wavelet solvers for hp fem discretizations in 3d using hexahedral elements in this paper we investigate the discretization of an elliptic boundary value problem in 3d by means of the hp version of the finite element method using a mesh of hexahedrons the corresponding linear system is solved by a preconditioned conjugate gradient method the construction of the preconditioner is based on an inexact additive overlapping schwarz method the remaining subproblems are treated by a tensor product based preconditioner this preconditioner uses a basis transformation into a basis which is stable in l 2 and h 1 several numerical examples show the efficiency of the proposed method

extension operators on tensor product structures in two and three dimensions in this paper a uniformly elliptic second order boundary value problem in two dimensions is discretized by the p version of the finite element method an inexact dirichlet dirichlet domain decomposition preconditioner for the system of linear algebraic equations is investigated the ingredients of such a preconditioner are a preconditioner for the schur complement a preconditioner for the subdomains and an extension operator operating from the edges of the elements into their interior using methods of multiresolution analysis we propose a new method to compute the extension efficiently we prove that this type of extension is optimal i e the h 1 omega norm of the extended function is bounded by the h 0 5 partial omega norm of the given function numerical experiments show the optimal performance of the described extension

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

inexact solvers on element interfaces for the p and h p finite element method parallel and iterative solvers for large scale linear system arising from finite element discretization based on domain decomposition decompose the approximation spaces into a number of subspaces the corresponding subproblems are solved in these subspaces and the solutions are corrected in an iterative process the subproblems are much smaller in size and much easier to be solved one of these subspaces is spanned by high order element side modes in two dimensions and high order element face modes in three dimensions solving the subproblems on the element interfaces is a major cost in each step of iteration to reduce computational cost various inexact solvers on element interfaces are proposed in this paper with theoretical analysis and numerical illustrations

schwarz methods with local refinement for the p version finite element method in some applications the accuracy of the numerical solution of an elliptic problem needs to be increased only in certain parts of the domain in this paper local refinement is introduced for an overlapping additive schwarz algorithm for the p version finite element method both uniform and variable degree refinements are considered the resulting algorithm is highly parallel and scalable

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

nonhomogeneous boundary value problems the diffusion equation is solved under stochastic nonhomogeneity using eigen function expansion and the georges method the statistical moments of the solution process are computed through the two previously mentioned techniques and proved to be the same a general solution is obtained under general initial and boundary conditions a random source composed of deterministic and stochastic parts is taken into consideration the stochastic part is then restricted to a generalized gaussian field mainly modulated white noise a special case is considered under constant noise level and constant average noise a numerical case study concerning pollution in a stream is solved and a parametric study is achieved through various figures

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

a parallel implementation of an iterative substructuring algorithm for problems in three dimensions numerical results from a parallel implementation of a class of iterative substructuring algorithms are reported the algorithms are for solving self adjoint elliptic partial differential equations in three dimensions results are given for several variants of the algorithm in the first variant exact interior solvers are used in the second one multigrid v cycle is used to solve the interior problems approximately the results are compared with theoretical behavior of the algorithm reported in previous work a numerical experiment involving the equations of linear elasticity is also included

preconditioning on element interfaces for the p version finite element method and spectral element method in the preconditioned iterative method based on nonoverlapping domain decompositions the preconditioner is usually of block diagonal type to calculate its action on a residual vector it is required to solve a linear system of algebraic equations with the coefficient matrix being the schur complement block corresponding to the interface to reduce the computational cost the schur complement blocks in the preconditioners can be replaced with other simpler matrices which are called the interface preconditioners in this paper we discuss various interface preconditioners for the p version finite element method and spectral element method in r2 and r3 emphases are placed on 1 the lower and upper bounds of the condition numbers for the preconditioned schur complement blocks using the interface preconditioners which characterize the enlargement of the overall condition numbers and 2 the algorithms to efficiently implement the interface preconditioners numerical results are presented for the condition numbers using various interface preconditioners

an iterative and parallel solver based on domain decomposition for the h p version of the finite element method in this paper we propose a new interative and parallel solver based on domain decomposition for the h p version of the finite element method in two dimensions it improves our previous work in two aspects 1 a subdomain may contain several super elements of the coarse mesh thus can be of arbitrary shape and size this makes the solver more efficient and more flexible in computational practice 2 the p version components i e the high order side and internal modes in every element are treated separately which results in better parallelism

preconditioned spectral multi domain discretization of the incompressible navier stokes equations in this paper we present a preconditioned multi domain algorithm applied to the elliptic kernels arising from the spectral collocation of the incompressible navier stokes equations in three space dimensions with one homogeneous direction the technique based on the iterative solution of the schur complement matrix allows for efficient numerical solution of the operators in complex geometries consisting of a collection of non overlapping rectangular subdomains the method is shown to be nearly optimal in terms of condition number behavior in a double path of refinement strategy i e whenever both the number of legendre modes and the number of subdomains are significantly increased it is thus well suited for engineering applications in the fields of direct numerical simulation and large eddy simulation of turbulence

