explicit residual based a posteriori error estimation for finite element discretizations of the helmholtz equation computation of the constant and new measures of error estimator quality abstract this paper continues the study of an explicit residual based a posteriori error estimator developed for finite element discretizations of the helmholtz equation in the context of time harmonic exterior acoustics problems in previous papers 1 3 the error estimator was derived and subsequently applied to h adaptive computations of model problems in two dimensions in the present paper a methodology is established for computing error estimates these error estimates are then analyzed in detail the error estimates are made possible by computing the scaling constant which exists in error estimators of this type in our case this constant is difficult to obtain an algorithm for computing the constant is described it is noted that the error indicators used for adaptive computations do not require knowledge of this constant several measures are used to analyze the quality of the error estimates providing a complete description of the error estimator in the context of these problems we compute global effectivity indices and utilize the local element wise effectivity index in computing three additional measures the first of these is the robustness index the others are the modified robustness index and the distribution index which are introduced for the first time it is found that the global effectivity index varies with problem parameters in particular the wave number robustness indices also vary due to the local error definition and wave number values of the distribution index are much better however indicating the suitability of the error estimator in guiding efficient adaptive mesh refinement the analysis of the error estimator is carried out on relatively coarse meshes typical of those used in practical engineering computations

neumann boundary value problems for a time fractional diffusion wave equation in a half plane the time fractional diffusion wave equation with the caputo derivative of the order 0 a 2 is considered in a half plane two types of neumann boundary condition are examined the mathematical condition with the prescribed boundary value of the normal derivative and the physical one with the prescribed boundary value of the matter flux

fractional order legendre collocation method for solving fractional initial value problems in this paper we present a numerical algorithm for solving second order fractional initial value problems this numerical algorithm is based on a fractional legendre collocation spectral method the governing fractional differential equation is converted into a nonlinear system of algebraic equations the error analysis of the proposed numerical algorithm is presented comparisons with other numerical methods shows that the proposed algorithm is more accurate and simpler to implement several examples are discussed to illustrate the efficiency and accuracy of the present scheme

recovery procedures in error estimation and adaptivity part ii adaptivity in nonlinear problems of elasto plasticity behaviour the first part of this paper 1 discusses in some detail the use of recovery procedures for stress strain or gradients for efficient error estimator and the methods of arriving at adaptive h refinement in this paper we extend such procedure to deal with elasto plasticity and discuss in detail some special features of such computation

discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

a semi local spectral hp element solver for linear elasticity problems summary we develop an efficient semi local method for speeding up the solution of linear systems arising in spectral hp element discretization of the linear elasticity equations the main idea is to approximate the element wise residual distribution with a localization operator we introduce in this paper and subsequently solve the local linear system additionally we decouple the three directions of displacement in the localization operator hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver this approach is effective for both nodal and modal bases in the spectral hp element method but here we focus on the modal hierarchical basis in numerical tests we verify that there is no loss of accuracy in the semi local method and we obtain good parallel scalability and substantial speed up compared to the original formulation in particular our tests include both structure only and fluid structure interaction problems with the latter modeling a 3d patient specific brain aneurysm copyright 2014 john wiley sons ltd

strategies for computing goal oriented a posteriori error measures in non linear elasticity we investigate the characteristics and performance of goal oriented a posteriori error measures for a class of non linear elasticity models while restriction is made to small strain theory the chosen error measure of the displacement field can be global or local probing the chosen quantity in a specific spatial point the error is computable with the aid of the solution of a dual problem whose data depend on the error measure the main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution the chosen strategies are compared in terms of accuracy ease of implementation reliability and cost efficiency a well known numerical example the cook s membrane is used for the numerical evaluations

mathematical modeling of different types of instabilities in time fractional reaction diffusion systems abstract in this article we investigate possible scenarios of pattern formations in reaction diffusion systems with time fractional derivatives linear stability analysis is performed for different values of derivative orders results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations most attention is paid to two models a fractional order reaction diffusion system with bonhoeffer van der pol kinetics and to the brusselator model

