higher order boundary infinite elements abstract higher order approximations are implemented in a novel approach to infinite element formulations for exterior problems of time harmonic acoustics this approach is based on a functional which provides a general framework for domain based computation of exterior problems two prominent features of this formulation simplify the task of discretization the lack of integration over the unbounded domain and weak enforcement of continuity between finite elements and infinite elements consequently the infinite elements mesh the interface only and need not match the finite elements on the interface various infinite element approximations for two dimensional configurations with circular interfaces are presented numerical results demonstrate the good performance of this scheme

scattering by a hybrid asymptotic finite element method abstract we describe a hybrid asymptotic finite element method with application to the solution of the helmholtz equation for large wavenumber thus this method is suitable to compute the acoustic field radiated or scattered by acoustically large objects the method is based on patching a short wavelength asymptotic expansion of the outer field to an fem interpolation of the inner field the patching takes place on an artificial boundary continuity of the field and its normal derivative across the artificial boundary is enforced weakly in a variational setting the variational principle is described as is an element like implementation strategy two example applications are included diffraction by a truncated wedge and radiation by a source on a wave bearing surface

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

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

diffraction from simple shapes by a hybrid asymptotic finite element method the application of a hybrid asymptotic finite element method to the problem of scattering from two dimensional submerged objects is considered the hybrid method is based on patching a short wavelength asymptotic expansion of the scattered field to a finite element interpolation of the near field in patching the diffracted field shape functions with unknown amplitude are forced to agree smoothly with the solution in the near field along a curve at a prescribed distance from the diffraction points a new hybrid finite element on this artificial boundary represents the effect of the outer domain on the solution within this new boundary this allows the replacement of the original boundary value problem with an asymptotically equivalent boundary value problem the domain of which is small and efficiently discretized the method is applied to diffraction by a blunted wedge which in this context represents a degenerate prism the hybrid scattering solution shall be compared to an analytic field representat

non radial solutions for some semilinear elliptic equations on the disk abstract starting with approximate solutions of the equation u w u 3 on the disk with zero boundary conditions we prove that there exist true solutions nearby one of the challenges here lies in the fact that we need simultaneous and accurate control of both the inverse dirichlet laplacean and nonlinearities we achieve this with the aid of a computer using a banach algebra of real analytic functions based on zernike polynomials besides proving existence and symmetry properties we also determine the morse index of the solutions

studies of domain based formulations for computing exterior problems of acoustics a strategy must be devised to compute solutions over domains which are unbounded exact impedance conditions at an artificial external boundary are specified by the dtn method yielding an equivalent problem that is suitable for domain based computation the dtn boundary condition is non reflective giving rise to exact and thereby unique solutions the truncated dtn operator which is employed in practice fails to inhibit the reflection of higher modes so that non unique solutions may occur at their harmonics simple expressions determine a sufficient number of terms in the truncated operator for unique solutions at any given wave number there are three characteristic length scales in the computational problem the radius of the artificial boundary the geometry of the body represented by the internal boundary and the mesh size

gevrey regularity for navier stokes equations under lions boundary conditions abstract the navier stokes system is considered in a compact riemannian manifold gevrey class regularity is proven under lions boundary conditions in 2d for the rectangle cylinder and hemisphere and in 3d for the rectangle the cases of the 2d sphere and 2d and 3d torus are also revisited

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

on an alleged philosophers stone concerning variational principles with subsidiary conditions in general the lagrange multiplier method lmm is used to incorporate subsidiary conditions into variational principles the lagrange multipliers represent an additional field of independent variables the attempt to satisfy subsidiary conditions without employing additional independent unknowns has led to the development of simplified variational principles svp they are characterized by expressing lagrange multipliers in terms of original field variables by means of the euler lagrange equations for the multipliers providing their physical interpretation in the first part of the theoretical investigation systems with infinitely many degrees of freedom are studied it is shown that the euler lagrange equations of a lmm based on a modification of the principle of minimum of potential energy pmipe do not hold unconditionally that is for arbitrary subsidiary conditions for the corresponding svp the second part of the theoretical investigation is concerned with systems with a finite number of degrees of freedom the finite element method fem is employed to discuss and compare the characteristics of the lmm and of the corresponding svp contrary to the former the latter are found to be problem dependent several shortcomings of the svp are listed including the possibility of obtaining an infinite sequence of singular coefficient matrices in the process of a systematic mesh refinement consequently convergence of finite element solutions to the true solution in the limit of finite element representations is not guaranteed the theoretical findings are corroborated by the results of a detailed numerical study