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

beta expansions natural extensions and multiple tilings associated with pisot units from the works of rauzy and thurston we know how to construct multiple tilings of some euclidean space using the conjugates of a pisot unit beta and the greedy beta transformation in this paper we consider different transformations generating expansions in base beta including cases where the associated subshift is not sofic under certain mild conditions we show that they give multiple tilings we also give a necessary and sufficient condition for the tiling property generalizing the weak finiteness property w for greedy beta expansions remarkably the symmetric beta transformation does not satisfy this condition when beta is the smallest pisot number or the tribonacci number this means that the pisot conjecture on tilings cannot be extended to the symmetric beta transformation closely related to these multiple tilings are natural extensions of the transformations which have many nice properties they are invariant under the lebesgue measure under certain conditions they provide markov partitions of the torus they characterize the numbers with purely periodic expansion and they allow determining any digit in an expansion without knowing the other digits

domain decomposition method for the h p version finite element method domain decomposition method for the h p version of the finite element method in two and three dimensions are discussed using the framework of additive schwarz method various iterative methods are described with their condition numbers estimated further to reduce the cost for solving the sub problems on element interfaces different inexact interface solvers are proposed the effects on the overall condition number as well as their efficient implementation are analyzed

the schwarz algorithm for spectral methods recently the schwarz alternating method has been successfully coupled to spatial discretizations of spectral type in order to solve boundary value problems in complex geometries with infinite order accuracy in this paper a simple version of the method is considered a proof of its convergence is given in the energy norm exploiting the properties of discrete harmonic polynomials and a discrete maximum principle for spectral methods more general situations can be handled theoretically in one space dimension

the problem of selecting the shape functions for a p type finite element abstract the paper addresses the question of the optimal selection of the shape functions for p type finite elements and discusses the effectivity of the conjugate gradient and multilevel iteration method for solving the corresponding linear system the selection of the shape functions is of major importance for the performance of the solver based on iterative methods neither the theory nor practice of the optimal selection of the shape functions is available yet we have seen that the condensation approach which has obvious advantages from the point of parallel computations is a very effective tool for keeping the condition number under the control and is especially advantageous for the conjugate gradient method

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

a domain decomposition algorithm for elliptic problems in three dimensions most domain decomposition algorithms have been developed for problems in two dimensions one reason for this is the difficulty in devising a satisfactory easy to implement robust method of providing global communication of information for problems in three dimensions several methods that work well in two dimension do not perform satisfactorily in three dimensions a new iterative substructuring algorithm for three dimensions is proposed it is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain the condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains the new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions

a unified approach to boundary perception edges textures and illusory contours a model consisting of a multistage system which extracts and groups salient features in the image at different spatial scales or frequencies is used in the first stage a gabor wavelet decomposition provides a representation of the image which is orientation selective and has optimal localization properties in space and frequency this decomposition is useful in detecting significant features such as step and line edges at different scales and orientations in the image following the wavelet transformation local competitive interactions are introduced to reduce the effects of noise and changes in illumination interscale interactions help in localizing the line ends and corners and play a crucial role in boundary perception the final stage groups similar features aiding in boundary completion the different stages can be identified with processing by simple complex and hypercomplex cells in the visual cortex of mammals experimental results demonstrate the performance of this model in detecting boundaries both real and illusory in real and synthetic images

an iterative substructuring method for the p version of the boundary element method for hypersingular integral operators in three dimensions we study preconditioners for the p version of the boundary element method for hypersingular integral equations in three dimensions the preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions we consider a so called wire basket preconditioner and a non overlapping additive schwarz method based on the complete natural splitting i e with respect to the nodal edge and interior functions as well as an almost diagonal preconditioner in any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size for all these methods we prove that the resulting condition numbers are bounded by c 1 log p here p is the polynomial degree of the ansatz functions and c is a constant which is independent of p and the mesh size of the underlying boundary element mesh numerical experiments supporting these results are reported

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

spectral element methods for large scale parallel navier stokes calculations abstract we analyze the computational complexity of a recently developed two level iteration scheme for spectral element solution of the time dependent incompressible navier stokes equations in complex domains we present several algorithmic advances which significantly enhance the scalability of this approach including implementation of an advanced combine operation for degrees of freedom on subdomain edges parallel solution of the fine grained coarse grid problem and implementation of local low order finite element preconditioners for the find grid problem timings on the 512 node intel delta machine show that the combined improvements lead to a fourfold reduction in navier stokes solution time for the particular case of a three dimensional boundary layer calculation consisting of one million degrees of freedom

the lifting of polynomial traces revisited we construct a lifting operator of polynomial traces on an interval that is stable in appropriate sobolev norms next we extend this result to the case of traces vanishing at the endpoints of the interval this has two applications the interpolation of polynomial spaces and the evaluation by discrete formulas of fractional order sobolev norms on polynomials

iterative substructuring methods for spectral elements problems in three dimensions based on numerical quadrature abstract iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems two preconditioners for p version finite element methods based on continuous piecewise qp functions are considered for second order elliptic problems in three dimensions these special methods can also be viewed as spectral element methods the first iterative method is designed for the galerkin formulation of the problem the second applies to linear systems for a discrete model derived by using gauss lobatto legendre quadrature for both methods it is established that the condition number of the relevant operator grows only in proportion to 1 log p 2 these bounds are independent of the number of elements into which the given region has been divided their diameters as well as the jumps in the coefficients of the elliptic equation between elements results of numerical computations are also given which provide upper bounds on the condition numbers as functions of p and which confirms the correctness of our theory