on numerical cubatures of singular surface integrals in boundary element methods we present and analyze methods for the accurate and efficient evaluation of weakly cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3 the class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three dimensional domains by the boundary integral equation method the possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point this rules out weighted quadrature approaches with a fixed singular weight for weakly singular integrals it is shown that duffy s triangular coordinates leadalways to a removal of the kernel singularity also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero these are based on the rabinowitz richter estimates in connection with an asymptotic estimate of domains of analyticity in 2 it is further shown that the modified extrapolation approach due to lyness is in the weakly singular case always applicable corresponding error and asymptotic work estimates are presented for the weakly singular as well as for cauchy and hypersingular integrals which e g arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates these rules are hierarchically constructed from finite part integration formulas in radial and gaussian formulas in angular direction again we show how the rabinowitz richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size

maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions a semilinear second order parabolic equation is considered in a regular and a singularly perturbed regime for this equation we give computable a posteriori error estimates in the maximum norm semidiscrete and fully discrete versions of the backward euler crank nicolson and discontinuous galerkin dg r methods are addressed for their full discretizations we employ elliptic reconstructions that are respectively piecewise constant piecewise linear and piecewise quadratic for r 1 in time we also use certain bounds for the green s function of the parabolic operator

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

an improved acoustic fourier boundary element method formulation using fast fourier transform integration effective use of the fourier series boundary element method fbem for everyday applications is hindered by the significant numerical problems that have to be overcome for its implementation in the fbem formulation for acoustics some integrals over the angle of revolution arise which need to be calculated for every fourier term these integrals were formerly treated for each fourier term separately in this paper a new method is proposed to calculate these integrals using fast fourier transform techniques the advantage of this integration method is that the integrals are simultaneously computed for all fourier terms in the boundary element formulation the improved efficiency of the method compared to a gaussian quadrature based integration algorithm is illustrated by some example calculations the proposed method is not only usable for acoustic problems in particular but for fourier bem in general

a numerical verification method for solutions of boundary value problems with local uniqueness by banach s fixed point theorem in this paper we propose a method to prove the existence and the local uniqueness of solutions to infinite dimensional fixed point equations using computers choosing a set which possibly includes a solution we transform it by an approximate linearization of the operator appearing in the equation then we calculate the radii of the transformed set in order to check sufficient conditions for banach s fixed point theorem this method is applied to elliptic problems and numerical examples are given

episturmian words and episturmian morphisms infinite episturmian words are a generalization of sturmian words which includes the arnoux rauzy sequences we continue their study and that of episturmian morphisms begun previously in relation with the action of the shift operator palindromic and periodic factors of these words are described we consider in particular the case where these words are generated by morphisms and introduce then a notion of intercept generalizing that of sturmian words finally we prove that the frequencies of the factors in a strong sense do exist for all episturmian words

an alternative approach to norm bound computation for inverses of linear operators in hilbert spaces abstract in the present paper we propose a computer assisted procedure to prove the invertibility of a linear operator in a hilbert space and to compute a verified norm bound of its inverse a number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates the approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications several verification examples that confirm the actual effectiveness of the proposed procedure are reported

elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems we derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations the space discretization uses finite element spaces that are allowed to change in time our main tool is an appropriate adaptation of the elliptic reconstruction technique introduced by makridakis and nochetto we derive novel a posteriori estimates for the norms of l 0 t l2 and the higher order spaces l 0 t h1 and h1 0 t l2 with optimal orders of convergence

maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction diffusion problem a singularly perturbed semilinear reaction diffusion equation posed in the unit square is discretized on arbitrary nonuniform tensor product meshes we establish a second order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter no mesh aspect ratio assumption is made numerical results are presented that support our theoretical estimate

discrete boundary finite element schemes for an exterior problem for the time harmonic maxwell s equation abstract a general boundary integral formulation using galerkin procedure is applied to compute the scattering electric field produced by the diffraction of an incident electromagnetic wave by a perfectly conducting obstacle this electric field satisfies the three dimensional time harmonic maxwell s equations for which the skin currents and charges are to be approximated using boundary finite element method with the help of linear and quadratic finite elements of lagrange type we introduce an approximate surface on which the discrete formulation is defined and construct approximate surface currents and charges in addition we study the existence and the uniqueness of the solution of the discrete problem and develop some error estimates for the currents and charges numerical results are also presented in order to validate our numerical approach

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

vectorial and parallel processing in stress analysis with the boundary element method the purpose of this short paper is to describe an experience at converting a scalar serial elastic axisymmetric bem code to a vector parallel environment on an ibm this is a single zone bem computer program

a posteriori error estimates for the crank nicolson method for parabolic equations we derive optimal order a posteriori error estimates for time discretizations by both the crank nicolson and the crank nicolson galerkin methods for linear and nonlinear parabolic equations we examine both smooth and rough initial data our basic tool for deriving a posteriori estimates are second order crank nicolson reconstructions of the piecewise linear approximate solutions these functions satisfy two fundamental properties i they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori and ii they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation the resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem as a consequence we provide alternative proofs for known a priori rates of convergence for the crank nicolson method

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

a method of verified computations for solutions to semilinear parabolic equations using semigroup theory this paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial boundary value problem of semilinear parabolic equations the main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution in the formulation used in this paper the initial boundary value problem is transformed into a fixed point form using an analytic semigroup the sufficient condition is derived from banach s fixed point theorem this paper also introduces a recursive scheme to extend a time interval in which the validity of the solution can be verified as an application of this method the existence of a global in time solution is demonstrated for a certain semilinear parabolic equation

infinite words with linear subword complexity abstract in section 1 we study the relations among some combinatorial properties of infinite words especially in the case of infinite words with linear subword complexity the main result of section 1 concerns a permutation property of infinite words with linear complexity in section 2 we investigate the special case of the sturmian infinite words the main result is that a sturmian infinite word associated to a real number contains no k th powers iff has a continued fraction expansion with bounded partial quotients in section 3 it is shown that some of the results of sections 1 and 2 are optimal

scaling integration method for singular integrals and its application to bem abstract a new accurate and efficient algorithm is developed for the numerical evaluation of a class of singular integrals which arise in the bem boundary element method analyses the algorithm is based on an idea of scaling integration that utilizes a proportional relationship inherent in the numerical computation of the singular integrals over flat triangular elements the present algorithm scaling integration method avoids the direct numerical evaluation of the integrand in the neighborhood of singularities it is shown and demonstrated that the method is simpler faster and more accurate than standard gaussian integration algorithms an application of the method to the bem analysis is also described and the results are favorably compared with the exact solution

relation between powers of factors and the recurrence function characterizing sturmian words in this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of sturmian words as a by product we give a new proof of the theorem describing the index of a sturmian word in terms of the continued fraction expansion of its slope this theorem was independently proved in a carpi a de luca special factors periodicity and an application to sturmian words acta inform 36 2000 983 1006 and d damanik d lenz the index of sturmian sequences european j combin 23 2002 23 29

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

abelian powers and repetitions in sturmian words abstract richomme saari and zamboni 2011 39 proved that at every position of a sturmian word starts an abelian power of exponent k for every k 0 we improve on this result by studying the maximum exponents of abelian powers and abelian repetitions an abelian repetition is an analogue of a fractional power in sturmian words we give a formula for computing the maximum exponent of an abelian power of abelian period m starting at a given position in any sturmian word of rotation angle by considering all possible abelian periods m we recover the result of richomme saari and zamboni as an analogue of the critical exponent we introduce the abelian critical exponent a s of a sturmian word s of angle as the quantity a s lim sup k m m lim sup k m m where k m resp k m denotes the maximum exponent of an abelian power resp of an abelian repetition of abelian period m the superior limits coincide for sturmian words we show that a s equals the lagrange constant of the number this yields a formula for computing a s in terms of the partial quotients of the continued fraction expansion of using this formula we prove that a s 5 and that the equality holds for the fibonacci word we further prove that a s is finite if and only if has bounded partial quotients that is if and only if s is power free for some real number concerning the infinite fibonacci word we prove that i the longest prefix that is an abelian repetition of period f j j 1 has length f j f j 1 f j 1 1 2 if j is even or f j f j 1 f j 1 2 if j is odd where f j is the j th fibonacci number ii the minimum abelian period of any factor is a fibonacci number further we derive a formula for the minimum abelian periods of the finite fibonacci words we prove that for j 3 the fibonacci word f j of length f j has minimum abelian period equal to f j 2 if j 0 1 2 mod 4 or to f 1 j 2 if j 3 mod 4

a singularity cancellation technique for weakly singular integrals on higher order surface descriptions accurate integration of singular and near singular functions is critical to the accuracy of the method of moments solution to surface integral equations while this problem has been widely addressed for flat geometries its extensions to higher order surface descriptions have been limited this letter provides a systematic prescription for the application of the rules for weakly singular integrals on higher order surfaces here we present implementation details and several demonstrative results that compare the accuracy and convergence of the integration rules

fractional powers in sturmian words abstract given an infinite sturmian word s we calculate the function l m which gives the length of the longest factor of s having period m the expression of l m makes use of the continued fraction of the irrational associated with s

non singular boundary integral equation implementation the basis of the boundary integral equation bie development is the singular field solution for the kelvin point load problem the presence of the mathematical singularity in an otherwise non singular problem has created over the years a variety of numerical coping strategies the current paper extends some of the earlier works in a simple manner that eliminates the mathematical singularity through analytical integration of the critical terms the resulting formulation is fully non singular and is shown to be amenable to simple low order gaussian integration the critical class of problems involving thin sections and or highly graded bie meshes now can be solved without previous problems and concerns the accuracy of the numerical integrations used in creating the linear bie equations extension of the formulation approach to remove hypersingular formulation problems in fracture mechanics is seen to be a future beneficiary of the semi analytical bie implementation reported herein

accurate integration of singular kernels in boundary integral formulations for helmholtz equation boundary integral formulations for the 2d helmholtz equation involve kernels in the form of modified bessel functions accurate schemes for evaluating integrals of the kernels and their derivatives are presented special attention is paid to integrals involving singular and near singular kernels both boundary and domain integrals are considered it is shown that with the use of series expansion functions for the modified bessel functions the boundary integrals can be evaluated analytically in the neighbourhood of the singularity for domain integrals the behaviour of the kernels in the vicinity of the singularity is used to construct accurate numerical quadrature schemes a transient heat conduction problem is formulated as a helmholtz equation solved and compared against analytic solution to demonstrate the effectiveness of these schemes in relation to traditional methods references are made to previous work to advocate the utility of the boundary integral method for non linear and time transient problems

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

quasi optimal and robust a posteriori error estimates in for the approximation of allen cahn equations past singularities quasi optimal a posteriori error estimates in l 0 t l 2 are derived for the finite element approximation of allen cahn equations the estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface the error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming nonconforming mixed and discontinuous galerkin methods numerical experiments illustrate the theoretical results