partition functions of classical heisenberg spin chains with arbitrary and different exchange the classical heisenberg model has been effective in modelling exchange interactions in molecular magnets in this model the partition function is important as it allows the calculation of the magnetization and susceptibility for an ensemble of n spin sites this typically involves integrals in 2n dimensions here for two three and four spin nearest neighbour open linear heisenberg chains these integrals are reduced to sums of known functions using a result due to gegenbauer for the case of the three and four spin chains the sums are equivalent in form to the results of joyce the general result for an n spin chain is also obtained

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

reexamination of the calculation of two center two electron integrals over slater type orbitals i coulomb and hybrid integrals in this paper which constitutes the first part of the series we consider calculation of two center coulomb and hybrid integrals over slater type orbitals general formulas for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters direct integration over the coordinates of one of the electrons leaves us with the set of overlaplike integrals which are evaluated by using two distinct methods the first one is based on the transformation to the ellipsoidal coordinates system and the second utilizes a recursive scheme for consecutive increase of the angular momenta in the integrand in both methods simple one dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulas it is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well behaved functions calculated recursively with decent speed special attention is paid to the numerical stability of the algorithms applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including at most 7i type functions l 6

efficient storage scheme for precalculated wigner 3 j 6 j and gaunt coefficients efficient storage schemes are presented for storing clebsch gordan wigner 3j and 6j symbols as well as gaunt coefficients which are the integral over three spherical harmonics use is hereby made of the large number of symmetries which these symbols exhibit computer codes have been written and benchmarked against well known published programs which usually use recursion relations for the evaluation it is shown that our codes can be an order of magnitude or more faster in execution speed maintaining full double precision accuracy

mapped wave envelope elements for acoustical radiation and scattering wave envelope elements provide an alternative to boundary element methods for the solution of exterior wave problems applications to date have been to axisymmetric geometries and have involved the use of large but finite wave envelope meshes compatibly matched to denser near field meshes of conventional elements in the present paper the wave envelope idea is extended by the inclusion of an element mapping taken directly from infinite element methodology and by modifications to the residual formulation taken in conjunction these two measures permit the outer boundary of the wave envelope region to be moved to infinity while preserving the boundedness of the acoustic mass and stiffness integrals the present formulation is developed initially with reference to a simple one dimensional model and is then applied in two and three dimensions comparisons between computed results and those obtained from analytic solutions where applicable and from boundary element codes demonstrate the effectiveness of the method

numerical solution for exterior problems we solve the helmholtz equation in an exterior domain in the plane a perfect absorption condition is introduced on a circle which contains the obstacle this boundary condition is given explicitly by bessel functions we use a finite element method in the bounded domain an explicit formula is used to compute the solution out of the circle we give an error estimate and we present relevant numerical results

fast and accurate evaluation of wigner 3j 6j and 9j symbols using prime factorisation and multi word integer arithmetic we present an efficient implementation for the evaluation of wigner 3j 6j and 9j symbols these represent numerical transformation coefficients that are used in the quantum theory of angular momentum they can be expressed as sums and square roots of ratios of integers the integers can be very large due to factorials we avoid numerical precision loss due to cancellation through the use of multiword integer arithmetic for exact accumulation of all sums a fixed relative accuracy is maintained as the limited number of floating point operations in the final step incur rounding errors only in the least significant bits time spent to evaluate large multiword integers is in turn reduced by using explicit prime factorization of the ingoing factorials thereby improving execution speed comparison with existing routines shows the efficiency of our approach and we therefore provide a computer code based on this work

three steps modified levenberg marquardt method with a new line search for systems of nonlinear equations three steps modified levenberg marquardt method for nonlinear equations was introduced by yang 2013 this method uses the addition of the levenberg marquardt lm step and two approximate lm steps as the trial step at every iteration using trust region technique the global and biquadratic convergence of the method is proved by yang the main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique since the search direction of yang method may be not a descent direction standard line searches cannot be used directly in this paper we propose a new nonmonotone third order armijo type line search technique which guarantees the global convergence of this method while we use an adaptive lm parameter it is proved that the convergence order of the new method is biquadratic numerical results show the new algorithm is efficient and promising