finite element error estimates for 3d exterior incompressible flow with nonzero velocity at infinity we consider a stationary incompressible navier stokes flow in a 3d exterior domain with nonzero velocity at infinity in order to approximate this flow we use the stabilized p1 p1 finite element method proposed by rebollo numer math 79 283 319 1998 following an approach by guirguis and gunzburger model math anal numer 21 445 464 1987 we apply this method to the navier stokes system with oseen term in a truncated exterior domain under a pointwise boundary condition on the artificial boundary this leads to a discrete problem whose solution approximates the exterior flow as is shown by error estimates

on the approximation solvability of a class of strongly nonlinear elliptic problems on unbounded domains a class of strongly nonlinear boundary value problems posed on unbounded regions is considered a nonlocal coupling of the linearized far field exterior to an auxiliary boundary allows for approximations to be defined on domains of finite extent constructive existence results for bounded domains are then extended by employing an exhausting sequence of approximating domains in particular well posedness is seen to be equivalent to unique approximation solvability with the rate of convergence dependent upon the radius of the auxiliary boundary application to a model of proteins immersed in an electrolyte solution is made

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

the finite element method with nonuniform mesh sizes for unbounded domains the finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains consider the following model problem du f in w sup c u g on partialw partialu partialr 1 r u o 1 r as r vertical barxvertical bar infinity where w sup c is the complement in r sup 3 three dimensional eucidean space of a bounded set w with smooth boundary partialw f and g are smooth functions and f has bounded support this problem is approximately solved by introducing an artifical bound g sub r near infinity e g a sphere of sufficiently large radius r the intersection of this sphere with w sup c is denoted by w sub r sup c and the given problem is replaced by du sub r f in w sup c sub r u sub r g on partialw partialu sub r partialr 1 r u sub r 0 on g sub r this problem is then solved approximately by the finite element method resulting in an approximate solution u sub r sup h for each h 0 in order to obtain a reasonably small error for u u sub r sup h u u sub r u sub r u sub r sup h it is necessary to make r large this necessitates the solution of amore large number of linear equations so that this method is often not very good when a uniform mesh size h is employed it is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by ch sup 3 with c independent of h and r less

polyharmonic functions on trees in this paper we introduce and study polyharmonic functions on trees we prove that the discrete version of the classical riquier problem can be solved on trees next we show that the discrete version of a characterization of harmonic functions due to globevnik and rudin holds for bi harmonic functions on trees furthermore on a homogeneous tree we characterize the polyharmonic functions in terms of integrals with respect to finitely additive measures distributions of certain functions depending on the poisson kernel we define polymartingales on a homogeneous tree and show that the discrete version of a characterization of polyharmonic functions due to almansi holds for polymartingales we then show that on homogeneous trees there are 1 1 correspondences among the space of nth order polyharmonic functions the space of nth order polymartingales and the space of n tuples of distributions finally we study the boundary behavior of polyharmonic functions on homogeneous trees whose associated distributions satisfy various growth conditions

a note on the generalized conjugate gradient method each iterate generated by the generalized conjugate gradient method of concus and golub lecture notes in economics and mathematical systems 134 springer verlag new york 1976 and widlund siam j numer anal 15 1978 pp 801 812 is shown to be the best approximation to the solution from a certain affine subspace although not from the natural afline krylov subspace this property is used to improve the error bounds given by widlund and hageman luk and young siam j numer anal 17 1980 pp 852 873

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

fast numerical solution of the biharmonic dirichlet problem on rectangles a new method for the numerical solution of the first biharmonic dirichlet problem in a rectangular domain is presented for an n times n mesh the complexity of this algorithm is on the order of

absorbing boundary conditions difference operators and stability in this paper we present a review of some of the methods currently used for solving the absorbing boundary problem for the two dimensional scalar wave equation we show the relationship between the methods of lindman and clayton and engquist through this relationship we can derive discretizations of any rational approximation to the one way wave equation we prove that for all the cases considered here which can be solved in a manner similar to lindman s approach the bo unds imposed on the courant number for stability at the boundary are no more severe than the bound 1 fi required for stability of the interior scheme these bounds are however necessary but not sufficient we also compare the methods reviewed numerically it is demonstrated that lindman s scheme is no better than a sixth order approximation of halpern and trefethen for low order approximations higdon s one dimensional equations are satisfactory but as the order increases the two dimensional form of the equations as derived by halpern and trefethen is preferable 0 1992 academic pws

nonreflecting boundary conditions in elastodynamics for finite element methods based upon off surface boundary integral equations in this work an off surface boundary integral osbi method is presented as a mesh termination scheme for solving large or infinite domain problems of elastodynamics the boundary integral equation is discretized using finite element shape functions and the neumann boundary condition term is solved for in terms of the dirichlet boundary condition term this expression is then substituted into associated finite element formulation for the interior problem comparisons are made using the new osbi technique the dtn method of givoli and keller and several other local nonreflecting boundary conditions the proposed boundary condition is shown to be accurate is well suited for use with finite element methods and is competitive with the dtn method

a finite element method for 3d exterior oseen flows error estimates stationary oseen flows in a 3d exterior domain are discretized by applying a p1 p1 stabilized finite element method to the oseen system in a truncated exterior domain with an implicit pointwise artificial boundary condition on the truncating boundary error estimates are proved for this discretization the paper extends an approach which was introduced by guirguis and gunzburger rairo model math anal numer 21 1987 pp 445 464 for exterior stokes flows the stabilized p1 p1 method in question was introduced by rebollo numer math 79 1998 pp 283 319

on the use of boundary conditions for variational formulations arising in financial mathematics the general intractability of derivative security valuation models to present techniques both analytic and numerical arguably remains one of the preeminant problem of mathematical finance it is the focus of this paper to examine the applicability of a promising recent development namely radial basis functions rbf to the problem of option valuation a black scholes framework is considered for american and european options written on a one and two risky assets the performance of rbf and finite differencing algorithms are examined with respect to artificial boundary conditions computational domain domain decomposition and mesh scaling

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements in this paper we analyze a galerkin procedure based on a combination of finite and spectral elements for approximating a time harmonic acoustic wave scattered by a bounded inhomogeneity the finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition which consists in a linear integral equation this integral equation is discretized by a spectral galerkin approximation method we provide error estimates for the galerkin method propose fully discrete schemes based on elementary quadrature formulas and show that the perturbation due to this numerical integration gives rise to a quasi optimal rate of convergence we also suggest a method for implementing the algorithm using the preconditioned gmres method and provide some numerical results

solution of biharmonic equations with application to radar imaging abstract in a radar imaging problem using broad band low frequency waves we encounter the problem of solving poisson s equation over a very large rectangular grid typically five thousand times thousand pixels in addition no information about boundary values is available in order to select suitable solutions we solve the poisson equation under the side condition that some criterion function usually a sobolev norm should be minimized under appropriate smoothness assumptions this problem may be reformulated as a boundary value problem for the biharmonic equation numerical techniques are investigated for this problem we also include the results of some numerical experiments

a lagrange multiplier method for the interface equations from electromagnetic applications in the vector scalar potential formulation for three dimensional electromagnetic field applications one arises at a system of interface equations in r 3 coupling the two potentials in this paper a lagrange multiplier method for solving the coupled equations is presented and it is shown that the resulting lagrange multiplier operator defined on the interface is not only symmetric positive definite but also spectrally equivalent to the identity operator with respect to an appropriate inner product hence the lagrange multiplier equation can be solved efficiently using iterative solution schemes such as the conjugate gradient iterations the exterior problem is approximated by truncating the equation on a bounded domain finite element methods are chosen for the approximation of interior and truncated exterior problems as well as for the lagrange multiplier problem error estimates are proved for the finite element approximations which guarantee convergence as the mesh is refined an alternative operat

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

a third order boundary condition for the exterior stokes problem in three dimensions approximation de l operateur de stokes sur un domaine exterieur a trois dimensions par un probleme tronque sur un sous domaine fini introduction d une condition aux limites artificielle d ordre 3 comportement du probleme tronque et de sa discretisation dans un espace a elements finis estimations d erreurs

on the dependence of the reflection operator on boundary conditions for biharmonic functions the biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the stokes flows as well as in a radar imaging problem we discuss the reflection formulas for the biharmonic functions u x y r2 subject to different boundary conditions on a real analytic curve in the plane the obtained formulas generalizing the celebrated schwarz symmetry principle for harmonic functions have different structures in particular in the special case of the boundary 0 y 0 reflections are point to point when the given on 0 conditions are u nu 0 u u 0 or nu n u 0 and point to a continuous set when u n u 0 or nu u 0 on 0

a combined method for solving elliptic problems on unbounded domains abstract a noncomforming combined method of the ritz galerkin and finite element methods is applied in solving unbounded domain problems of the equation u u 0 the ritz galerkin method is used in an exterior subdomain and the traditional finite element method is used in a bounded subdomain errors of numerical solutions have been derived in the sobolev norms and a significant coupling relation n 1 o ln h as h 0 is found where n 1 is the total number of asymptotic particula ritz galerkin method and h is the maximal boundary length of triangular elements used in the finite element method numerical experiments have been carried out for a model problem on unbounded domains by using the nonconforming combination and the coupling strategy 1 showing that only two or three terms i e n 1 2 or 3 of particular solutions near the infinity are in general cases required for good approximate solutions in engineering application

the finite element method with non uniform mesh sizes applied to the exterior helmholtz problem the finite element method with non uniform mesh sizes is employed to approximately solve helmholtz type equations in unbounded domains the given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary this approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases this results in a great reduction in the number of equations to be solved it is proved that optimal error estimates hold inl 2 h 1 andl provided that certain relationships hold between the frequency mesh size and outer radius

local convergence of the exact and inexact iterative aggregation method for linear systems and markov operators the iterative aggregation method for the solution of linear systems is extended in several directions to operators on banach spaces to the method with inexact correction i e to methods where the inner linear system is in turn solved iteratively and to the problem of finding stationary distributions of markov operators local convergence is shown in all cases convergence results apply to the particular case of stochastic matrices moreover an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled markov chains as well as for markov chains with other zero nonzero structures

an analysis of quadrature errors in second kind boundary integral methods this paper considers the effect of quadrature errors on galerkin methods for approximating the solution of second kind fredholm integral equations with weakly singular kernels defined on a smooth closed surface in three dimensions a quadrature strategy for galerkin methods that use piecewise poly nomials both continuous and discontinuous on a quadrilateral based mesh is proposed the error resulting from this further discretization is proved to be no larger than the original discretization error and moreover the amount of work needed to compute the resulting matrix for the corresponding linear system is shown to be proportional to the number of matrix entries

refinements of milnor s fibration theorem for complex singularities abstract let x be an analytic subset of an open neighbourhood u of the origin 0 in c n let f x 0 c 0 be holomorphic and set v f 1 0 let b e be a ball in u of sufficiently small radius e 0 centred at 0 c n we show that f has an associated canonical pencil of real analytic hypersurfaces x with axis v which leads to a fibration of the whole space x b e v over s 1 its restriction to x s e v is the usual milnor fibration f f while its restriction to the milnor tube f 1 d b e is the milnor le fibration of f each element of the pencil x meets transversally the boundary sphere s e b e and the intersection is the union of the link of f and two homeomorphic fibres of over antipodal points in the circle furthermore the space x obtained by the real blow up of the ideal re f im f is a fibre bundle over rp 1 with the x as fibres these constructions work also to some extent for real analytic map germs and give us a clear picture of the differences concerning milnor fibrations between real and complex analytic singularities

convergence of the t matrix approach in scattering theory ii the t matrix numerical scheme is widely used in practice convergence of this scheme was not proved a proof of convergence is given in this paper

vanishing polyhedron and collapsing map in this paper we give a detailed proof of the fact that the milnor fiber x t of an analytic complex isolated singularity function defined on a reduced n equidimensional analytic complex space x is a regular neighborhood of a polyhedron p t subset x t of real dimension n 1 moreover we describe the degeneration of x t onto the special fiber x 0 by giving a continuous collapsing map psi t x t rightarrow x 0 which sends p t to 0 and which restricts to a homeomorphism x t backslash p t rightarrow x 0 backslash 0

three dimensional approximate local dtn boundary conditions for prolate spheroid boundaries we propose a new class of approximate local dtn boundary conditions to be applied on prolate spheroidal shaped exterior boundaries when solving problems of acoustic scattering by elongated obstacles these conditions are a exact for the first modes b easy to implement and to parallelize c compatible with the local structure of the computational finite element scheme and d applicable to exterior ellipsoidal shaped boundaries that are more suitable in terms of cost effectiveness for surrounding elongated scatterers we investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions we also compare their performance to the second order absorbing boundary condition bgt2 designed by bayliss gunzburger and turkel when expressed in prolate spheroid coordinates the analysis reveals that in the low frequency regime the new second order dtn condition dtn2 retains a good level of accuracy regardless of the slenderness of the boundary in addition the dtn2 boundary condition outperforms the bgt2 condition such superiority is clearly noticeable for large eccentricity values

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods