1982b jarrett hall 1982 gave some examples of small designs based on cyclic and generalized cyclic methods of con struction which they suggested could provide useful alternatives to nearest neighbour designs and methods as an example of a balanced nested row column design consider the following design given by singh dey 1979 for five treatments 0 1 2 3 4 arranged in five blocks of two rows and two columns

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

numerical algorithm for the calculation of nonsymmetric dipolar and rotating monopolar vortex structures a numerical iteration scheme is presented for the calculation of coherent vortex structures steady solutions of the euler vorticity equation are found using a variational characterization for dipolar and monopolar vortices as relative equilibria of the poisson system the variational principle for the vorticity is solved by a numerical method for nonconvex optimization besides the variational principle for the vorticity an optimization process is used for the multipliers that appear in the description the free boundary is solved implicitly in the iteration process

a fast parallel poisson solver on irregular domains applied to beam dynamics simulations we discuss the scalable parallel solution of the poisson equation within a particle in cell pic code for the simulation of electron beams in particle accelerators of irregular shape the problem is discretized by finite differences depending on the treatment of the dirichlet boundary the resulting system of equations is symmetric or mildly nonsymmetric positive definite in all cases the system is solved by the preconditioned conjugate gradient algorithm with smoothed aggregation sa based algebraic multigrid amg preconditioning we investigate variants of the implementation of sa amg that lead to considerable improvements in the execution times we demonstrate good scalability of the solver on distributed memory parallel processor with up to 2048 processors we also compare our iterative solver with an fft based solver that is more commonly used for applications in beam dynamics

splitting techniques for the pseudospectral approximation of the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is considered using the uzawa algorithm the spectral system is decoupled into a helmholtz equation and an equation with a pseudo laplace operator it is proven that the eigenvalues of the pseudo laplacian are real and negative except one zero eigenvalue belonging to the constant mode the pseudospectral method avoids spurious modes since the pressure is approximated with lower order two degrees lower polynomials than the velocity only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used for discretization for the iterative solution of the spectral system an effective preconditioner is necessary here effective finite difference preconditioners for the spectral pseudo laplacian are presented

efficient symmetric positive definite second order accurate monolithic solver for fluid solid interactions we introduce a robust and efficient method to simulate strongly coupled monolithic fluid rigid body interactions we take a fractional step approach where the intermediate state variables of the fluid and of the solid are solved independently before their interactions are enforced via a projection step the projection step produces a symmetric positive definite linear system that can be efficiently solved using the preconditioned conjugate gradient method in particular we show how one can use the standard preconditioner used in standard fluid simulations to precondition the linear system associated with the projection step of our fluid solid algorithm overall the computational time to solve the projection step of our fluid solid algorithm is similar to the time needed to solve the standard fluid only projection step the monolithic treatment results in a stable projection step i e the kinetic energy does not increase in the projection step numerical results indicate that the method is second order accurate in the l norm and demonstrate that its solutions agree quantitatively with experimental results

a cartesian grid embedded boundary method for the heat equation and poisson s equation in three dimensions we present an algorithm for solving poisson s equation and the heat equation on irregular domains in three dimensions our work uses the cartesian grid embedded boundary algorithm for 2d problems of johansen and colella a cartesian grid embedded boundary method for poisson s equation on irregular domains j comput phys 147 2 1998 60 85 and extends work of mccorquodale colella and johansen a cartesian grid embedded boundary method for the heat equation on irregular domains j comput phys 173 2001 620 635 our method is based on a finite volume discretization of the operator on the control volumes formed by intersecting the cartesian grid cells with the domain combined with a second order accurate discretization of the fluxes the resulting method provides uniformly second order accurate solutions and gradients and is amenable to geometric multigrid solvers

spectral multigrid methods for domain decomposition problems using patching techniques abstract we consider spectral domain decomposition problems that use patching techniques the resulting discrete systems are solved by the very efficient spectral multigrid methods for relaxation we employ a preconditioned richardson iteration effective preconditioning is achieved by one or more steps of a line relaxation method for the corresponding finite difference scheme numerical results are presented for rectangular and l shaped domains furthermore a cross point problem is considered

a class of energy stable high order finite difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems we present a class of energy stable high order finite difference interface closures for grids with step resolution changes these grids are commonly used in adaptive mesh refinement of hyperbolic problems the interface closures are such that the global accuracy of the numerical method is that of the interior stencil the summation by parts property is built into the stencil construction and implies asymptotic stability by the energy method while being non dissipative we present one dimensional closures for fourth order explicit and compact pade type finite differences tests on the scalar one and two dimensional wave equations the one dimensional navier stokes solution of a shock and two dimensional inviscid compressible vortex verify the accuracy and stability of this class of methods

a new algorithm for charge deposition for multiple grid method for pic simulations in r z cylindrical coordinates a mesh of nodes used in particle in cell simulations may be refined in some regions to obtain better local spatial resolution without adding excessive computational cost each refinement can be seen as a finer grid in a multiple grid system the standard bilinear weighting method in two dimensional r z cylindrical coordinates using a multiple grid system is detailed and an inadequacy in the method is presented a new algorithm of particle to node weighting in a multiple grid system is then described and compared to the standard bilinear weighting method in two dimensional r z cylindrical coordinates this description pays particular attention to the nodes near the interface between grids of differing cell size the volume weighting method used in this study is geometrically similar to the standard bilinear pic method in cartesian coordinates but weights in proportion to the volumes of revolution of the areas about the z axis rather than the areas themselves as the volume weighting method has the axisymmetric geometry built in it is a natural system to use in this particular case a full particle in cell pic simulation of metal plasma immersion ion implantation and deposition mepiiid of a flat stage using three grids of cell size 0 5x0 5 1x1 and 2x2mm is performed and results are compared to those obtained from a single grid simulation with the same parameters the multiple grid method compares very well to the single grid simulation

a spectral multigrid method for the stokes problem in streamfunction formulation abstract suitable spectral multigrid components for the stokes problem in streamfunction formulation are presented we split the stokes problem into a system of two equations with the laplace operator we discretize by means of a collocation or pseudo spectral method preconditioning with finite differences is employed numerical results for the spectral multigrid method are presented

spectrally accurate numerical incompressible flows with deformed boundaries a finite volume preconditioning method spectrally accurate solutions of the stokes problem have been computed with finite volume preconditioning of a chebyshev collocation method it has been validated on two analytical test cases and applied to the resolution of the navier stokes equations in a deformed driven cavity configuration this numerical tool is ready for application to flows with deformed free surface

numerical methods for the transition to wavy taylor vortices abstract numerical methods for the analysis of the nonaxisymmetric stability of two dimensional axisymmetric flows are discussed the relative merits of a fully spectral approach are compared with methods which are spectral in one direction and finite difference in the other the results are applied to the onset of waviness in taylor vortices it is shown that the methods described are capable of finding accurate eigenvalues for taylor numbers up to fifty times critical

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

geometric discretization of the multidimensional dirac delta distribution application to the poisson equation with singular source terms abstract we present a discretization method for the multidimensional dirac distribution we show its applicability in the context of integration problems and for discretizing dirac distributed source terms in poisson equations with constant or variable diffusion coefficients the discretization is cell based and can thus be applied in a straightforward fashion to quadtree octree grids the method produces second order accurate results for integration superlinear convergence is observed when it is used to model dirac distributed source terms in poisson equations the observed order of convergence is 2 or slightly smaller the method is consistent with the discretization of dirac delta distribution for codimension one surfaces presented in 1 2 we present quadtree octree construction procedures to preserve convergence and present various numerical examples including multi scale problems that are intractable with uniform grids

a robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries in this paper an improved finite volume scheme to discretize diffusive flux on a non orthogonal mesh is proposed this approach based on an iterative technique initially suggested by khosla p k khosla s g rubin a diagonally dominant second order accurate implicit scheme computers and fluids 2 1974 207 209 and known as deferred correction has been intensively utilized by muzaferija s muzaferija adaptative finite volume method for flow prediction using unstructured meshes and multigrid approach ph d thesis imperial college 1994 and later fergizer and peric j h fergizer m peric computational methods for fluid dynamics springer 2002 to deal with the non orthogonality of the control volumes using a more suitable decomposition of the normal gradient our scheme gives accurate solutions in geometries where the basic idea of muzaferija fails first the performances of both schemes are compared for a poisson problem solved in quadrangular domains where control volumes are increasingly skewed in order to test their robustness and efficiency it is shown that convergence properties and the accuracy order of the solution are not degraded even on extremely skewed mesh next the very stable behavior of the method is successfully demonstrated on a randomly distorted grid as well as on an anisotropically distorted one finally we compare the solution obtained for quadrilateral control volumes to the ones obtained with a finite element code and with an unstructured version of our finite volume code for triangular control volumes no differences can be observed between the different solutions which demonstrates the effectiveness of our approach

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

multigrid sinc methods a galerkin method using whittaker cardinal or sinc functions as basis functions is described for the solution of boundary value problems when the solution is analytic in the interior of the domain the error of approximation using 2n 1 points is o e c n 1 2 even if derivatives of the solution are singular at the boundaries a multigrid method with overall complexity o nlogn is used to solve the discrete equations this paper contains a description of the multigrid sinc algorithm along with some preliminary numerical results for two point boundary value problems

nested youden square designs summary youden square designs nested in two or three blocks are discussed these designs are useful in experimental settings with scarce materials involving many treatments

improved spectral multigrid methods for periodic elliptic problems abstract the spectral multigrid method for periodic elliptic problems is examined several modifications are introduced including a midpoint pseudospectral discretization which eliminates the need for filtering the highest fourier mode and new relaxation schemes for isotropic and anisotropic problems numerical results are presented demonstrating substantial increases in efficiency and accuracy over previous methods

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

relaxation schemes for spectral multigrid methods the effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined iterative methods are an attractive alternative to direct methods because fourier transform techniques enable the discrete matrix vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations preconditioning is found to be essential for acceptable rates of convergence preconditioners based on second order finite difference methods are used a comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary the investigations show that iterations based on incomplete lu decompositions provide the most efficient methods for solving these algebraic systems

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

finite difference ghost point multigrid methods on cartesian grids for elliptic problems in arbitrary domains in this paper we present a numerical method for solving elliptic equations in an arbitrary domain described by a level set function with general boundary conditions dirichlet neumann robin etc on cartesian grids using finite difference discretization and non eliminated ghost values a system of n i n g equations in n i n g unknowns is obtained by finite difference discretization on the n i internal grid points and second order interpolation to define the conditions for the n g ghost values the resulting large sparse linear system is then solved by a multigrid technique the novelty of the papers can be summarized as follows general strategy to discretize the boundary condition to second order both in the solution and its gradient a relaxation of inner equations and boundary conditions by a fictitious time method inspired by the stability conditions related to the associated time dependent problem with a convergence proof for the first order scheme an effective geometric multigrid which maintains the structure of the discrete system at all grid levels it is shown that by increasing the relaxation step of the equations associated to the boundary conditions a convergence factor close to the optimal one is obtained several numerical tests including variable coefficients anisotropic elliptic equations and domains with kinks show the robustness efficiency and accuracy of the approach

on semi infinite spectral elements for poisson problems with re entrant boundary singularities abstract a spectral element method is described which enables poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi infinite rectangular regions two choices of trial functions are considered namely the eigenfunctions of the differential operator and chebyshev polynomials the coefficients in the series expansions are obtained by imposing weak c1 matching conditions across element interfaces singularities at re entrant corners are treated by a post processing technique which makes use of the known asymptotic behaviour of the solution at the singular point accurate approximations are obtained with few degrees of freedom

a direct spectral collocation poisson solver in polar and cylindrical coordinates in this paper we present a direct spectral collocation method for the solution of the poisson equation in polar and cylindrical coordinates the solver is applied to the poisson equations for several different domains including a part of a disk an annulus a unit disk and a cylinder unlike other poisson solvers for geometries such as unit disks and cylinders no pole condition is involved for the present solver the method is easy to implement fast and gives spectral accuracy we also use the weighted interpolation technique and nonclassical collocation points to improve the convergence

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

a gpu based multi level algorithm for boundary value problems abstract a novel geometric single grid multi level gsgml algorithm is presented for the numerical solution of boundary value problems bvps the algorithm is specifically designed for parallel scalability with high occupancy of streaming processors inside many core architectures like the graphics processing unit gpu the algorithm consists of iterative superposed operations on a single grid and is composed of two full grid routines a restriction and a coarsened interpolation relaxation the restriction is a local average of scalar or vector fields and the interpolation relaxation is a simultaneous application of coarsened finite difference operators and interpolations in complementary subsets of the grid the routines are scheduled in a saw like refining cycle convergence to machine precision is achieved repeating the full cycle using residuals and superposition the algorithm is attractive for its computational simplicity and provides a competitive gpu based fast solver for elliptic partial differential equations like the poisson helmholtz equation applications shown in this work include the deformation of two dimensional grids the computation of three dimensional streamlines for a singular trifoil knot vortex and the calculation of three dimensional electric potentials in heterogeneous dielectric media

spectral multigrid techniques for the navier stokes equations abstract we present a spectral multigrid method for the effective solution of the navier stokes equations for incompressible fluids here we consider the vorticity streamfunction formulation discretization in time is performed by a second order semi implicit scheme where the diffusive term is treated in an implicit manner while the convective term is evaluated explicitly for the discretization of the resulting stokes type problem we introduce a new spectral collocation method this scheme is highly accurate and allows efficient solution by spectral multigrid methods we propose an efficient finite difference preconditioner numerical results are presented which substantiate the usefulness of our concept

analysis of chebyshev collocation methods for parabolic equations time discretizations of both the heat equation and the advection diffusion equation in two space variables are analyzed for space approximation the pseudospectral chebyshev method which enjoys infinite order of accuracy for smooth solutions is used the equation is collocated at the gauss chebyshev points and derivatives are computed by the pseudospectral differencing technique at any time interval the pseudospectral solution which is a polynomial of degree n is advanced in time using the implicit theta method for the diffusive part of the equation while the advective term is dealt with explicitly we prove unconditional stability and optimal error bounds depending on both n and delta t the time step in the norms of the weighted sobolev spaces the method considered here is the most commonly used spectral method for parabolic equations to solve the linear system arising at any time interval efficient iterative techniques with scaling are presented

short note controlling self force errors at refinement boundaries for amr pic we analyze the source of the self force errors in the node centered adaptive mesh refinement particle in cell amr pic algorithm and propose a method for reducing those self forces our approach is based on a method of charge deposition due to mayo a mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam journal of numerical analysis 21 2 1984 285 299 that can reduce the self force error to any specified degree of accuracy

splitting techniques for the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is presented using the uzawa algorithm the spectral system is decoupled into helmholtz equations for the velocity components and an equation with the pseudo laplacian for the pressure both the velocity components and the pressure are approximated by polynomials of the same degree n compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well posed the resulting discrete velocity field is identically divergence free only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used here we further compare our treatment with an inhomogeneous neumann condition for the pressure this scheme was proposed in g karniadakis m israeli and s a orszag j comp phys 97 1991 pp 414 443 the comparison shows that the neumann condition yields somewhat better approximations for the pressure in the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure it is numerically shown that a stable third order method in time can be achieved in the case of periodic boundary conditions this result is confirmed by fourier analysis

analysis of the cell centred finite volume method for the diffusion equation abstract this paper presents a rigorous theoretical analysis of the cell centred finite volume method for poisson s equation we review the traditional taylor series expansion technique which suggests that the cell centred method is inconsistent on nonuniform grids which is not confirmed by numerical experiments we then present an analysis of the method which confirms that the solution error does indeed reduce as the cell size is reduced this is supported by numerical calculations

spectral methods for exterior elliptic problems this paper deals with spectral approximations for exterior elliptic problems in two dimensions as in the conventional finite difference or finite element methods it is found that the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions we introduce a spectral boundary treatment at infinity which is compatible with the infinite order interior spectral scheme computational results are presented to demonstrate the spectral accuracy attainable although we deal with a simple laplace problem throughout the paper our analysis covers more complex and general cases

line relaxation for spectral multigrid methods abstract spectral multigrid methods for elliptic problems with dirichlet and periodic boundary conditions are examined the multigrid convergence is improved by certain line relaxation techniques which are suggested by a smoothing analysis furthermore several choices of relaxation parameters are compared numerical results are presented showing the gain in efficiency and accuracy over previous methods

a node centered local refinement algorithm for poisson s equation in complex geometries this paper presents a method for solving poisson s equation with dirichlet boundary conditions on an irregular bounded three dimensional region the method uses a nodal point discretization and adaptive mesh refinement amr on cartesian grids and the amr multigrid solver of almgren the discrete laplacian operator at internal boundaries comes from either linear or quadratic shortley weller extrapolation and the two methods are compared it is shown that either way solution error is second order in the mesh spacing error in the gradient of the solution is first order with linear extrapolation but second order with shortley weller examples are given with comparison with the exact solution the method is also applied to a heavy ion fusion accelerator problem showing the advantage of adaptivity

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

self sorting mixed radix fast fourier transforms abstract some examples are given of the uses of the fft algorithm in numerical weather prediction the algorithm is derived by means of matrix factorization in such a way that the many possible variants can be obtained in a unified manner particular attention is paid to self sorting variants of the algorithm but skeleton fortran routines are included for all the canonical forms the fft works by decomposing the discrete fourier transform into a sequence of short transforms algorithms for these short transforms are presented and used to derive operation counts for the whole algorithm the implementation of the fft on vector computers is described and in the final section it is demonstrated how savings can be achieved in the case of two dimensional transforms

chebyshev expansion methods for the solution of elliptic partial differential equations abstract in a recent paper haidvogel and zang j comput phys 30 1979 167 described a method based on an expansion in chebyshev polynomials for the solution of poisson s equation in a rectangle they also indicated how the method might be extended to more general equations in this note we give a comparison of the method with the global element method delves and hall j inst math appl 23 1979 223 and give numerical results for those problems treated by haidvogel and zang

boundary and interface conditions within a finite element preconditioner for spectral methods abstract the performances of a finite element preconditioner in the iterative solution of spectral collocation schemes for elliptic boundary value problems is investigated it is shown how to make the preconditioner cheap by adi iterations and how to take advantage of the finite element properties in enforcing neumann and interface conditions in the spectral schemes

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

boundary conditions in chebyshev and legendre methods we discuss two different ways of treating non dirichlet boundary conditions in chebyshev and legendre collocation methods for second order differential problems an error analysis is provided the effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated

some methods of construction of balanced incomplete block designs with nested rows and columns summary a number of methods of construction of balanced incomplete block designs with nested rows and columns are developed using the method of differences

a 3d spectral multigrid method we consider a 3d spectral multigrid method for 3d elliptic problems with arbitrary anisotropies the smoothing method is a preconditioned richardson iteration where a pseudospectral discretization is used on the right hand side and a finite difference discretization on the left hand side in each step of this iteration a finite difference problem is solved approximately by plane relaxation sweeps the plane relaxation requires the solution of 2d finite difference problems these problems are solved approximately using 2d multigrid cycles

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

konvergenzaussagen fur kollokationsverfahren bei elliptischen randwertaufgaben we examine the convergence of collocation with polynomials for elliptic problems the general theory is imbedded in thel p theory from which known results of existence and regularity are adopted convergence results are deduced in the cases of periodic dirichlet and mixed boundary conditions it becomes obvious that the order of convergence essentially depends on the smoothness of the solution