a domain decomposition method based on weighted interior penalties for advection diffusion reaction problems we propose a domain decomposition method for advection diffusion reaction equations based on nitsche s transmission conditions the advection dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries we prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method the convergence of the resulting domain decomposition method is proved and this result holds true uniformly with respect to the diffusion parameter the numerical scheme that we propose here can thus be applied straightforwardly to diffusion dominated advection dominated and hyperbolic problems some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method and we compare our method with some other domain decomposition techniques for advection diffusion equations

a robust nitsche s formulation for interface problems abstract in this work we propose a novel weighting for the interfacial consistency terms arising in a nitsche variational form we demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method consequently we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of a elements with arbitrarily small volume fractions b large material heterogeneities and c both large heterogeneities as well as arbitrarily small elements we then highlight the accuracy and efficiency of the proposed formulation through numerical examples focusing particular attention on interfacial quantities of interest

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

error analysis of nitsche s mortar method optimal a priori and a posteriori error estimates are derived for three variants of nitsche s mortar finite elements the analysis is based on the equivalence of nitsche s method and the stabilised mixed method nitsche s method is defined so that it is robust with respect to large jumps in the material and mesh parameters over the interface numerical results demonstrate the robustness of the a posteriori error estimators

optimal a priori estimates for interface problems we consider a priori estimates in weighted norms for interface problems with piecewise constant diffusion constants which do not depend on the ratio between the constants our result generalizes an estimate of lemrabet to arbitrary dimensions and includes curved boundaries furthermore we discuss criteria for the existence of a uniform poincare estimate in weighted norms in the affirmative case we obtain a robust finite element error bound in weighted norms finally we present numerical experiments including a case with no uniform poincare constant

unconventional schemes for a class of ordinary differential equations with applications to the korteweg de vries equation an unconventional numerical method for solving a restrictive and yet often encountered class of ordinary differential equations is proposed the method has a crucial what we callreflexive property and requires solving one linear system per time step but is second order accurate a systematical and easily implementable scheme is proposed to enhance the computational efficiency of such methods whenever needed applications are reported on how the idea can be applied to solve the korteweg de vries equation discretized in space

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

the coiflet galerkin method for linear volterra integral equations this paper deals with the application of the wavelet galerkin method based on coiflets as a basis for solving linear volterra integral equations vies the main contribution of this work is that some new connection coefficients are introduced and a suitable algorithm is developed for their solution once they have been computed they can be stored and applied to any linear vie the convergence properties of the coiflet galerkin method are analyzed some test examples are presented to illustrate the performance of the method with respect to the error norms and cpu time

boundary lagrange multipliers in finite element methods error analysis in natural norms a finite element procedure for circumventing the babuska brezzi condition in mixed formulations with lagrange multipliers defined on the boundary is presented residual terms constructed from the euler lagrange equations are added to the classical galerkin formulation in order to attain coercivity in a mesh dependent norm convergence is proven for the primal variable and the multiplier in the natural mesh independent norm of the problem generalizing results of a previous paper

application of taylor series in obtaining the orthogonal operational matrix in this research first we explicitly obtain the relation between the coefficients of the taylor series and jacobi polynomial expansions then we present a new method for computing classical operational matrices derivative integral and product for general jacobi orthogonal functions polynomial and rational this method can be used for many classes of orthogonal functions

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

a bubble stabilized finite element method for dirichlet constraints on embedded interfaces we examine a bubble stabilized finite element method for enforcing dirichlet constraints on embedded interfaces by embedded we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh as such the robust imposition of dirichlet constraints using a lagrange multiplier field is not trivial to focus issues we consider a simple one sided problem that is representative of a wide class of evolving interface problems the bulk field is decomposed into coarse and fine scales giving rise to coarse scale and fine scale one sided sub problems the fine scale solution is approximated with bubble functions permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method importantly the method is simple to implement readily extends to multiple dimensions obviates the need to specify any free stabilization parameters and can lead to a symmetric positive definite system of equations the performance of the method is then examined through several numerical examples the accuracy of the lagrange multiplier is compared to results obtained using a local version of the domain integral method the variational multiscale approach proposed herein is shown to both stabilize the lagrange multiplier and improve the accuracy of the post processed fluxes copyright 2006 john wiley sons ltd

robust imposition of dirichlet boundary conditions on embedded surfaces summary we develop both stable and stabilized methods for imposing dirichlet constraints on embedded three dimensional surfaces in finite elements the stable method makes use of the vital vertex algorithm of bechet et al bechet et al 2009 int j numer meth engng 78 8 931 954 albeit with a modified set of discontinuous basis functions for the lagrange multiplier field on the embedded surface the stabilized method extends the work of dolbow and harari dolbow and harari 2009 int j numer meth engng 78 2 229 252 to three dimensional surfaces algorithmic and implementational details of both methods are provided several three dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches the results indicate that both methods yield optimal rates of convergence in various quantities of interest with the primary differences being in the surface flux the utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques published by john wiley sons ltd copyright c 2000 john wiley sons ltd

discontinuous galerkin finite element methods for interface problems a priori and a posteriori error estimations discontinuous galerkin dg finite element methods were studied by many researchers for second order elliptic partial differential equations and a priori error estimates were established when the solution of the underlying problem is piecewise h 3 2 epsilon smooth with epsilon 0 however elliptic interface problems with intersecting interfaces do not possess such a smoothness in this paper we establish a quasi optimal a priori error estimate for interface problems whose solutions are only h 1 alpha smooth with alpha in 0 1 and hence fill a theoretical gap of the dg method for elliptic problems with low regularity the second part of the paper deals with the design and analysis of robust residual and recovery based a posteriori error estimators theoretically we show that the residual and recovery estimators studied in this paper are robust with respect to the dg norm i e their reliability and efficiency bounds do not depend on the jump provided that the distribution of coefficients is locally quasi monotone

integrable and superintegrable systems associated with multi sums of products we construct and study certain liouville integrable superintegrable and non commutative integrable systems which are associated with multi sums of products

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

stable imposition of stiff constraints in explicit dynamics for embedded finite element methods summary we investigate various strategies to enforce the kinematics at an embedded interface for transient problems within the extended finite element method in particular we focus on explicit time integration of the semi discrete equations of motion and extend both dual and primal variational frameworks for constraint enforcement to a transient regime we reiterate the incompatibility of the dual formulation with purely explicit time integration and the severe restrictions placed by the courant friedrichs levy condition on primal formulations we propose an alternate consistent formulation for the primal method and derive an estimate for the stabilization parameter which is more amenable in an explicit dynamics framework importantly the use of the new estimate circumvents the need for any tolerances as an interface approaches an element boundary we also show that with interfacial constraints existing mass lumping schemes can lead to prohibitively small critical time steps accordingly we propose a mass lumping procedure which provides a more favorable estimate these techniques are then demonstrated on several benchmark numerical examples where we compare and contrast the accuracy of the primal methods against the dual methods in enforcing the constraints copyright 2012 john wiley sons ltd

efficient algorithms to solve singular integral equations of abel type in the present paper we obtain the approximate solution of abel s integral equation by using the following powerful efficient but simple methods i he s homotopy perturbation method hpm ii modified homotopy perturbation method mhpm iii adomian decomposition method adm and iv modified adomian decomposition method madm the validity and applicability of these techniques are illustrated through various particular cases which demonstrate their efficiency and simplicity in solving these types of integral equations compared with the other existing methods

numerical solution of fractional pantograph differential equations by using generalized fractional order bernoulli wavelet in the current study new functions called generalized fractional order bernoulli wavelet functions gfbwfs based on the bernoulli wavelets are defined to obtain the numerical solution of fractional order pantograph differential equations in a large interval for the concept of fractional derivative we will use caputo sense by using riemann liouville fractional integral operator first the generalized fractional order bernoulli wavelets are constructed then these functions and their properties are employed to derive the gfbwfs operational matrices of fractional integration and pantograph the operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations finally some examples are included for demonstrating the validity and applicability of our method

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

highly efficient invariant conserving explicit runge kutta schemes for nonlinear hamiltonian differential equations abstract a unified framework of invariant conserving explicit runge kutta schemes for the nonlinear hamiltonian odes and pdes are proposed by utilizing the invariant energy quadratization technique first the nonlinear hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant for the nonlinear ode the reformulation is then discretized using a class of relaxation runge kutta schemes for the nonlinear pde the reformulation is first discretized in the space direction by adopting the fourier pseudo spectral discretization which preserves the semi discrete quadratic conservation laws then the semi discrete system is integrated in the time direction using the explicit relaxation runge kutta method the obtained method can conserve different quadratic conservation laws to machine accuracy which improves the numerical stability during long time computations besides the proposed method keeps the same convergence rate of the standard runge kutta scheme when computed with time step size rescaling and gives a rate of convergence reduced by at most one when computed without step size rescaling numerical experiments for several odes and pdes are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis

residual free bubbles for embedded dirichlet problems abstract we examine a stabilized method employing residual free bubbles for enforcing dirichlet constraints on embedded finite element interfaces in particular we focus attention on problems where the underlying mesh is not explicitly fitted to the geometry of the interface the residual free bubble problem is derived for a simple case and extensions are discussed we show that under certain conditions stabilization only requires knowledge of the residual free bubble on the interface we then examine methods to approximate the residual free bubble a series of benchmark tests are used to demonstrate the accuracy and robustness of the method comparisons are made to nitsche s method employing an eigenvalue estimate for the global stability parameter particular emphasis is placed on the accuracy of flux evaluations on the interface finally we employ the method to simulate an evolving interface problem motivated by resin transfer molding

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

integrability properties of kahanʼs method we present several novel examples of integrable quadratic vector fields for which kahan s discretization method preserves integrability our examples include generalized suslov and ishii systems nambu systems riccati systems and the first painleve equation we also discuss how manin transformations arise in kahan discretizations of certain vector fields

fourier series direct method for variational problems a direct method for solving variational problems using fourier series is presented an operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations illustrative examples are also given

a wavelet based computational method for solving stochastic ito volterra integral equations this paper presents a computational method based on the chebyshev wavelets for solving stochastic ito volterra integral equations first a stochastic operational matrix for the chebyshev wavelets is presented and a general procedure for forming this matrix is given then the chebyshev wavelets basis along with this stochastic operational matrix are applied for solving stochastic ito volterra integral equations convergence and error analysis of the chebyshev wavelets basis are investigated to reveal the accuracy and efficiency of the proposed method some numerical examples are included

on methods for stabilizing constraints over enriched interfaces in elasticity summary enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non smooth problems internal features such as boundaries are represented in such methods by using discontinuous enrichment of the standard finite element basis within such frameworks however imposition of interface constraints and or constitutive relations can cause unexpected difficulties depending upon how relevant fields are interpolated on un gridded interfaces this work address the stabilized treatment of constraints in an enriched finite element context both the lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields we demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner in a bubble stabilized approach fine scale degrees of freedom are added over elements supporting the interface the variational form can be shown to have a similar form to a second approach we consider nitsche s method with the exception that the stabilization terms follow directly from the bubble functions in this work we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two dimensional elasticity we examine several benchmark problems in elasticity and show that only nitsche s method and the bubble stabilization approach produce stable traction fields over internal boundaries we also demonstrate a novel difference between the penalty method and nitsche s method in that the latter passes the patch test exactly regardless of the stabilization parameter s magnitude results for more complicated geometries and triple interface junctions are also presented copyright 2008 john wiley sons ltd

a weighted nitsche stabilized method for small sliding contact on frictional surfaces abstract we propose a weighted nitsche framework for small sliding frictional contact problems on three dimensional interfaces the proposed method inherits the advantages of both augmented lagrange multiplier and penalty methods while also addressing their shortcomings algorithmic details of the traction update and consistent linearization in the presence of nitsche terms are provided several benchmark numerical experiments are conducted and the results are compared with existing studies the results are encouraging and indicate accurate satisfaction of the non interpenetration constraint stable tractions and asymptotic quadratic convergence of the newton raphson method

the shifted boundary method for embedded domain computations part ii linear advection diffusion and incompressible navier stokes equations abstract we propose a new embedded finite element method for the linear advection diffusion equation and the laminar and turbulent incompressible navier stokes equations the proposed method belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates we include the full analysis of stability and convergence of the method in the linear advection diffusion equation and a battery of tests for the case of laminar and turbulent incompressible navier stokes equations we also discuss the conservation properties of the proposed method in all cases