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

modeling and computation of shakedown problems for nonlinear hardening materials abstract the paper is concerned with shakedown analysis of perfectly plastic nonlinear kinematic hardening linear unlimited kinematical hardening and cyclic kinematical hardening materials a 3 d overlay model being able to describe nonlinear kinematic hardening material behavior is reviewed and a static shakedown theorem is formulated which is shown to be an extension of the melan s static shakedown theorems for both perfectly plastic and linear unlimited kinematic hardening materials different numerical algorithms are presented which efficiently solve the nonlinear discretized optimization problems using the finite element method the influence of kinematic hardening on shakedown limits is discussed it is shown how the exactness of the numerical shakedown limits can efficiently be improved by using the adaptive mesh refinement a possibility for analytical determination of shakedown limits in case of alternating plasticity is demonstrated by a square plate with a central circular hole under biaxial uniform loading the shakedown behaviour of materials with cyclic kinematical hardening has been investigated by an incremental computation method the theoretical and numerical aspects discussed are illustrated by the examples given in the paper

pattern formation in a fractional reaction diffusion system abstract we investigate pattern formation in a fractional reaction diffusion system by the method of computer simulation of the model of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and space fractional derivatives we further compare the patterns obtained by computer simulation with those obtained by simulation of the similar system without fractional derivatives as a result we are able to show that nonlinearity plays the main role in structure formation and fractional derivative terms change the transient dynamics so when the order of time derivative increases and approaches the value of 1 5 the special structure formation switches to homogeneous oscillations in the case of space fractional derivatives the decrease of the order of these derivatives leads to more contrast dissipative structures the variational principle is used to find the approximate solution of such fractional reaction diffusion model in addition we provide a detailed analysis of the characteristic dissipative structures in the system under consideration

the p version of the finite element method compared to an adaptive h version for the deformation theory of plasticity a p version of the finite element method is applied to the deformation theory of plasticity and the results are compared to a state of the art adaptive h version it is demonstrated that even for nonlinear elliptic problems the p version is a very efficient discretization strategy

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

error estimates and automatic time step control for nonlinear parabolic problems i in this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem the method for time discretization is the so called discontinuous galerkin method based on using piecewise polynomials of degree q geqq 0 we consider in this note the case q 0 corresponding to a variant of the backward euler method we prove a new almost optimal error estimate and present a related new algorithm for automatic time step control this algorithm is very simple but yet is efficient and gives control of the global error

discrete mittag leffler kernel type fractional difference initial value problems and gronwall s inequality abstract in this article we studied the caputo and riemann liouville type discrete fractional difference initial value problems with discrete mittag leffler kernels the existence and uniqueness of the solution is proved by using banach contraction principle the linear type equations are used to prove new discrete fractional versions of the gronwall s inequality the nabla discrete laplace transform is used to obtain solution representations the proven gronwall s inequality under a new defined lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions numerical examples are discussed to demonstrate the reliability of the theoretical results

analysis of the zienkiewicz zhu a posteriori error estimator in the finite element method an a posteriori error estimator for finite element analysis proposed by zienkiewicz and zhu is analysed and shown to be effective and convergent in addition we analyse wider classes of estimators of which the zienkiewicz zhu estimator is a special case it is shown that some of these estimators will be asymptotically exact numerical evidence is presented supporting the analysis

trisections of 4 manifolds the study of n dimensional manifolds has seen great advances in the last half century in dimensions greater than four surgery theory has reduced classification to homotopy theory except when the fundamental group is nontrivial where serious algebraic issues remain in dimension 3 the proof by perelman of thurston s geometrization conjecture 1 allows an algorithmic classification of 3 manifolds the work of freedman 2 classifies topological 4 manifolds if the fundamental group is not too large also gauge theory in the hands of donaldson 3 has provided invariants leading to proofs that some topological 4 manifolds have no smooth structure that many compact 4 manifolds have countably many smooth structures and that many noncompact 4 manifolds in particular 4d euclidean space r 4 have uncountably many however the gauge theory invariants run into trouble with small 4 manifolds such as those with the same homology groups as the 4d sphere s 4 in particular the smooth 4d poincare conjecture the last remaining case of that hallowed conjecture is still open in higher dimensions the smooth poincare conjecture is sometimes true in the following sense in dimensions 3 5 6 12 and 61 a homotopy sphere is diffeomorphic to the standard one and in all other known cases there are increasingly many exotic smooth structures on the topological sphere however it is possible that there may be more high dimensional cases with no exotic spheres the gauge theory invariants are very good at distinguishing smooth 4 manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic what is missing is the equivalent of the higher dimensional s cobordism theorem a key to the successes in higher dimensions the s cobordism theorem states that if m 0 m and m 1 m are the two boundary components of an m 1 dimensional manifold w and if 1 1email kirby at math berkeley edu 1 xref corresp 1 1

pq adaptive solid finite elements for three dimensional plates and shells this article presents a pq adaptive finite element procedure for three dimensional computation of thin walled structures like plates and shells the approach is based on a hexahedral element with high order shape functions the polynomial degree of the shape functions in the three spatial directions of the element is adjusted by means of an adaptive procedure to this end a hierarchical error indicator is applied to automatically control the polynomial degree q in thickness and p in the in plane direction of plate and shell like structures the efficiency and reliability of the proposed adaptive method is demonstrated using two numerical examples

fast tensor product solvers for optimization problems with fractional differential equations as constraints fractional differential equations have recently received much attention within computational mathematics and applied science and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms an optimization problem with constraints given by fractional differential equations is considered which in its discretized form leads to a high dimensional tensor equation to reduce the computation time and storage the solution is sought in the tensor train format we compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned krylov solvers or tailored alternating schemes the competitiveness of these approaches is presented using several examples with constant and variable coefficients

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

transcendental bernstein series for solving nonlinear variable order fractional optimal control problems abstract this paper deals with finding an approximate solution of variable order fractional optimal control problems v focps based on lagrange multiplier optimization technique in combination with the new operational matrix of variable order vo fractional derivatives to carry out the proposed approach we firstly develop the well known bernstein polynomials to the new series of functions namely transcendental bernstein series tbs then we implement these basis functions to approximate the solutions of v focps in fact the series expansion in tbs with unknown free coefficients and control parameters is the novel idea for solving the fractional systems with less terms of approximation the convergence analysis of our proposed method will be guaranteed by proving a new theorem for the tbs to investigate the practical computational efficiency and accuracy of the presented method some numerical examples are provided the experimental results confirm the applicability of the method and a good agreement between the approximate and exact solutions

generalized taylor s formula abstract in this paper a new generalized taylor s formula of the kind f x j 0 n a j x a j r n x where a j r x a 0 1 is established such an expression is precisely the classical taylor s formula in case of 1 in addition detailed expressions for r n x and a j that involving caputo fractional derivatives are well constructed some applications including approximation of functions and solutions of fractional differential equations are also given

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

on some recent adventures into improved finite element cfd methods for convective transport abstract the quest continues for computational fluid dynamics cfd algorithms that are accurate and efficient for convection dominated applications including shocks travelling fronts and wall layers the boundary value optimal galerkin weak statement invariably requires manipulation either in the test space or in an augmented form for the conservation law system to handle the disruptive character introduced by the discretized first order convection term an incredible variety of methodologies have been derived and examined to address this issue in particular seeking achievement of accurate and monotone discrete approximate solutions in an efficient implementation the utk cfd research group continues its search on examining the breadth of approaches leading to development of a consistent encompassing theoretical statement exhibiting quality performance included herein are adventures into generalized taylor series lax wendroff methods characteristic euler flux resolutions sub grid embedded high degree lagrange bases with static condensation and assembled stencil fourier analysis optimization for finite element weak statement implementations for appropriate model problems including steady convection diffusion and pure unsteady convection and benchmark navier stokes definitions recent advances have lead to candidate accurate monotone methods with linear basis efficiency this contribution highlights the theoretical developments and presents quantitative documentation of achievable high quality solutions

numerical analysis of the primal problem of elastoplasticity with hardening the finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time step in an elastoplastic evolution problem in this paper we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of hencky plasticity the finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp piecewise constant trial functions an a priori error estimate is established for the fully discrete method which shows linear convergence as the mesh size tends to zero provided the exact displacement field u is smooth near the boundary of the plastic domain which is unknown a priori it is most likely that u is non smooth in this situation automatic mesh refinement strategies are believed to improve the quality of the finite element approximation we suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate this estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above the constants depend on the hardening involved and become larger for decreasing hardening

a posteriori error estimation for variational problems with uniformly convex functionals the objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations we consider variational problems of the form inf f v g v where f v r is a convex lower semicontinuous functional g y r is a uniformly convex functional v and y are reflexive banach spaces and v y is a bounded linear operator we show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and thus can be justified via the duality theory

front propagation in hyperbolic fractional reaction diffusion equations from the continuous time random walk scheme and assuming a l evy waiting time distribution typical of subdiffusive transport processes we study a hyperbolic reaction diffusion equation involving time fractional derivatives the linear speed selection of wave fronts in this equation is analyzed when the reaction diffusion dimensionless number and the fractional index satisfy a certain condition we find fronts exhibiting an unphysical behavior they travel faster in the subdiffusive than in the diffusive regime

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

the p version of the finite element method for three dimensional curved thin walled structures in this paper we present an implementation of a three dimensional p version for structural problems of solids with almost arbitrarily curved surfaces applying the blending function method complex structures can often be modelled by a few p elements being the basis for a higher order approximation numerical examples will demonstrate that the p version with anisotropic ansatz spaces allows to predict the structural behaviour of three dimensional plates and shells with approximately the same amount of degrees of freedom as in the two dimensional case yet significantly more accurate due to the three dimensional model furthermore it is advantageous to compute complex structures exclusively with three dimensional discretizations as no special elements are needed to model the transition from dimensionally reduced formulations like plates or shells to fully three dimensional solid elements using the p version with anisotropic ansatz spaces the whole structure can be efficiently discretized with solid elements even if the aspect ratio of the elements becomes very large copyright 2001 john wiley sons ltd

adaptive multiresolution antenna modeling using hierarchical mixed order tangential vector finite elements hierarchical mixed order tangential vector finite elements tvfes for tetrahedra are attractive for accurate and efficient finite element or hybrid finite element boundary integral fe bi simulation of complicated electromagnetic problems they provide versatility in the geometrical modeling of physical structures guarantee solutions free of spurious modes and allow a local increase of resolution by combination of mixed order tvfes of different orders within a computational domain regions with higher order mixed order tvfes can be selected a priori or be found adaptively via methods for a posteriori error estimation or indication this paper demonstrates the merits of various explicit residual methods for a posteriori error indication via adaptive refinement of fe bi solutions of electromagnetic radiation problems using hierarchical mixed order tvfes for tetrahedra

adaptive finite element methods for hydrodynamic lubrication with cavitation we present an adaptive finite element method for a cavitation model based on reynolds equation a posteriori error estimates and adaptive algorithms are discussed and numerical examples illustrating the theory are supplied

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

diffusive effects on dispersion in excitable media perturbation and numerical methods are used to study traveling and standing wave solutions for a system of reaction diffusion equations modeled after excitable media and the effect of variation of the diffusion coefficients it is shown that traveling waves and standing waves can coexist and that as a certain diffusion coefficient is varied a branch of traveling waves can bifurcate from a branch of standing waves providing a transition between only traveling solutions in one limit and only standing waves in the other limit key words traveling waves standing waves bifurcation dispersion reaction diffusion equations singular perturbation

automatic mesh refinement in limit analysis a strategy for automatic mesh refinement in limit analysis is combined with a recently developed computational method in the absence of estimates of the local error the strategy can be based on the deformations and on slack in the yield condition the approach is tested on standard problems in plane strain including the classical punch problem very accurate results are obtained with the use of moderate computational power

