on the construction of deflation based preconditioners in this article we introduce new bounds on the effective condition number of deflated and preconditioned deflated symmetric positive definite linear systems for the case of a subdomain deflation such as that of nicolaides siam j numer anal 24 1987 pp 355 365 these theorems can provide direction in choosing a proper decomposition into subdomains if grid refinement is performed keeping the subdomain grid resolution fixed the condition number is insensitive to the grid size subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication numerical experiments for a steady state convection diffusion problem are included

first order system least squares for the helmholtz equation this paper develops a multilevel least squares approach for the numerical solution of the complex scalar exterior helmholtz equation this second order equation is first recast into an equivalent first order system by introducing several field variables a combination of scaled l2 and h 1 norms is then applied to the residual of this system to create a least squares functional it is shown that in an appropriate hilbert space the homogeneous part of this functional is equivalent to a squared graph norm that is a product norm on the space of individual variables this equivalence to a norm that decouples the variables means that standard finite element discretization techniques and standard multigrid solvers can be applied to obtain optimal performance however this equivalence is not uniform in the wavenumber k which can signal degrading performance of the numerical solution process as k increases to counter this difficulty we obtain a result that characterizes the error components causing performance degradation we do this by defining a finite dimensional subspace of these components on whose orthogonal complement k uniform equivalence is proved for this functional and an analogous functional that is based only on l2 norms this subspace equivalence motivates a nonstandard multigrid method that attempts to achieve optimal convergence uniformly in k we report on numerical experiments that empirically confirm k uniform optimal performance of this multigrid solver we also report on tests of the error in our discretization that seem to confirm optimal accuracy that is free of the so called pollution effect

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

deflation of conjugate gradients with applications to boundary value problems a method for improving the convergence of the standard conjugate gradient method is given this method involves the use of auxiliary subspaces it is shown how such subspaces may be constructed for boundary value problems and an analysis of convergence for second order problems is presented

multilevel projection based nested krylov iteration for boundary value problems we propose a multilevel projection based method for acceleration of krylov subspace methods the projection is constructed in a similar way as in deflation but shifts small eigenvalues to the largest one instead of to zero in contrast with deflation however the convergence rate of a krylov method combined with this new projection method is insensitive to the inaccurate solve of the galerkin matrix which with some particular choice of deflation subspaces is closely related to the coarse grid solve in multigrid or domain decomposition methods such an insensitivity allows the use of inner iterations to solve the galerkin problem an application of a krylov subspace method to the associated galerkin system with the projection preconditioner leads to a multilevel nested krylov iteration in this multilevel projection krylov subspace method information about small eigenvalues to be projected is contained implicitly in the galerkin system associated with the matrix of the linear system to be solved these small eigenvalues from a krylov method point of view are responsible for slow convergence in terms of projection methods this is conceptually similar to multigrid but different in the sense that in multigrid the projection is done by the smoother furthermore with the only condition being that the deflation matrices are full rank we have in principle more freedom in choosing the deflation subspace intergrid transfer operators used in multigrid are some of the possible candidates we present numerical results from solving the poisson equation and the convection diffusion equation both in two dimensions the latter represents the case where the related matrix of coefficients is nonsymmetric by using a simple piecewise constant interpolation as the basis for constructing the deflation subspace we obtain the following results i h independent convergence for the poisson equation and ii almost independent of h and the peclet number for the convection diffusion equation

preconditioning harmonic unsteady potential flow calculations this paper considers finite element discretizations of the helmholtz equation and its generalization arising from harmonic acoustic perturbations to a nonuniform steady potential flow a novel elliptic positive definite preconditioner with a multigrid implementation is used to accelerate the iterative convergence of krylov subspace solvers both theory and numerical results show that for a model 1 d helmholtz test problem the preconditioner clusters the discrete system s eigenvalues and lowers its condition number to a level independent of grid resolution for the 2 d helmholtz equation grid independent convergence is achieved using a quasi minimal residual krylov solver significantly outperforming the popular symmetric successive over relaxation preconditioner impressive results are also presented on more complex domains including an axisymmetric aircraft engine inlet with nonstagnant mean flow and modal boundary conditions

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

a deflated version of the conjugate gradient algorithm we present a deflated version of the conjugate gradient algorithm for solving linear systems the new algorithm can be useful in cases when a small number of eigenvalues of the iteration matrix are very close to the origin it can also be useful when solving linear systems with multiple right hand sides since the eigenvalue information gathered from solving one linear system can be recycled for solving the next systems and then updated

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

deflated preconditioned conjugate gradient solvers for the pressure poisson equation extensions and improvements extensions and improvements to a deflated preconditioned conjugate gradient technique for the solution of the pressure poisson equation within an incompressible flow solver are described in particular the use of the technique for embedded grids for cases where volume of fluid or level set schemes are required and its implementation on parallel machines are considered several examples are included that demonstrate a considerable reduction in the number of iterations and a remarkable insensitivity to the number of groups regions chosen and or to the way the groups are formed copyright 2010 john wiley sons ltd

stabilised vs stable mixed methods for incompressible flow abstract the accuracy of low order mixed finite element methods for the incompressible navier stokes equations is investigated in this work some numerical experiments suggest that the lowest order stabilised p 1 p 0 method linear velocity constant pressure is more efficient than the alternative a priori stable non conforming crouzeix raviart p 1 p 0 approach the relative accuracy of stabilised p 1 p 0 and p 1 p 1 linear velocity continuous linear pressure is assessed herein

quadrature formulae and functions of exponential type we obtain certain generalizations of the trapezoidal rule and the euler maclaurin formula that involve derivatives in the case of quadrature of functions of exponential type over infinite intervals we find conditions under which existence of the improper integral and convergence of the approximating series become equivalent

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

a residual based finite element method for the helmholtz equation a new residual based finite element method for the scalar helmholtz equation is developed this method is obtained from the galerkin approximation by appending terms that are proportional to residuals on element interiors and inter element boundaries the inclusion of residuals on inter element boundaries distinguishes this method from the well known galerkin least squares method and is crucial to the resulting accuracy of this method in two dimensions and for regular bilinear quadrilateral finite elements it is shown via a dispersion analysis that this method has minimal phase error numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods it is found that even for unstructured meshes this method retains a high level of accuracy copyright 2000 john wiley sons ltd

multigrid simulation for high frequency solutions of the helmholtz problem in heterogeneous media the helmholtz problem is hard to solve in heterogeneous media in particular when the wave number is real and large the problem is neither coercive nor hermitian symmetric this article concerns the v cycle multigrid mg method for high frequency solutions of the helmholtz problem since we need to choose at least 10 12 grid points per wavelength for stability the coarse grid problem is still large to solve the coarse grid problem efficiently a nonoverlapping domain decomposition method is adopted without introducing another coarser subspace correction various numerical experiments have shown that the convergence rate of the resulting mg method is independent on the grid size and the wave number provided that the coarse grid problem is fine enough for the solution to capture characteristics of the physical problem

residual free bubbles for the helmholtz equation the galerkin method enriched with residual free bubbles is considered for approximating the solution of the helmholtz equation two dimensional tests demonstrate the improvement over the standard galerkin method and the galerkin least squares method using piecewise bilinear interpolations

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

an algebraic multigrid based shifted laplacian preconditioner for the helmholtz equation a preconditioner defined by an algebraic multigrid cycle for a damped helmholtz operator is proposed for the helmholtz equation this approach is well suited for acoustic scattering problems in complicated computational domains and with varying material properties the spectral properties of the preconditioned systems and the convergence of the gmres method are studied with linear quadratic and cubic finite element discretizations numerical experiments are performed with two dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder the proposed preconditioner is particularly effective for low frequency and mid frequency problems

accuracy properties of the wave ray multigrid algorithm for helmholtz equations helmholtz equations with their highly oscillatory solutions play an important role in physics and engineering these equations present the main computational difficulties typical to acoustic electromagnetic and other wave problems they are often accompanied by radiation boundary conditions and are considered on infinite domains solving them numerically using standard procedures including multigrid is too expensive the wave ray multigrid algorithm efficiently solves the helmholtz equations and naturally incorporates the radiation boundary conditions important accuracy properties of the wave ray solver are discussed in this paper using various mode analyses we show that with the right choice of parameters this algorithm can obtain an approximation to the differential solution with accuracy that equals the accuracy of the target grid discretization moreover the boundary conditions can be introduced with any desired accuracy our theoretical conclusions are confirmed by numerical experiments

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

an algebraic variational multiscale multigrid method based on plain aggregation for convection diffusion problems a new computational framework referred to as algebraic variational multiscale multigrid method is proposed representing the initial step of our work on merging the variational multiscale method with algebraic multigrid projection methods this new approach allows for a separation of resolved scales into various scale groups in a purely algebraic way that is with no need to utilize further discretizations beyond the basic one by this means it conveniently facilitates the application of modeling terms either to all scale groups or only to selected scale groups for instance only to the fine scales of the problem variational multigrid techniques form the basis of our new approach after presenting the cornerstones of the framework for an abstract problem formulation the close relationship of the variational multiscale method to projection methods in general and to variational multigrid techniques are discussed the algebraic variational multiscale multigrid method is then particularly analyzed for convection diffusion problems the present initial study focuses on exploiting the methodical aspects of the new framework by developing a fine scale discontinuity capturing approach to diminish oscillations at sharp layers our technique including a fine scale discontinuity capturing term is applied to numerical examples of strongly convection dominated convection diffusion problems the results demonstrate that it enables the diminuation of local oscillations at parabolic boundary and interior layers by about 60 80 percent without any notable smearing effect

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

analysis of acceleration strategies for restarted minimal residual methods we provide an overview of existing strategies which compensate for the deterioration of convergence of minimum residual mr krylov subspace methods due to restarting we evaluate the popular practice of using nearly invariant subspaces to either augment krylov subspaces or to construct preconditioners which invert on these subspaces in the case where these spaces are exactly invariant the augmentation approach is shown to be superior we further show how a strategy recently introduced by de sturler for truncating the approximation space of an mr method can be interpreted as a controlled loosening of the condition for global mr approximation based on the canonical angles between subspaces for the special case of krylov subspace methods we give a concise derivation of the role of ritz and harmonic ritz values and vectors in the polynomial description of krylov spaces as well as of the use of the implicitly updated arnoldi method for manipulating krylov spaces

a quadrature formula for entire functions of exponential type a quadrature formula over a semi infinite interval for entire functions of exponential type is established an alternative approach to a known expansion formula and an extension of it are also presented

iterative solution applied to the helmholtz equation complex deflation on unstructured grids abstract extensions of deflation techniques developed for the poisson and navier equations aubry et al 2008 mut et al 2010 lohner et al 2011 aubry et al 2011 1 2 3 4 are presented for the helmholtz equation numerous difficulties arise compared to the previous case after discretization the matrix is now indefinite without sommerfeld boundary conditions or complex with them it is generally symmetric complex but not hermitian discarding optimal short recurrences from an iterative solver viewpoint saad 2003 5 furthermore the kernel of the operator in an infinite space typically does not belong to the discrete space the choice of the deflation space is discussed as well as the relationship between dispersion error and solver convergence similarly to the symmetric definite positive spd case subdomain deflation accelerates convergence if the low frequency eigenmodes are well described however the analytic eigenvectors are well represented only if the dispersion error is low cpu savings are therefore restricted to a low to mid frequency regime compared to the mesh size which could be still relevant from an application viewpoint given the ease of implementation

deflated preconditioned conjugate gradient solvers for the pressure poisson equation a deflated preconditioned conjugate gradient technique has been developed for the solution of the pressure poisson equation within an incompressible flow solver the deflation is done using a region based decomposition of the unknowns making it extremely simple to implement the procedure has shown a considerable reduction in the number of iterations for grids with large graph depth the savings exceed an order of magnitude furthermore the technique has shown a remarkable insensitivity to the number of groups regions chosen and to the way the groups are formed

on testing monotone tendencies abstract in certain problems it may be expected that a regression function has a substantial overall tendency to be monotone and yet we may not be certain that all of the restrictions imposed by a simple order are satisfied distribution theory for liklihood ratio tests of homogeneity of a collection of normal means when the collection is decreasing on the average and for testing decreasing on the average as a null hypothesis is presented the restriction decreasing on the average is less restrictive than the usual monotone restriction and allows the data to give rise to reversals over short ranges of values of the parameter set it is closely related to the starshaped ordering restriction