new strategies for some issues of numerical manifold method in simulation of crack propagation summary aiming to solve in a unified way continuous and discontinuous problems in geotechnical engineering the numerical manifold method introduces two covers namely the mathematical cover and the physical cover in order to reach the goal some issues in the simulation of crack propagation have to be solved among which are the four issues to be treated in this study 1 to reduce the rank deficiency induced by high degree polynomials as local approximation a new variational principle is formulated which suppresses the gradient dependent dofs 2 to evaluate the integrals with singularity of 1 r a new numerical quadrature scheme is developed which is simpler but more efficient than the existing duffy transformation 3 to analyze kinked cracks a sign convention for argument in the polar system at the crack tip is specified which leads to more accurate results in a simpler way than the existing mapping technique and 4 to demonstrate the mesh independency of numerical manifold method in handling strong singularity a mesh deployment scheme is advised which can reproduce all singular locations of the crack with regard to the mesh corresponding to the four issues typical examples are given to demonstrate the effectiveness of the proposed schemes copyright 2013 john wiley sons ltd

numerical verification for existence of a global in time solution to semilinear parabolic equations abstract this paper presents a method of numerical verification for the existence of a global in time solution to a class of semilinear parabolic equations such a method is based on two main theorems in this paper one theorem gives a sufficient condition for proving the existence of a solution to the semilinear parabolic equations with the initial point t t 0 if the sufficient condition does not hold the other theorem is used for enclosing the solution for time t 0 0 in a neighborhood of a numerical solution numerical results of obtaining a global in time solution for a certain semilinear parabolic equation are also given

a semi analytical algorithm for the evaluation of the nearly singular integrals in three dimensional boundary element methods the nearly singular integrals occur in the boundary integral equations when the source point is close to an integration element as compared to its size but not on the element in this paper the concept of a relative distance from a source point to the boundary element is introduced to describe possible influence of the singularity of the integrals then a semi analytical algorithm is proposed for evaluating the nearly strongly singular and hypersingular integrals in the three dimensional bem by using integration by parts the nearly singular surface integrals on the elements are transformed to a series of line integrals along the contour of the element the singular behavior which appears as factor is separated from remaining regular integrals consequently standard numerical quadrature can provide very accurate evaluation of the resulting line integrals the semi analytical algorithm is applied to analyzing the three dimensional elasticity problems such as very thin walled structures meanwhile the displacements and stresses at the interior points very close to its bounding surface are also determined efficiently the results of the numerical investigation demonstrate the accuracy and effectiveness of the algorithm

accurate numerical integration of singular boundary element kernels over boundaries with curvature use of the green function for the solution of boundary value problems frequently results in singular integral equations algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method bem they are based upon the use of appropriately weighted gaussian quadrature formulae together with numerical geometrical transformations of the region of integration the use of high order subdomain expansion functions for interpolation over nonplanar elements allows boundary curvature to be accommodated in particular the handling of green functions with logarithmic and r 1 behaviour are detailed volume integrals with r 2 singularity are outlined operations are performed on a simplex thus resulting in generality and ease of automation this scheme has been incorporated into boundary element method software and successfully applied to a variety of problems

on the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems we consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial boundary conditions our estimation technique uses a full discrete numerical scheme which is based on the galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space in our technique the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role combining these estimates with an argument for the discretized inverse operator and a contraction property of the newton type formulation we derive an a posteriori estimate of the norm for the infinite dimensional operator in numerical examples we show that the proposed method should be more efficient than the existing method moreover as an application we give some prototype results for numerical verification of solutions of nonlinear parabolic problems which confirm the actual usefulness of our technique

on critical exponents in fixed points of non erasing morphisms let s be an alphabet of size t let f s s be a non erasing morphism let w be an infinite fixed point of f and let e w be the critical exponent of w we prove that if e w is finite then for a uniform f it is rational and for a non uniform f it lies in the field extension q r l 1 l where r l 1 l are the eigenvalues of the incidence matrix of f in particular e w is algebraic of degree at most t under certain conditions our proof implies an algorithm for computing e w

some notes on singular integral techniques in boundary element analysis the paper presented here reviews the numerical techniques used currently to calculate the singular integrals and nearly singular integrals in the boundary element analysis some incorrect algorithms published before are discussed and a new numerical technique to calculate the nearly singular integral is developed the numerical results show a significant improvement in both accuracy and efficiency compared to the traditional adaptive gaussian quadrature and subdivision techniques

an invariant imbedding method for singular integral evaluation on finite domains the invariant imbedding technique is used to develop versatile and powerful integration formulae for the evaluation of singular integrals on finite domains encountered in continuum problems in engineering and applied science this method is considerably simpler than the several ad hoc integral evaluation methods presented in the recent literature moreover the method provides new insight into the central role played by the residue in the evaluation of principal value integrals this relationship is embodied as a gauge condition the main emphasis of the paper is on developing a general approach from which specific cases follow several examples chosen from different areas of engineering computation are provided in order to show the advantage of this approach

a meshless integral method based on regularized boundary integral equation a meshless integral method based on the regularized boundary integral equation is developed and applied to two dimensional linear elasticity the governing integral equation is obtained from the weak form of elasticity over a local sub domain and the moving least squares approximation is used for meshless function approximation the method is built on the lbie method proposed by atluri and co workers and several key improvements are introduced in this work that significantly enhance the accuracy and robustness of the method the most critical improvement is the use of the subtraction technique to remove the strong singularity that results in a regularized governing integral equation the technique is straightforward and efficient and is much simpler and easier compared to other singularity removal techniques a special numerical integration is employed for the calculation of integrals with weak singularity which further improves accuracy the collocation method is employed to enforce the essential boundary conditions exactly which is simple and computationally efficient the natural boundary conditions are incorporated in the system governing equation and require no special handling numerical tests show that the meshless integral method is accurate and robust the effects of weight function support domain sub domain and monomial basis are investigated and discussed

seismic soil structure interaction analysis by direct boundary element methods abstract upon establishing the mathematical framework necessary for a proper understanding of the analytical theory a regularized form of the conventional direct boundary integral equation formulation for three dimensional elastodynamics is presented for a general anisotropic medium founded on the basis of a full decomposition of the greens functions into regular and singular parts the alternative boundary integral equation format is compact and without demand ingmathematical and numerical complexities such as cauchy principal values extended to deal with general seismic soil structure interaction problems in semi infinite media the formulation is implemented computationally together with a rigorous treatment of singular dynamic multi layered viscoelastic half space greens functions and interfacial boundary tractions arising in typical soil structure foundation configurations a set of new benchmark numerical results are included

elliptic reconstruction and a posteriori error estimates for parabolic problems it is known that the energy technique for a posteriori error analysis of finite element discretizations of parabolic problems yields suboptimal rates in the norm l infty 0 t l 2 omega in this paper we combine energy techniques with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order and regularity for piecewise polynomials of degree higher than one this technique may be regarded as the dual a posteriori counterpart of wheeler s elliptic projection method in the a priori error analysis

a posteriori error analysis of a fully discrete finite element method for the wave equation abstract a fully discrete finite element method is proposed for linear wave equations the spatial discretization is carried out using p l conforming element while the temporal discretization is obtained using p 1 continuous discontinuous galerkin method borrowing a novel time reconstruction operator combined with the technique of elliptic reconstruction a reliable posteriori error bound is developed based on this error bound two adaptive algorithms in time and space separately are proposed a series of numerical experiments are reported to illustrate the performance of the error bound and the adaptive algorithms proposed

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

estimating quadrature errors for an efficient method for quasi singular boundary integral abstract the numerical resolution of the boundary integral equations applied to the differential equations of laplace helmholtz and maxwell requires the handling of quasi singular integrals with different order of singularity the numerical approximation of the integral equations of different kinds is made by boundary finite elements in this paper we present a complete survey for estimating quadrature errors for the numerical techniques proposed by huang and cruse q huang t a cruse some notes on singular integral techniques in boundary element analysis int j numer methods eng 36 15 1993 2643 2659 to calculate the quasi singular integrals to validate the accuracy and efficiency of these techniques and approve our study some numerical examples are presented and discussed

reusable intrinsic sample point risp algorithm for the efficient numerical integration of three dimensional curved boundary elements this paper presents an algorithm that provides an order of magnitude gain in the computational performance of the numerical integration of the boundary integral equations for three dimensional analysis existing algorithms for numerical integration have strategically clustered integration sample points based on the relative proximity of the load points to the boundary element being integrated using element subdivision or element co ordinate transformation the emphasis in these techniques has been on minimizing the number of sample points required to obtain a given level of accuracy the present algorithm while closely following the spirit of these earlier approaches employs a discrete number of sets of predetermined customized near optimum sample point quantities associated with the intrinsic boundary element the ability created by this approach to reuse sample point geometric information of the actual element allows for the realization of substantive computational economy this algorithm provides accurate and efficient numerical results both when load points are far from and when they are on the boundary element being integrated numerical results are provided to demonstrate the substantial economy achieved through the use of the present algorithm

sturmian words and words with a critical exponent let s be a standard sturmian word that is a fixed point of a non trivial homomorphism associated to the infinite word s is a unique irrational number with 0 0 it contains a fractional power with exponent greater than e here is a constant that depends on the constant is given explicitly using these results we are able to give a short proof of mignosi s theorem and give an exact evaluation of the maximal power that can occur in a standard sturmian word

validated continuation for equilibria of pdes one of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor corrector continuation techniques in the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output we introduce a new technique that combines the information obtained from the predictor corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation using the cahn hilliard and swift hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone

use of simple solutions for boundary integral methods in elasticity and fracture analysis the simple solutions or indirect method of analysing cauchy and hypersingular integrands in the gradient flux or traction boundary integral equation bie is applied to linear elastic fracture analysis because of the geometric singularity of the crack surface application of the simple solutions formulas on the crack face requires integration over a temporary closure surface rather than the remainder of the body closure surface constructions are exhibited for crack surfaces allowing the gradient bie to be applied as a constraint equation on a crack surface where the primary bie is degenerate computational results are given for two benchmark fracture problems

a posteriori error estimates in the maximum norm for parabolic problems we derive a posteriori error estimates in the l infty 0 t l infty omega norm for approximations of solutions to linear parabolic equations using the elliptic reconstruction technique introduced by makridakis and nochetto and heat kernel estimates for linear parabolic problems we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations we then establish a posteriori bounds for a fully discrete backward euler finite element approximation the elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators

the continuation approach a general framework for the analysis and evaluation of singular and near singular integrals the continuation approach presented in this paper is a simple and unified framework for analyzing and computing a large class of singular and near singular integrals such as those that arise in the boundary element method the analysis formulates singular and near singular integrals consistently as instances of a single phenomenon with all types of algebraic singularities treated in a similar way for domains of arbitrary dimension singular integrals are viewed merely as continuations of nonsingular but perhaps near singular ones by placing the singularity of the integrand outside the integration domain and taking the limit as it approaches the domain the technique exploits the functional homogeneity shared by the green s functions of many physical problems for flat surfaces this allows the integral to be mapped to the contour of the integration domain curved surfaces are handled by projecting the integration domain to a tangent hyperplane where the singular components of the integral can be

critical exponents of infinite balanced words abstract over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2 2 2 over an alphabet of size 4 we construct an infinite balanced word with critical exponent 5 5 4 over larger alphabets we give some candidates for balanced words found computationally having small critical exponents we also explore a method for proving these results using the automated theorem prover walnut

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings

hierarchical structures in sturmian dynamical systems the paper is concerned with hierarchical stuctures in subshifis over a finite alphabet in particular we present a hierarchy based approach to sturmian systems this approach is then used to characterize the linearly repetitive sturmian systems among the sturmian systems by uniform positivity of certain weights more generally we discuss various bounds on weights and their relationship finally we combine hierarchies and bounds on a weights to study a thermodynamic formalism for lattice gas models associated to arbitrary linearly repetitive subshifts