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

mixed spectral hp element formulation for nonlinear elasticity we present a mixed formulation for high order spectral hp element methods and investigate its stability in large strain analysis of nearly incompressible solids specifically we employ the displacement pressure formulation and use a pressure projection method along with general jacobi polynomial bases to derive the discrete equations we first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements subsequently we investigate numerically the validity of ellipticity and inf sup conditions for various test problems including cases with large deformation we find that the inf sup condition is satisfied even when the pressure interpolation order op is equal to the displacement order ou but to resolve the locking free behavior it is required that op ou 1 and this choice also gives the best convergence rate in large deformation analysis surprisingly the results are more stable for ou op ou k because the ellipticity in kernel is better satisfied with higher k as ou increases compared to the displacement only formulation the mixed formulation seems to have superior accuracy and it is overall more efficient especially for problems with high bulk modulus

a loop theorem for duality spaces and fibred ribbon knots in this paper we prove a version of the loop theorem for surfaces in the boundary of a 3 dimensional duality space i e a space which resembles a 3manifold only in that it satisfies the appropriate form of poincar6 lefschetz duality over some field of untwisted coefficients our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance and as the main application of our theorem we show that if a fibred knot in the 3 sphere is a ribbon knot then its monodromy extends over a handlebody we approach the loop theorem via the study of planar coverings of a surface as in the original paper of papakyriakopoulos 11 and the subsequent work of maskit 9 w contains a simple geometric treatment of these matters the main result of w is that a duality space actually satisfies duality with twisted coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal in w 4 the results of w167 2 and 3 together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring are used to prove the loop theorem for 3 dimensional duality spaces for the reader s convenience a proof of the algebraic temma is included as an appendix w contains the application to fibred ribbon knots mentioned above in w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4 manifolds in w we apply our methods to another aspect of knot concordance and show that for any concordance with a rationally anisotropic fibred knot see 7 at one end the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups for torus knots this question was raised by scharlemann 14

on the adaptive finite element method of steady state rolling contact for hyperelasticity in finite deformations in this paper we present an adaptive finite element method for steady state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with coulomb friction a general formulation of rolling contact geometry is derived from the point of view of differential geometry solvability conditions for the rolling contact problems are discussed we use newton s method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem we provide a numerical example to illustrate the method

mathematical modeling of time fractional reaction diffusion systems we study a fractional reaction diffusion system with two types of variables activator and inhibitor the interactions between components are modeled by cubical nonlinearity linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives it is shown that by combining the fractional derivatives index with the ratio of characteristic times it is possible to find the marginal value of the index where the oscillatory instability arises the increase of the value of fractional derivative index leads to the time periodic solutions the domains of existing periodic solutions for different parameters of the problem are obtained a computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented for the fractional reaction diffusion systems it is established that there exists a set of stable spatio temporal structures of the one dimensional system under the neumann and periodic boundary conditions the characteristic features of these solutions consist of the transformation of the steady state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system

the p version of the finite element method in incremental elasto plastic analysis summary whereas the higher order versions of the finite element method p and hp versions are fairly well established as highly efficient methods for monitoring and controlling the discretization error in linear problems little has been done to exploit their benefits in elasto plastic structural analysis in this paper we discuss which aspects of incremental elasto plastic finite element analysis are particularly amenable to improvements by the p version these theoretical considerations are supported by several numerical experiments first we study an example for which an analytical solution is available it is demonstrated that the p version performs very well even in cycles of elasto plastic loading and unloading not only as compared with the traditional h version but also with respect to the exact solution finally an example of considerable practical importancethe analysis of a cold working lug is presented which demonstrates how the modelling tools offered by higher order finite element techniques can contribute to an improved approximation of practical problems

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

on error estimates and adaptivity in elastoplastic solids applications to the numerical simulation of strain localization in classical and cosserat continua the a posteriori error estimates based on the post processing approach are introduced for elastoplastic solids the standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy this is known to represent a natural metric for the class of elastoplastic problems of evolution in addition the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates a posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem as a model problem the constitutive equations describing a generalization of standard j2 elastoplasticity within the cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post instability region the proposed a posteriori error estimates are appropriately modified to account for the cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems superior behaviour of the cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material resulting in convergent solutions with an h refinement and almost uniform error distribution in all considered error norms

constitutive relation error estimator for transient finite element analysis the concept of error in the constitutive relation allows defining a consistent error measurement for dynamic finite element simulations a solution which satisfies exactly the kinematic constraints the initial conditions and the equilibrium equation is built from the solution to the discretised problem this construction utilizes as best as possible the properties of the integration scheme used to obtain a satisfactory effectivity index the quality of the numerical solution is assessed by the way in which the constitutive relations are satisfied results from various tests conducted on a one degree of freedom problem and on a 1d example display the estimator s good behaviour the effectivity index remains at a reasonable value and on the basis of the first test a comparison of the error in the constitutive relation with different estimators drawn from the literature is presented

characterizing dehn surgeries on links via trisections we summarize and expand known connections between the study of dehn surgery on links and the study of trisections of closed smooth 4 manifolds in particular we propose a program in which trisections could be used to disprove the generalized property r conjecture including a process that converts the potential counterexamples of gompf scharlemann and thompson into genus four trisections of the standard 4 sphere that are unlikely to be standard we also give an analog of the casson gordon rectangle condition for trisections that obstructs reducibility of a given trisection

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

a posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem the reciprocal theorem of betti and rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities including local variables like single displacements and stresses the proposed error estimator is evaluated solving the set of equations for an additional right hand side simply applying the energy norm error estimators two times furthermore a general h adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables the efficiency of the current approach is demonstrated for plate and shell examples

an explicit a posteriori error indicator for electromagnetic finite element boundary integral analysis within the finite element method fem there is always an error associated with the solution quantifying this error is a very important aspect of fe analysis since it can increase confidence in a solution and can be used to drive adaptive refinements of the discretization optimizing the quality of the solution relative to the number of degrees of freedom and in turn reducing computational cost this paper presents an a posteriori error indicator for time harmonic electromagnetic analysis in three dimensions when using a galerkin fe approach with curl conforming elements and including a nonlocal boundary integral boundary condition as well as a dominant mode port boundary condition the error indicator is residual based of an explicit nature and constitutes a bound on an error energy measure the error indicator is evaluated by presenting results relating to common microwave engineering applications which shows that it can be used successfully to estimate the relative error distribution and that it can successfully serve within an adaptive algorithm

a kronecker product splitting preconditioner for two dimensional space fractional diffusion equations abstract we study preconditioned iterative methods for the linear system arising in the numerical discretization of a two dimensional space fractional diffusion equation our approach is based on a formulation of the discrete problem that is shown to be the sum of two kronecker products by making use of an alternating kronecker product splitting iteration technique we establish a class of fixed point iteration methods theoretical analysis shows that the new method converges to the unique solution of the linear system moreover the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real the basic iteration is accelerated by a krylov subspace method like gmres the corresponding preconditioner is in a form of a kronecker product structure and requires at each iteration the solution of a set of discrete one dimensional fractional diffusion equations we use structure preserving approximations to the discrete one dimensional fractional diffusion operators in the action of the preconditioning matrix numerical examples are presented to illustrate the effectiveness of this approach

existence of solutions of abstract fractional integrodifferential equations of sobolev type this paper deals with the study of existence of solutions of nonlinear fractional integrodifferential equations of sobolev type with nonlocal condition in banach spaces the results are obtained by using resolvent operators fractional calculus and fixed point technique an example is provided to illustrate the theory

constitutive relation errors for f e analysis considering visco plasticity and damage a posteriori error estimators based on constitutive relation residuals are developed to measure the quality of finite element computations of structures which exhibit plastic viscoplastic behaviour with softening all the classical error sources involved in the computation over the studied time interval are taken into account the space discretization the mesh the time discretization and the iterative technique used to solve the non linear discrete problem copyright 2001 john wiley sons ltd

efficiency of a posteriori bem error estimates for first kind integral equations on quasi uniform meshes in the numerical treatment of integral equations of the first kind using boundary element methods bem the author and e p stephan have derived a posteriori error estimates as tools for both reliable computation and self adaptive mesh refinement so far efficiency of those a posteriori error estimates has been indicated by numerical examples in model situations only this work affirms efficiency by proving the reverse inequality based on best approximation on inverse inequalities and on stability of the discretization and complementary to our previous work an abstract approach yields a converse estimate this estimate proves efficiency of an a posteriori error estimate in the bem on quasi uniform meshes for symm s integral equation for a hypersingular equation and for a transmission problem

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

stability of the static spike autosolitons in the gray scott model we performed an asymptotic linear stability analysis of the static spike autosolitons ass self sustained solitary inhomogeneous states in the gray scott model of an autocatalytic chemical reaction we found that in one dimension these ass destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough in higher dimensions the one dimensional static spike ass are always unstable with respect to corrugation and wriggling the higher dimensional radially symmetric static spike ass may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow

approximate solutions of nonlinear fractional equations fractional exponential that are invariant under fractional derivatives elementary and special fractional functions are introduced approximate solutions to fractional burgers equation by using the homotopy perturbation method are obtained furthermore real integral representations for some h functions are found that may be very helpful in numerical computations

a posteriori error estimators in the finite element method in the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method this aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations a class of abstract estimators for finite elements of orderp 1 in re n n 2 3 based on exploiting the superconvergence phenomenon are analyzed

variational representation of the projection dynamics and random motion of highly dissipative systems we propose the method of investigation of highly dissipative systems which are based on the approximation of the attractor by some manifold the projection dynamic equations for the general form of such manifold of the dissipative dynamic system are obtained variational principles for the projection dynamics are considered on the basis of the projection dynamic equations we investigate the influence of the random forces on the behaviour of the system

mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard fem analysis of 3 d cohesive fractures summary a robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures this class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small sized critical regions the proposed approach is based on the division of a simulation into a suitable number of sub simulations where adaptive mesh refinement is performed only once based on refinement window s around crack front process zone s the initialization of newton raphson nonlinear iterations at the start of each sub simulation is accomplished by solving a linear problem based on a secant stiffness rather than a volume mapping of nonlinear solutions between meshes the secant stiffness is evaluated using material state information stored read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework moreover a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software data transfer between sub simulations is not required in the simplified strategy the computational efficiency accuracy and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3 d copyright 2016 john wiley sons ltd

a robust and accurate led bgk solver on unstructured adaptive meshes starting from the bgk model of the boltzmann equation we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes the proposed numerical approach is composed of two steps an initial reconstruction step and a gas evolution step in the initial reconstruction step an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge in the gas evolution step the local integral solution of the bgk model is used to compute numerical fluxes at a cell interface this approach provides an alternative to riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in godunov type schemes a classich refinement adaptive procedure is implemented to increase the spatial resolution of high speed unsteady flow characteristics such as shock waves contact discontinuities or expansion waves with minimal computational costs and memory overheads this procedure involves mesh enrichment coarsening steps to either insert nodes on an edge center in high gradient regions or delete nodes in over resolved regions numerical results of several test cases for unsteady compressible inviscid flows are presented in order to verify the accuracy and robustness of the current numerical approach the computed results are compared with analytical solutions experimental data the results of structured mesh calculations and the results obtained by widely used flux splitting methods

a posteriori error estimation for elasto plastic problems based on duality theory abstract in this paper we introduce a new approach to a posteriori error estimation for elasto plastic problems based on the duality theory of the calculus of variations we show that in spite of the prevailing view duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts the error in the constitutive law and the error in the equilibrium equations this representation is both physically meaningful and computationally important the duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the galerkin orthogonality condition in particular they encompass the familiar smoothening or gradient averaging techniques commonly used in practice we prove that the duality error estimators are equivalent to the error when the approximate solution is the exact solution of the discretised problem moreover in the case of linear elasticity the known residual type error estimators can be obtained from the duality error estimators numerical results for a model elasto plastic problem show the accuracy of the duality error estimators

p fem for finite deformation powder compaction the simulation of powder compaction problems die compaction and cold isostatic pressing is considered herein by an implicit high order p version finite element method in this class of problems use is made of a finite strain viscoplasticity model with evolution equations for internal variables developed for the highly compressible behavior in powder compaction processes the classical approach of implicit finite elements applies the combination of backward euler integration scheme and the multilevel newton algorithm to solve the system of differential algebraic equations resulting from the space discretized weak formulation by means of p version finite elements this approach requires on gauss point level a robust stress algorithm the challenging investigations are the incorporation of the applied highly non linear viscoplasticity model into a p version finite element formulation using follower load applications several axisymmetric numerical examples show the feasibility and good performance of this p version approach

adaptive streamline diffusion methods for compressible flow using conservation variables we consider the streamline diffusion finite element method applied to compressible flow using conservation variables we propose some adaptive algorithms and present related numerical results

iterative solutions for a coupled system of fractional differential integral equations with two point boundary conditions abstract in this paper we deal with a coupled system of fractional differential integral equations with two point boundary conditions the maximal and minimal solutions are obtained by using the monotone iterative technique combined with the method of upper and lower solutions meantime the error estimate of the solutions are given in the end some examples are given to illustrate the results

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

fractional diffusion equations coupled by reaction terms we investigate the behavior for a set of fractional reaction diffusion equations that extend the usual ones by the presence of spatial fractional derivatives of distributed order in the diffusive term these equations are coupled via the reaction terms which may represent reversible or irreversible processes for these equations we find exact solutions and show that the spreading of the distributions is asymptotically governed by the same the long tailed distribution furthermore we observe that the coupling introduced by reaction terms creates an interplay between different diffusive regimes leading us to a rich class of behaviors related to anomalous diffusion

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

a framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition part ii computational aspects computational aspects of the formulation of elastoplasticity proposed in part i of this work employing a multiplicative decomposition and based on the principle of maximum plastic dissipation are examined in detail in particular the implementation of the proposed hyperelastic j2 flow theory is shown to reduce to a trivial modification of the classical radial return algorithm which is amenable to exact linearization this results in a closed form expression for the consistent elastoplastic tangent moduli the algorithmic treatment of the general case is considered subsequently it is emphasized that objective rates and incrementally objective algorithms play no role in the present approach a number of numerical experiments are presented that illustrate the excellent performance of the proposed formulation

a model adaptive hanging node concept based on a new non linear solid shell formulation in this contribution we propose an innovative approach to automatically refine shell finite element meshes in thickness direction the procedure employs a new eight node solid shell element formulation based on the concept of reduced integration with hourglass stabilization apart from the important advantages of computational efficiency and robustness this element formulation offers the possibility to work with a variable mesh density in thickness direction in the present paper we suggest an adaptive approach using the hanging node concept to avoid additional mesh distortion among different stress and strain based error criteria the warping of the cross section and the residual error in the solution of the strong form of the balance of linear momentum have been proven to be suitable error measures the former represents the deviation of the realistically non linear dependence of the characteristic strain coefficient from its corresponding linear distribution applications show that the mesh refines as expected in loading domains in the neighbourhood of bearings or regions with strongly accumulating inelastic deformation on the other hand the discretization remains rather coarse in uncritical parts of the structure which decreases the computational effort noticeably

finite difference spectral approximation for a time space fractional equation on two and three space dimensions abstract we consider the numerical method for solving a time space fractional equation whose solution has a weak initial singularity l 1 scheme on graded mesh is used to capture the initial weak singularity it is shown that the method can recover order 2 convergence by choosing some appropriate grading parameters where 0 1 is the order of temporal caputo fractional derivative a fully discrete scheme is proposed by combining spectral approximation for the discretization of space in 2d and 3d cases stability and convergence of the fully discrete scheme are rigorously established numerical results are presented to show the sharpness of the error estimate

a p version finite element approach for two and three dimensional problems of the j2 flow theory with non linear isotropic hardening in this paper an implementation of a two and three dimensional p version approach to the j 2 flow theory with non linear isotropic hardening for small displacements and small strains is presented based on higher order quadrilateral and hexahedral element formulations a newton raphson iteration scheme combined with a radial return algorithm is applied to find approximate solutions for the underlying physically non linear model problem curved boundaries are taken care of with the blending function method allowing an accurate representation of geometry with only a few p elements numerical examples demonstrate that the p version supplies efficient and accurate approximations to this class of physically non linear problems

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner