diffusion snakes introducing statistical shape knowledge into the mumford shah functional we present a modification of the mumford shah functional and its cartoon limit which facilitates the incorporation of a statistical prior on the shape of the segmenting contour by minimizing a single energy functional we obtain a segmentation process which maximizes both the grey value homogeneity in the separated regions and the similarity of the contour with respect to a set of training shapes we propose a closed form parameter free solution for incorporating invariance with respect to similarity transformations in the variational framework we show segmentation results on artificial and real world images with and without prior shape information in the cases of noise occlusion or strongly cluttered background the shape prior significantly improves segmentation finally we compare our results to those obtained by a level set implementation of geodesic active contours

a genetic search for optimal multigrid components within a fourier analysis setting in this paper fourier analysis is used for finding efficient multigrid components the individual multigrid components for several discrete partial differential operators are chosen automatically by a genetic optimization method from a set of multigrid components such as different smoothers coarse grid correction components cycle types number of smoothing iterations and relaxation parameters an optimal three grid fourier convergence factor corrected for computational complexity is obtained by the genetic search the resulting methods can be tuned for optimal efficiency or toward robustness the analysis results are verified by numerical experiments

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

black box multigrid for nonsymmetric problems 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 previous work this defect was overcome and a code written in which one needed only to specify the logically rectangular positive definite matrix problem the code did everything else necessary to set up the auxiliary coarser problems to achieve a multigrid solution in this paper the previous work is extended to a single nonsymmetric equation

the role of polymorphism in the characterisation of complexity by soft types soft type assignment systems sta sta and sta b characterise by means of reduction of terms computation in complexity classes ptime np and pspace respectively all these systems are inspired by linear logic and include polymorphism similar to the one of system f we show that the presence of polymorphism gives the undecidability of typechecking and type inference we also show that reductions in decidable monomorphic versions of these systems also capture the same complexity classes in a way sufficient for the traditional complexity theory the translations we propose show in addition that the monomorphic systems to serve as a programming language require some metalanguage support since the program which operates on data has form and type which depend on the size of the input

a continuous time approximation to the unstable first order autoregressive process the case without an intercept we consider a first order autoregression with i i d errors and a fixed initial condition the asymptotic distribution of the normalized least squares estimator as the sampling interval converges to zero is shown to be the same as the exact distribution of the continuous time estimator in an ornstein uhlenbeck process this asymptotic distribution permits explicit consideration of the effect of the initial condition the appropriate moment generating function is derived and used to tabulate the limiting distribution and probability density functions the moments and some power functions the adequacy of this asymptotic approximation is found to be excellent for values of the autoregressive parameter near one and any fixed initial condition copyright 1991 by the econometric society

application of the finite difference heterogeneous multiscale method to the richards equation 1 this paper extends the finite difference heterogeneous multiscale method fdhmm to simulate transient unsaturated water flow problems in random porous media the numerical method is based on the use of two different schemes for the original equation at different grid levels which allows numerical results at a lower cost than solving the original equations the main feature of fdhmm is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain the generated code is verified by applying the linearization model of the richards equation considering two different constitutive relationships this method is applied to several test examples with different soil textures and boundary conditions both the dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by fdhmm the numerical experiments demonstrate that fdhmm can effectively simulate the transient unsaturated water flow in the specific soils the numerical experiments also demonstrate that fdhmm can achieve accurate global mass balance and is a globally convergent algorithm and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method

a comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems in l adams and z li siam j sci comput 24 2002 pp 463 479 a multigrid method was designed specifically for interface problems that have been discretized using the methods described in l adams and z li siam j sci comput 24 2002 pp 463 479 and in z li and k ito siam j sci comput 23 2001 pp 339 361 for elliptic interface problems using the maximum principle preserving schemes in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 a new method was introduced that utilizes a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse grid matrices are m matrices this paper compares the immersed interface multigrid methods introduced in l adams and z li siam j sci comput 24 2002 pp 463 479 and l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 with algebraic multigrid which uses no geometric information to set up the multigrid components for coarse grid correction we show that algebraic multigrid is a robust solver for our test problems it outperforms the method in l adams and z li siam j sci comput 24 2002 pp 463 479 and performs nearly as well as the method in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 which is shown to be the most efficient for all problem parameters and sizes

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

a multigrid preconditioner for stabilised discretisations of advection diffusion problems abstract this paper is concerned with the iterative solution of linear systems arising from stabilised discretisations of advection diffusion problems on stretched finite element grids using nonuniform grids of this type leads in general to very badly conditioned matrix problems we therefore consider using gmres in conjunction with a multigrid mg preconditioning strategy in particular we show that in order to achieve the grid size independent convergence which is characteristic of mg methods it is essential to use an effective stabilisation strategy at each level of the mg structure

a semicoarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients the purpose of this paper is to present a semicoarsening multigrid algorithm for solving the finite difference discretization of symmetric and nonsymmetric two and three dimensional elliptic partial differential equations with highly discontinuous and anisotropic coefficients the discrete equations are assumed to be defined on a logically rectangular grid obtained possibly through grid generation for a problem defined on an irregular domain the basic algorithm is described along with some modifications which are designed to improve its efficiency and robustness for certain types of problem cases fortran codes that implement the two and three dimensional semicoarsening multigrid algorithms are described briefly and numerical results are presented

acceleration and stabilization properties of minimal residual smoothing technique in multigrid we analyze the standard multigrid method accelerated by a minimal residual smoothing mrs technique we show that mrs can accelerate the convergence of the slow residual components thus accelerates the overall multigrid convergence we prove that under certain hypotheses mrs stabilizes the divergence of certain slow residual components and thus stabilizes the divergent multigrid iteration the analysis is customarily conducted on the two level method

a multigrid based preconditioned krylov subspace method for the helmholtz equation with pml abstract in this paper we generalize the complex shifted laplacian preconditioner to the complex shifted laplacian pml preconditioner for the helmholtz equation with perfectly matched layer helmholtz pml equation the helmholtz pml equation is discretized by an optimal 9 point difference scheme and the preconditioned linear system is solved by the krylov subspace method especially by the biconjugate gradient stabilized method bi cgstab the spectral analysis of the linear system is given and a new matrix based interpolation operator is proposed for the multigrid method which is used to approximately invert the preconditioner the numerical experiments are presented to illustrate the efficiency of the preconditioned bi cgstab method with the multigrid based on the new interpolation operator also numerical results are given for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in erlangga et al 2006 10

on three grid fourier analysis for multigrid in this paper we present three grid fourier analysis for multigrid methods due to the recursive structure of a multigrid iteration this analysis can be deduced from the well known two grid fourier analysis the coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three grid analysis we apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation the main focus lies on possible improvements by carefully chosen galerkin operators and or by an additional acceleration with restarted gmres gmres m numerical test calculations validate the theoretical predictions

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

fourier analysis of a robust multigrid method for convection diffusion equations we consider a two grid method for solving 2d convection diffusion problems the coarse grid correction is based on approximation of the schur complement as a preconditioner of the schur complement we use the exact schur complement of modified fine grid equations we assume constant coefficients and periodic boundary conditions and apply fourier analysis we prove an upper bound for the spectral radius of the two grid iteration matrix that is smaller than one and independent of the mesh size the convection diffusion ratio and the flow direction i e we have a strong robustness result numerical results illustrating the robustness of the corresponding multigrid w cycle are given

comparison of second and fourth order discretizations for multigrid poisson solvers we combine a compact high order difference approximation with multigrid v cycle algorithm to solve the two dimensional poisson equation with dirichlet boundary conditions this scheme along with several different orderings of grid space and projection operators is compared with the five point formula to show the dramatic improvement in computed accuracy on serial and vector machines

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

robust norm equivalencies for diffusion problems additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem these solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition to obtain a robust iterative scheme it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem in this paper we present such a robust norm equivalence for the model problem w u with a scalar diffusion coefficient in c r 2 our estimates involve only very weak information about w and the results are applicable for a large class of diffusion coefficients namely we require to be in the muckenhoupt class a 1 a function class well studied in harmonic analysis the presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems

the undecidability of the semi unification problem abstract the semi unification problem sup is a natural generalization of both first order unification and matching the problem arises in various branches of computer science and logic although several special cases of sup are known to be decidable the problem in general has been open for several years we show that sup in general is undecidable by reducing what we call the boundedness problem of turing machines to sup the undecidability of this boundedness problem is established by a technique developed in the mid 1960s to prove related results about turing machines

fast multigrid solution of the advection problem with closed characteristics the numerical solution of the advection diffusion problem in the inviscid limit with closed characteristics is studied as a prelude to an efficient high reynolds number flow solver it is demonstrated by a heuristic analysis and numerical calculations that using upstream discretization with downstream relaxation ordering in a multigrid cycle with appropriate residual weighting leads to an efficient solution process upstream finite difference approximations to the advection operator are derived whose truncation terms approximate physical laplacian viscosity thus avoiding spurious solutions to the homogeneous problem when the artificial diffusivity dominates the physical viscosity a brandt and i yavneh j comput phys 93 1991 pp 128 143

matrix dependent multigrid homogenization for diffusion problems for problems with strongly varying or discontinuous diffusion coefficients we present a method to compute coarse scale operators and to approximately determine the effective diffusion tensor on the coarse scale level the approach is based on techniques that are used in multigrid such as matrix dependent prolongations and the construction of coarse grid operators by means of the galerkin approximation in numerical experiments we compare our multigrid homogenization method with continuous homogenization renormalization and simple averaging approaches

a simple proof of a theorem of statman abstract in this note we reprove a theorem of statman that deciding the equality of two first order typable terms is not elementary recursive statman 1982 the basic idea of our proof like that of statman s is the henkin quantifier elimination procedure henkin 1963 however our coding is much simpler and easy to understand

high accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids abstract a fourth order compact finite difference scheme and a multigrid method are employed to solve the two dimensional convection diffusion equations with boundary layers the computational domain is first discretized on a nonuniform stretched grid to resolve the boundary layers a grid transformation technique is used to map the nonuniform grid to a uniform one the fourth order compact scheme is applied to the transformed uniform grid a multigrid method is used to solve the resulting linear system numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and dicretized by the fourth order compact scheme

multigrid geometric active contour models geometric active contour models are very popular partial differential equation based tools in image analysis and computer vision we present a new multigrid algorithm for the fast evolution of level set based geometric active contours and compare it with other established numerical schemes we overcome the main bottleneck associated with most numerical implementations of geometric active contours namely the need for very small time steps to avoid instability by employing a very stable fully 2 d implicit explicit time integration numerical scheme the proposed scheme is more accurate and has improved rotational invariance properties compared with alternative split schemes particularly when big time steps are utilized we then apply properly designed multigrid methods to efficiently solve the occurring sparse linear system the combined algorithm allows for the rapid evolution of the contour and convergence to its final configuration after very few iterations image segmentation experiments demonstrate the efficiency and accuracy of the method

an immersed finite volume element method for 2d pdes with discontinuous coefficients and non homogeneous jump conditions an immersed finite volume element method is developed to solve 2d elliptic interface problems with a variable coefficient that has a finite jump across an interface the solution and the flux may also have a finite jump across the interface using the source removal technique an equivalent elliptic interface problem with homogeneous jump conditions is obtained the nodal basis functions are constructed to satisfy the homogeneous jump conditions near the interface and the usual finite element nodal basis functions are applied away from the interface the resulting linear problem is simple and easy to solve a proof of the error estimate in the energy norm is given numerical experiments demonstrate the convergence rates of the proposed method with the usual o h 2 in the l 2 the l norms and o h in the h 1 norm

on an adaptive multigrid solver for convection dominated problems the objective of this paper is to develop and describe an adaptive multigrid scheme for the numerical solution of convection diffusion reaction equations with dominating convection or reaction term specifically we propose a systematic construction of problem and scale dependent restriction prolongation and smoothing operators in combination with an adaptive mesh refinement strategy in particular adaptivity is not only to reduce computational complexity but also to stabilize standard galerkin discretizations so that adding artificial viscosity can be avoided the approach is suited for any choice of underlying multiresolution sequences in particular for standard finite element discretizations on nested but possibly nonuniform meshes moreover all concepts are in principle independent of the spatial dimension and avoid any special flow dependent enumeration of unknowns the performance of the scheme is illustrated by various examples in one and two spatial dimensions they are chosen so as to exhibit possibly many types of adverse or singular behavior in connection with constant and varying convection or the interplay between convective and reactive terms in the light of these tests those issues are identified that require further research

convergence analysis of a multigrid method for convection diffusion equations this paper is concerned with the convergence analysis of robust multigrid methods for convection diffusion problems we consider a finite difference discretization of a 2d model convection diffusion problem with constant coefficients and dirichlet boundary conditions for the approximate solution of this discrete problem a multigrid method based on semicoarsening matrix dependent prolongation and restriction and line smoothers is applied for a multigrid w cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion convection ratio for the contraction number of a multigrid v cycle a bound is proved which is uniform for a class of convection dominated problems the analysis is based on linear algebra arguments only

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

an optimal 25 point finite difference scheme for the helmholtz equation with pml in this paper we present an optimal 25 point finite difference scheme for solving the helmholtz equation with perfectly matched layer pml in two dimensional domain based on minimizing the numerical dispersion we propose the refined choice strategy for choosing optimal parameters of the 25 point finite difference scheme numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

conditional semicoarsening multigrid algorithm for the poisson equation on anisotropic grids a conditional semicoarsening multigrid algorithm is developed for the poisson equation on stretched grids the smoothing behaviour of the pointwise smoother as predicted by local fourier analysis is used to determine how to coarsen the grid this process ensures that the smoothed error can be adequately resolved on the coarse grid which results in an efficient multigrid algorithm that is simple to implement the use of a pointwise smoother increases the degree of parallelism of the algorithm the resulting algorithm is tested on several different grid types some with very high degrees of stretching the computational cost is found to scale almost linearly with the number of grid points and parallel efficiencies of over 90 are achieved even for very coarse grids

an algebraic two level preconditioner for asymmetric positive definite systems a two level linear algebraic solver for asymmetric positive definite systems is developed using matrices arising from stabilized finite element formulations to motivate the approach supported by an analysis of a representative smoother the parent space is divided into oscillatory and smooth subspaces according to the eigenvectors of the associated normal system using a mesh based aggregation technique which relies only on information contained in the matrix a restriction prolongation operator is constructed various numerical examples on both structured and unstructured meshes are performed using the two level cycle as the basis for a preconditioner results demonstrate the complementarity between the smoother and the coarse level correction as well as convergence rates that are nearly independent of the problem size copyright 2001 john wiley sons ltd

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

black box learning of multigrid parameters abstract this paper studies the optimality of the restriction and prolongation operators in the geometric multigrid method gmg gmg is used in solving discretized partial differential equation pde and it relies greatly on the restriction and prolongation operators many methods to find these operators were proposed but most of them have limited optimality proofs to study their optimality we introduce stochastic convergence functional which estimates the spectral radius of the iteration matrix for given gmg parameters we implement the gmg method in a modern machine learning framework that can automatically compute the gradients of the introduced convergence functional with respect to restriction and prolongation operators therefore we can minimize the proposed functional starting from some initial parameters and get better ones after some iterations of stochastic gradient descent to illustrate the performance of the proposed approach we carry out experiments on the discretized poisson equation helmholtz equation and singularly perturbed convection diffusion equation and demonstrate that proposed approach gives operators which lead to faster convergence

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions

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

on multiscale fe analyses of heterogeneous structures from homogenization to multigrid solvers heterogeneous structures like composites often need a fine scale resolution of micro effects which influence the macroscopic overall response this is of particular relevance in the fully non linear range of large strains and inelastic material response of the constituents suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales for big differences of micro and macro scales the argument of scale separation induces the application of homogenization methods such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine scale micro structure as well as the macroscopic boundary value problem in contrast small scale differences require full resolution of the heterogeneous structure effective solution methods for the resulting large scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques which may be considered as numerical multiscale approaches in both scenarios a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales the paper outlines new mesh bridging techniques in a deformation driven context for fully non linear response which exploit in a non trivial manner weak constraints on the average deformation in typical finite element patches the framework is based on an incremental variational structure of finite inelasticity the proposed new formulations provide variational based homogenization algorithms for physical multiscale scenarios and problem dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials copyright 2007 john wiley sons ltd

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

analysis and comparison of geometric and algebraic multigrid for convection diffusion equations the discrete convection diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes estimates of error reduction rates for both geometric multigrid gmg and algebraic multigrid amg are established on uniform rectangular meshes for a model problem our analysis shows that gmg with line gauss seidel smoothing and bilinear interpolation converges if h gg epsilon 2 3 and amg with the same smoother converges more rapidly than gmg if the interpolation constant beta in the approximation assumption of amg satisfies beta ll frac h sqrt epsilon alpha hs 1mm mbox where alpha big begin smallmatrix 1 h sqrt varepsilon 2 h ge sqrt varepsilon end smallmatrix on unstructured triangular meshes the performance of gmg and amg both as solvers and as preconditioners for gmres are evaluated numerical results show that gmres with amg preconditioning is a robust and reliable solver on both type of meshes

high order fem for multigrid homogenization the high order fem was applied to the multigrid based homogenization and proved to be an efficient method for upscaled solution approximation of problems with rapidly varying possibly noncontinuous coefficients as well as non coherent porous domains that may or may not depict a periodic micro structure and do not have to satisfy the separation condition our main contribution is an improved appropriate for bubble functions inter grid mapping consequently a fast exponential convergence of both displacements and stresses was observed in the conducted numerical experiments with respect to the number of degrees of freedom as well as the time of computation particularly the high accuracy of solution derivatives is an important advantage of this homogenization technique that is equivalent to the multiscale fem

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

the discontinuous petrov galerkin methodology for the mixed multiscale finite element method abstract we present the application of the discontinuous petrov galerkin dpg methodology for the mixed multiscale finite element method msfem the msfem upscaling technique relies on incorporating fine scale features through special in a sense optimized for approximability trial functions while the dpg methodology allows for the selection of the optimal test functions to provide stability of the fem approximation the special trial functions are computed online by the solution of local boundary value problems we improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton element interfaces only we have verified by numerical tests that the proposed improvement of msfem reduced both the approximation error and computational cost moreover it simplified the algorithm significantly the key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape the proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters using an arbitrary well posed functional setting with or without the dpg methodology

finite volume multiscale finite element method for solving the groundwater flow problems in heterogeneous porous media 1 in this paper we present a numerical method the finite volume multiscale finite element method fvmsfem for solving the groundwater flow problems in heterogeneous porous media spanning over many scales this method is based on an efficient coupling between the finite volume discretization and the multiscale finite element base functions it can efficiently capture the large scale structure of the solution on a coarse grid without resolving all the small scale features and is locally conservative the underlying idea is to estimate the macroscopic fluxes across the finite control volume interface segments which bring the local small scale information of the medium property to the large scales by employing the multiscale finite element base functions we describe the strategy for constructing such a method on the basis of the transient flow problems in porous media numerical experiments are carried out for groundwater flow in porous media with a random lognormal conductivity field to demonstrate the efficiency and accuracy of the developed method

an evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed problems in this paper we try to achieve h independent convergence with preconditioned gmres y saad and m h schultz siam j sci comput 7 1986 pp 856 869 and bicgstab h a van der vorst siam j sci comput 13 1992 pp 63 644 for two dimensional 2d singularly perturbed equations three recently developed multigrid methods are adopted as a preconditioner they are also used as solution methods in order to compare the performance of the methods as solvers and as preconditioners two of the multigrid methods differ only in the transfer operators one uses standard matrix dependent prolongation operators from j e dendy jr j comput phys 48 1982 pp 366 386 w hackbusch multi grid methods and applications springer berlin 1985 the second uses upwind prolongation operators developed in p m de zeeuw j comput appl math 33 1990 pp 1 27 both employ the galerkin coarse grid approximation and an alternating zebra line gauss seidel smoother the third method is based on the block lu decomposition of a matrix and on an approximate schur complement this multigrid variant is presented in a reusken a multigrid method based on incomplete gaussian elimination university of eindhoven the netherlands 1995 all three multigrid algorithms are algebraic methods the eigenvalue spectra of the three multigrid iteration matrices are analyzed for the equations solved in order to understand the convergence of the three algorithms furthermore the construction of the search directions for the multigrid preconditioned gmres solvers is investigated by the calculation and solution of the minimal residual polynomials for poisson and convection diffusion problems all solution methods are investigated and evaluated for finite volume discretizations on fine grids the methods have been parallelized with a grid partitioning technique and are compared on an mimd machine

multigrid for locally refined meshes a multilevel method for the solution of finite element schemes on locally refined meshes is introduced for isotropic diffusion problems the condition number of the two level method is bounded independently of the mesh size and the discontinuities in the diffusion coefficient the curves of discontinuity need not be aligned with the coarse mesh indeed numerical applications with 10 levels of local refinement yield a rapid convergence of the corresponding 10 level multigrid v cycle and other multigrid cycles which are more suitable for parallelism even when the discontinuities are invisible on most of the coarse meshes

convergence and performance of iterative methods for solving variable coefficient convection diffusion equation with a fourth order compact difference scheme abstract we conduct convergence analysis on some classical stationary iterative methods for solving the two dimensional variable coefficient convection diffusion equation discretized by a fourth order compact difference scheme several conditions are formulated under which the coefficient matrix is guaranteed to be an m matrix we further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods multigrid method and preconditioned gmres three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions the orderings of the grid points and the magnitude of the convection coefficients

gpu implementation of a helmholtz krylov solver preconditioned by a shifted laplace multigrid method a helmholtz equation in two dimensions discretized by a second order finite difference scheme is considered krylov methods such as bi cgstab and idr s have been chosen as solvers since the convergence of the krylov solvers deteriorates with increasing wave number a shifted laplace multigrid preconditioner is used to improve the convergence the implementation of the preconditioned solver on cpu central processing unit is compared to an implementation on gpu graphics processing units or graphics card using cuda compute unified device architecture the results show that preconditioned bi cgstab on gpu as well as preconditioned idr s on gpu is about 30 times faster than on cpu for the same stopping criterion

a grid based multilevel incomplete lu factorization preconditioning technique for general sparse matrices we design a grid based multilevel incomplete lu preconditioner gilum for solving general sparse matrices this preconditioner combines a high accuracy ilu factorization with an algebraic multilevel recursive reduction the gilum preconditioner is a compliment to the domain based multilevel block ilut preconditioner a major difference between these two preconditioners is the way that the coarse level nodes are chosen the approach of gilum is analogous to that of algebraic multigrid method the gilum approach avoids some controversial issues in algebraic multigrid method such as how to construct the interlevel transfer operators and how to compute the coarse level operator numerical experiments are conducted to compare gilum with other ilu preconditioners

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

a multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media 1 numerical solution of unsaturated water flow in heterogeneous porous media is difficult not only because of the spatial variability in the parameters used to characterize the relevant physical properties of the natural porous media but also because of the mathematical problems that arise in dealing with the nonlinearities in this study a multiscale finite element linearization scheme is presented for effectively simulating unsaturated flow in heterogeneous porous media spanning over many scales the central goal of this method is to obtain the large scale solution of richards equation with heterogeneous coefficients accurately and efficiently on a coarse grid without resolving all the small scale details the underlying idea is to employ the slodicka linear relaxation approximation scheme to address the nonlinear characteristics of the equation and to use the multiscale finite element base functions to account for the spatial variability in the equation coefficients we describe the principle for constructing such a method and give an algorithm for implementing it numerical experiments are carried out for the unsaturated flow equation with both periodic and randomly generated lognormal hydraulic conductivity to demonstrate the efficiency and accuracy of the proposed method

collocation coarse approximation in multigrid the two common approaches to defining coarse operators in multigrid numerical algorithms are discretization coarse approximation dca and petrov galerkin coarse approximation gca here a new approach called collocation coarse approximation cca is introduced which like gca is algebraically defined and able to cater to difficult features such as discontinuous coefficients but unlike gca allows explicit control over the coarse grid sparsity pattern stencil and therefore control over the computational complexity of the solver cca relies on certain basis functions for which the coarse approximation to the fine grid problem is exact numerical experiments for two dimensional diffusion problems including jumping coefficients demonstrate the potential of the resulting multigrid algorithms

schur complement multigrid a schur complement multigrid method for the solution of convection diffusion problems with strongly discontinuous coefficients is the focus of this paper this algorithm turns out to be robust and efficient for our test problems its convergence rate is in all cases superior to the standard multigrid method the schur complement multigrid method is applied to the problem of the decontamination of a heterogeneous porous medium by flushing we determine the typical decontamination time for several configurations and its functional dependence on the statistical properties of the hydraulic conductivity distribution of the medium

residual scaling techniques in multigrid ii practical applications this paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of a companion paper residual scaling techniques in multigrid i equivalence proof appl math comput 86 283 303 1997 we discuss the computational issues of some residual scaling techniques which have been proven mathematically equivalent a heuristic residual analysis technique based on the geometry of the grid points and the relaxation pattern is introduced to estimate the optimal residual scaling factor for a high order multigrid method we compare the performance of a typical pre optimization pre acceleration technique with a typical post optimization post acceleration technique and show that the pre optimization is preferable in both convergence and efficiency our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre optimization technique over the post optimization technique

parallel adaptive subspace correction schemes with applications to elasticity in this paper we give a survey on the three main aspects of the efficient treatment of pdes i e adaptive discretization multilevel solution and parallelization we emphasize the abstract approach of subspace correction schemes and summarize its convergence theory then we give the main features of each of the three distinct topics and treat the historical background and modern developments furthermore we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic pdes and especially of linear elasticity problems we report on numerical experiments for the adaptive parallel multilevel solution of some test problems namely the poisson equation and lame s equation here we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute which is achieved through hash storage techniques and space filling curves

multigrid based shifted laplacian preconditioning for the time harmonic elastic wave equation we investigate the numerical performance of an iterative solver for a frequency domain finite difference discretization of the isotropic elastic wave equation the solver is based on the shifted laplacian preconditioner originally designed for the acoustic wave equation this preconditioner represents a discretization of a heavily damped wave equation and can be efficiently inverted by a multigrid iteration however the application of multigrid to the elastic case is not straightforward because standard methods such as point jacobi fail to smooth the s wave wavenumber components of the error when high p to s velocity ratios are present we consider line smoothers as an alternative and apply local mode analysis to evaluate the performance of the various components of the multigrid preconditioner numerical examples in 2 d demonstrate the efficacy of our method

coarse grid correction for nonelliptic and singular perturbation problems the suitability of coarse grid correction operators for multigrid solution of discretized nonelliptic partial differential equations is studied it is found that the requirements for the orders of intergrid transfers and the order of approximation of the coarse grid operators differ in an important way from the well known rules pertaining to elliptic operators both galerkin coarsening and coarsening schemes based on the differential operator are examined and new coarse grid correction strategies are suggested in light of this analysis

resolving small scale structures in boussinesq convection by adaptive grid methods inviscid boussinesq convection is a challenging problem both analytically and numerically due to the complex dynamic development of small scales and the rapid loss of solution regularity the boussinesq convection pushes any numerical strategy to the limit in e and shu phys fluids 6 1994 49 a detailed numerical study of the boussinesq convection in the absence of viscous effects is carried out using filtered pseudospectral method and a high order accurate eno schemes in their computations very fine grids have to be used in order to resolve the small structures of the boussinesq fluid in this work we will develop an efficient adaptive grid method for solving the inviscid incompressible flows which can be useful in resolving extremely small structures with reasonably small number of grid points to demonstrate the effectiveness of the proposed method the boussinesq convection problem will be computed using the adaptive grid method

domain decomposition for multiscale pdes we consider additive schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic pdes with highly variable coefficients in contrast to standard analyses we do not assume that the coefficients can be resolved by a coarse mesh this situation arises often in practice for example in the computation of flows in heterogeneous porous media in both the deterministic and monte carlo simulated stochastic cases we consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid we perform a new analysis of the preconditioned matrix which shows rather explicitly how its condition number depends on the variable coefficient in the pde as well as on the coarse mesh and overlap parameters the classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements by contrast our new results show that with a good choice of subdomains and coarse space basis functions the preconditioner can still be robust even for large coefficient variation inside domains when the classical method fails to be robust in particular our estimates prove very precisely the previously made empirical observation that the use of low energy coarse spaces can lead to robust preconditioners we go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary i e two scale media model problems moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case we show also how the analysis extends in a straightforward way to multiplicative versions of the schwarz method

a simple cell centered multigrid method for 3d interface problems abstract a multigrid algorithm is presented for cell centered discretizations of interface problems instead of constructing the coarse grid operators by means of the galerkin approximation the coarse grid operators are obtained by discretization on the coarse grids the advantage of this approach is that we obtain m matrices on all grids and that the sparsity pattern of the fine grid matrix is retained on all grids moreover the coarse grid operators are very easy to construct numerical results of several test problems are presented

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

soft linear logic and polynomial time we present a subsystem of second order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms and vice versa