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

numerical grid generation in arbitrary surfaces through a second order differential geometric model abstract in this paper a set of second order partial differential equations for the generation of coordinates in a given surface has been developed and then solved numerically to demonstrate its potential for surface coordinate generation the proposed equations are not some arbitrarily chosen equations but are a consequence of the formulae of gauss and thus carry with them an explicit dependence on the geometric properties of the given surface furthermore these equations are easy to solve and require only the specification of the bounding curves to provide the dirichlet boundary conditions for their solution results of coordinate generation both in simply and doubly connected regions on some known surfaces with the option of coordinate redistribution have been presented extension of this technique to arbitrary surfaces seems to be straightforward

displacement rank and quasitriangular decomposition for r toeplitz matrices abstract an r toeplitz r t matrix can be regarded as a block toeplitz matrix of order mr from which an l shaped border of k r rows and columns is removed the concept of displacement rank is extended to r t matrices and it is shown that both an r t matrix and its inverse can be expressed as the sum of at most two products of r t matrices this result reduces to a standard lu decomposition in the case when an r t matrix becomes an arbitrary matrix an explicit expression for the inverse is given which contains three triangular matrices and one quasitriangular matrix furthermore the inverse of an r t matrix is completely determined by the elements in its first and last r rows and columns the results presented generalize known properties of toeplitz and block toeplitz matrices

toeplitz matrices with banded inverses abstract given a toeplitz matrix t with banded inverse i e t 1 ij 0 for j i p we show that the elements of t can be expressed in terms of the roots of a polynomial then using properties we have previously established we generalize this result appropriately to allow singular t and show that the converse also holds finally we give a sufficient condition for the decay of the elements of t as one moves away from the diagonal

hp interpolation of nonsmooth functions and an application to hp a posteriori error estimation the quasi interpolation operators of clement and scott zhang type are generalized to the setting of the hp fem finite element method new polynomial lifting and inverse estimates are presented the classical residual based a posteriori error estimator is generalized to the hp fem

inversion of certain extensions of toeplitz matrices it is shown how an algorithm for inverting toeplitz matrices using o n2 operations can be modified to deal with a certain type of extension named conjugate toeplitz matrices the block partitioned case is also included

eigenvalues of a certain class of tridiagonal matrices we make a short comment on a paper involving the minimal eigenvalues of a specialized version of 2 toeplitz tridiagonal matrices

polynomial liftings on a tetrahedron and applications to the h p version of the finite element method in three dimensions we construct a stable polynomial lifting on a tetrahedron given polynomial traces defined on its faces satisfying continuity conditions along the edges we apply this result to establish the optimal convergence rate of the h p version of the finite element method in dimension three assuming the solution to be regular

polynomial extension operators part ii consider the tangential trace of a vector polynomial on the surface of a tetrahedron we construct an extension operator that extends such a trace function into a polynomial on the tetrahedron this operator can be continuously extended to the trace space of h mathbf curl furthermore it satisfies a commutativity property with an extension operator we constructed in part i of this series such extensions are a fundamental ingredient of high order finite element analysis

additive schwarz method for the p version of the boundary element method for the single layer potential operator on a plane screen we analyze an additive schwarz preconditioner for the p version of the boundary element method for the single layer potential operator on a plane screen in the three dimensional euclidean space we decompose the ansatz space which consists of piecewise polynomials of degree p on a mesh of size h by introducing a coarse mesh of size h ge h after subtraction of the coarse subspace of piecewise constant functions on the coarse mesh this results in local subspaces of piecewise polynomials living only on elements of size h this decomposition yields a preconditioner which bounds the spectral condition number of the stiffness matrix by o log frac hhp 2 numerical results supporting the theory are presented

coupling spectral and finite elements for second order elliptic three dimensional equations this paper deals with the approximation of elliptic second order problems with a nonconforming domain decomposition method that allows for coupling spectral and finite element discretizations in three dimensions we present two kinds of matching conditions at the interfaces between the subdomains the detailed numerical analysis of the error committed upon the exact solution turns out to be optimal we describe also the implementation and give some numerical results confirming the theoretical predictions

generating polynomials for matrices with toeplitz or conjugate toeplitz inverses abstract a complex matrix a a ij has been called conjugate toeplitz if a ij c i 1 a i j where c denotes conjugation a necessary and sufficient condition is derived for a matrix h to have a conjugate toeplitz inverse the elements of h are generated from the coefficients of certain polynomials the result is simplified either when h is to have a toeplitz inverse or when h is banded and is to have a conjugate toeplitz inverse each of these consequences of the main theorem is a different generalization of a previous result on banded matrices having a toeplitz inverse

elliptic grid generation conclusions some modi cations for the restriction and prolongation process of multigrid methods have been presented that are necessary for the simulation of complex supersonic ow elds damping of the restricted residual error and blending between a second order central and a rst order upwind prolongation enable convergence in supersonic ows and resulted in strong reductions in cpu time the advantageof this kind of blending is the reduction to rst order prolongationat shock waves only

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

local high order regularization and applications to hp methods we develop a regularization operator based on smoothing on a spatially varying length scale this operator is defined for functions u l 1 and has approximation properties that are given by the local sobolev regularity of u and the local smoothing length scale additionally the regularized function satisfies inverse estimates commensurate with the approximation orders by combining this operator with a classical h p interpolation operator we obtain an h p clement type quasi interpolation operator i e an operator that requires minimal smoothness of the function to be approximated but has the expected approximation properties in terms of the local mesh size and polynomial degree as a second application we consider residual error estimates in h p boundary element methods that are explicit in the local mesh size and the local approximation order

the determination of companion matrices characterizing toeplitz and r toeplitz matrices abstract it is shown that the unknown elements in the companion matrix used to characterize a nonsingular toeplitz matrix are easily found from the elements of the inverse of the toeplitz matrix similar formulae are obtained for the two companion like matrices characterizing an r toeplitz matrix this case includes the result for a block toeplitz matrix

fast domain decomposition algorithm discretizations of 3 d elliptic equations by spectral elements the main obstacle for obtaining fast domain decomposition solvers for the spectral element discretizations of the 2 nd order elliptic equations was the lack of fast solvers for local internal problems on subdomains of decomposition and their faces as was recently shown by korneev rytov such solvers can be derived on the basis of the specific interrelation between the stiness matrices of the spectral and hierarchical p reference elements the coordinate polynomials of the latter are tensor products of the integrated legendre s polynomials this interrelation allows to apply to the spectral element discretizations fast solvers which in some basic features are quite similar to those developed for the discretizations by the hierarchical elements using these facts and the preceding results on the wire basket preconditioners we present an almost optimal in total arithmetical cost domain decomposition preconditioner solver for the spectral element discretizations of the 2 nd order elliptic equations in 3 d domains 2000 mathematics subject classification 65n22 65m30 65n55

a note on conjugate toeplitz matrices abstract two properties of conjugate toeplitz matrices are given 1 an expression for the elements inside the inverse matrix of an r toeplitz and of a conjugate toeplitz matrix is obtained in a direct way from elementary results about arbitrary partitioned matrices 2 the necessary conditions to apply the generalized trench algorithm for conjugate toeplitz matrices are discussed it is shown that there exist strongly invertible conjugate toeplitz matrices to which the algorithm may not be applied

the eigenproblem of a tridiagonal 2 toeplitz matrix abstract the characteristic polynomial of a tridiagonal 2 toeplitz matrix is shown to be closely connected to polynomials which satisfy the three point chebyshev recurrence relationship this is an extension of the well known result for a tridiagonal toeplitz matrix when the order of the matrix is odd the eigenvalues are found explicitly in terms of the chebyshev zeros the eigenvectors are found in terms of the polynomials satisfying the three point recurrence relationship

polynomial extensions of compatible polynomial traces in three dimensions abstract we describe the construction of polynomial extension operators of polynomial traces over the reference cube satisfying optimal stability conditions with respect to sobolev norms combined with hilbertian interpolation argument these results show the equivalence of sobolev norms and the interpolation norms on spaces of polynomials defined over a square

characterisation and properties of r toeplitz matrices abstract an r toeplitz rt matrix can be regarded as a block toeplitz matrix with order some multiple of r from which a bottom border and a right hand border have been removed it is shown that some known properties of toeplitz matrices extend to this new class in particular a necessary and sufficient condition is given in terms of a linear matrix equation for a nonsingular matrix to have rt form from which it follows that the elements of the inverse are completely determined by those in its first and last r rows and columns in addition some results are presented for cases when the inverse of an rt matrix has certain banded forms

cyclic displacements and decompositions of inverse matrices for cupl toeplitz matrices abstract in this paper we deal mainly with a class of cupl toeplitz matrices 32 without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor row first plus last right circulants furthermore the inverses of the cupl hankel matrices can be also obtained which benefits from the relation between the cupl toeplitz matrices and the cupl hankel matrices

elliptic grid generation based on laplace equations and algebraic transformations abstract an elliptic grid generation method is presented to generate boundary conforming grids in domains in 2d and 3d physical space and on minimal surfaces and parametrized surfaces in 3d physical space the elliptic grid generation method is based on the use of a composite mapping this composite mapping consists of a nonlinear transfinite algebraic transformation and an elliptic transformation the elliptic transformation is based on the laplace equations for domains or on the laplace beltrami equations for surfaces the algebraic transformation maps the computational space one to one onto a parameter space the elliptic transformation maps the parameter space one to one onto the domains or surfaces the composition of these two mappings is a differentiable one to one mapping from computational space onto the domains or surfaces and has a nonvanishing jacobian this composite mapping defines the grid point distribution in the interior of the domains or surfaces for domains and minimal surfaces the composite mapping obeys a nonlinear elliptic poisson system with control functions completely defined by the algebraic transformation the solution of the poisson systems is obtained by picard iteration and black box multigrid solvers for parametrized curved surfaces it is not necessary to define and solve a nonlinear elliptic poisson system instead a linear elliptic system and an inversion problem is solved to generate the grid in the interior of the surface

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