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

some new a priori estimates for second order elliptic and parabolic interface problems abstract we present some new a priori estimates of the solutions to the second order elliptic and parabolic interface problems the novelty of these estimates lies in the explicit appearance of the discontinuous coefficients and the jumps of coefficients across the interface

a lagrange multiplier method for the finite element solution of elliptic interface problems using non matching meshes in this paper we propose a lagrange multiplier method for the finite element solution of multi domain elliptic partial differential equations using non matching meshes the interface lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes e g we will consider global polynomials as multiplier the ideas are illustrated using poisson s equation as a model and the proposed method is shown to be stable and optimally convergent numerical experiments demonstrating the theoretical results are also presented

liouville integrability and superintegrability of a generalized lotka volterra system and its kahan discretization we prove the liouville and superintegrability of a generalized lotka volterra system and its kahan discretization

geometric properties of kahan s method we show that kahan s discretization of quadratic vector fields is equivalent to a runge kutta method when the vector field is hamiltonian on either a symplectic vector space or a poisson vector space with constant poisson structure the map determined by this discretization has a conserved modified hamiltonian and an invariant measure a combination previously unknown amongst runge kutta methods applied to nonlinear vector fields this produces large classes of integrable rational mappings in two and three dimensions explaining some of the integrable cases that were previously known

nitsche s method for general boundary conditions we introduce a method for treating general boundary conditions in the finite element method generalizing an approach due to nitsche 1971 for approximating dirichlet boundary conditions we use poisson s equations as a model problem and prove a priori and a posteriori error estimates the method is also compared with the traditional galerkin method the theoretical results are verified numerically

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

an effective scheme for solving system of fractional volterra fredholm integro differential equations based on the muntz legendre wavelets abstract in this study a class of wavelet techniques is used for finding approximate solutions of systems of fractional integro differential volterra fredholm fidvf equations based on the muntz legendre wavelets mlw for the suggested method operational matrices of the riemann liouville fractional rlf integral and caputo fractional cf derivative operators are obtained and used for converting the system of the integral equations into a system of linear or nonlinear algebraic equations using the lipschitz s condition for multivariate functions and the fixed point theorem the existence and uniqueness of the solution are shown and also convergence stability and error bound of the solution in interval 0 1 are investigated in this work at the end three examples are indicated and the results of the proposed method are compared with the first and second kind wavelet chebyshev methods

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

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

a construction of a large family of commuting pairs of integrable symplectic birational four dimensional maps we give a construction of completely integrable four dimensional hamiltonian systems with cubic hamilton functions applying to the corresponding pairs of commuting quadratic hamiltonian vector fields the so called kahan hirota kimura discretization scheme we arrive at pairs of birational four dimensional maps we show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on r 4 and possess two independent integrals of motion which are perturbations of the original hamilton functions and which are in involution with respect to the perturbed symplectic structure thus these maps are completely integrable in the liouville arnold sense moreover under a suitable normalization of the original pairs of vector fields the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion

numerical solution of differential equations by using chebyshev wavelet operational matrix of integration in this paper we present operational matrix of integration omi of chebyshev wavelets basis and the product operation matrix pom of it some comparative examples are included to demonstrate the superiority of operational matrix of chebyshev wavelets to those of legendre wavelets

a wavelet based approach for vibration analysis of framed structures in this paper an explicit time integration scheme is proposed for structural vibration analysis by using wavelet functions initially the differential equation of vibration governing sdof single degree of freedom systems has been solved by wavelet operators and later the proposed approach has been generalized for mdof multi degrees of freedom systems for this purpose two different types of wavelet functions have been exemplified including complex chebyshev wavelet functions and simple haar wavelet functions in the proposed approach a straightforward formulation has been derived from the numerical approximation of response through the wavelet definition emphasizing on frequency domain approximation a simple step by step algorithm has been implemented and improved to calculate the response of mdof systems moreover stability and accuracy of results have been evaluated the effectiveness of the proposed approach is demonstrated using three examples compared with some of the existing numerical integration schemes such as family of newmark b wilson q and central difference method in all the procedures computation time involved has also been considered finally it is concluded that the vibration analysis of structures is improved by lesser computation time and high accuracy of proposed approach particularly in large scaled systems

a penalty free nonsymmetric nitsche type method for the weak imposition of boundary conditions in this paper we show that the nonsymmetric version of nitsche s method for the weak imposition of boundary conditions is stable without penalty term for nonconforming elements we prove the same result for the symmetric formulation as well we prove optimal h 1 error estimates and l 2 error estimates that are suboptimal with half an order in h both the pure diffusion and the convection diffusion problems are discussed

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

analysis of backward euler extended finite element discretization of parabolic problems with moving interfaces abstract numerical experiments performed by several authors repeatedly confirmed that the extended finite element method combined with classical time stepping schemes such as backward euler or crank nicolson provides optimal rates of convergence when applied to problems with moving interfaces in spite of the rapid spread of this discretization approach a complete space and time error analysis is not available yet the objective of this work is to contribute to closing the gap between observations and rigorous analysis more precisely we preform a thorough space and time error analysis of different variants of the extended finite element method combined with backward euler time advancing scheme when applied to problems with moving interfaces our results prove that the discretization error measured in the natural norms decreases with the expected rate when space and time discretization steps are refined we also show that the space and time approximation properties are fully decoupled as a consequence high order approximation schemes could be developed and analyzed within the theoretical framework proposed here numerical experiments are finally addressed for the verification of the algorithms besides the interest of providing rigorous error bounds we believe that a general theoretical framework is extremely helpful as a guide for further developing and refining extended finite element methods for free interface problems

discretization algorithm for fractional order integral by haar wavelet approximation abstract a discretization algorithm is proposed by haar wavelet approximation theory for the fractional order integral in this paper the integration time is divided into two parts one presents the effect of the past sampled data calculated by the iterative method and the other presents the effect of the recent sampled data at a fixed time interval calculated by the haar wavelet this method can reduce the amount of the stored data effectively and be applied to the design of discrete time fractional order pid controllers finally several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm

geometry of the kahan discretizations of planar quadratic hamiltonian systems kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow for a planar quadratic canonical hamiltonian vector

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

a stable lagrange multiplier space for stiff interface conditions within the extended finite element method this paper introduces a new algorithm to define a stable lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method in contrast to earlier approaches we do not work with an interior penalty formulation as e g for nitsche techniques but impose the constraints weakly in terms of lagrange multipliers roughly speaking a stable and optimal discrete lagrange multiplier space has to satisfy two criteria a best approximation property and a uniform inf sup condition owing to the fact that the interface does not match the edges of the mesh the choice of a good discrete lagrange multiplier space is not trivial here we propose a new algorithm for the local construction of the lagrange multiplier space and show that a uniform inf sup condition is satisfied a counterexample is also presented i e the inf sup constant depends on the mesh size and degenerates as it tends to zero numerical results in two dimensional confirm the theoretical ones copyright

on the connection between the stabilized lagrange multiplier and nitsche s methods we derive nitsche s method for the domain decomposition through the stabilized lagrange multiplier method taking material parameters carefully into account this derivation naturally introduces parameter weighted average flux and stabilizing terms to nitsche s method we show stability and a priori analyses in the mesh dependent norms for both the stabilized method and nitsche s method and discuss connections between the proposed methods

chebyshev wavelets approach for nonlinear systems of volterra integral equations in this paper a new approach for solving nonlinear systems of volterra integral equations has been proposed the method is based on chebyshev wavelets approximations the method is described and after that the error is analyzed at the end some examples are presented to illustrate the ability and simplify of the method and the results reveal the effectiveness of the technique

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

a coupled eulerian lagrangian extended finite element formulation for simulating large deformations in hyperelastic media with moving free boundaries we present a coupled eulerian lagrangian cel formulation aimed at modeling the moving interface of hyperelastic materials undergoing large to extreme deformations this formulation is based on an eulerian description of kinematics of deformable bodies together with an updated lagrangian formulation for the transport of the deformation gradient tensor the extended finite element method xfem is used to discretize the mechanical equilibrium and deformation gradient transport equations in a two phase domain a mixed interpolation scheme biquadratic for the velocity and bilinear for the deformation gradient is adopted to improve the accuracy of the numerical formulation the interface describing the deformed shape of the body is represented by the level set function and is evolved using the grid based particle method the performance of the scheme is explored in two dimensions in the compressible regime for an adequate spatial and temporal discretization our numerical results are in good agreement with theory and with numerical results from the traditional lagrangian formulation in abaqus the advantage of the proposed formulation is that material motion is not coupled with that of the mesh this eliminates the issues of mesh distortion and the need for remeshing associated with lagrangian formulations when bodies undergo very large distortions it is therefore well adapted to describe the motion of complex fluids and soft matter whose physical properties are intermediate between conventional liquids and solids

solving fredholm integro differential equations using reproducing kernel hilbert space method in this study the numerical solution of fredholm integro differential equation is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the initial condition of the problem is satisfied the exact solution ux is represented in the form of series in the space w 2 2 a b in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 2 a b some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear fredholm integro differential equations

theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear lane emden type equations abstract in this article a new method is generated to solve nonlinear lane emden type equations using legendre hermite and laguerre wavelets we are interested to note that these wavelets will give same solutions with good accuracy theorems on convergence analysis are stated and proved on the spaces which are created by legendre hermite and laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis the main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients a very high level of accuracy reflects the reliability of this scheme for such problems