stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

american fractional lookback options valuation and premium decomposition this paper deals with valuation and premium decomposition of american fractional lookback options written on dividend paying assets for which exact formulas are unknown except for the perpetual case via a pde approach we derive laplace transforms of the values of lookback call and put options which can be decomposed into the associated european values plus the early exercise premiums using abelian theorems of laplace transforms we characterize asymptotic behaviors of the early exercise boundaries at a time to close to expiration and at infinite time to expiration based on the gaver stehfest inversion method combined with the newton method we develop a fast and accurate algorithm for computing both the option value and the early exercise boundary numerical analysis reveals some notable features of the american fractional lookback options

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

valuing american floating strike lookback option and neumann problem for inhomogeneous black scholes equation abstract this paper presents our study of american floating strike lookback options written on dividend paying assets the valuation of these options can be mathematically formulated as a free boundary inhomogeneous black scholes pde with a neumann boundary condition which we by using a mellin transform convert into a relatively simple ordinary differential equation with dirichlet boundary conditions we then use these results to derive an integral equation that can be used to calculate the price of american floating strike lookback options in addition we also used mellin transforms to derive the closed form of the perpetual case

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

prediction of apparent properties with uncertain material parameters using high order fictitious domain methods and pgd model reduction this contribution presents a numerical strategy to evaluate the effective properties of image based microstructures in the case of random material properties the method relies on three points i a high order fictitious domain method ii an accurate spectral stochastic model and iii an efficient model reduction method based on the proper generalized decomposition in order to decrease the computational cost introduced by the stochastic model a feedback procedure is proposed for an automatic estimation of the random effective properties with a given confidence numerical verifications highlight the convergence properties of the method for both deterministic and stochastic models the method is finally applied to a real 3d bone microstructure where the empirical probability density function of the effective behaviour could be obtained

investigation of the convergence of the mixed displacement pressure formulation for three dimensional poroelastic materials using hierarchical elements recently a mixed pressure displacement u p formulation based on biot s poroelasticity equations has been presented for porous materials this model leads to a reduction of the number of degrees of freedom required for the modeling of three dimensional porous media in comparison to classical displacement displacement u u formulations in this paper an extension of the u p formulation based on hierarchical elements is presented first a variant of the weak integral form of the u p formulation is presented and its numerical implementation using hierarchical elements is detailed together with the application of boundary and loading conditions numerical results are presented to show the accuracy and performance of the present approach in particular the importance of correctly capturing the coupling effects between the two phases is highlighted

theh p version of the finite element method for elliptic equations of order2m the problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data it has been shown in 3 4 5 that the solutions of the second order and fourth order equations belong to the spacesb 1 where the weighted sobolev norms of thek th derivatives are bounded bycd k l k l k l l 2 wherec andd are constants independent ofk in this case theh p version of the finite element method leads to an exponential rate of convergence measured in the energy norm 6 12 13 theh p version was implemented in the code probe1 18 and has been very successfully used in the industry we will discuss in this paper the generalization of these results for problems of order2m we will show also that the exponential rate can be achieved if the exact solution belongs to the spacesb 1 where the weighted sobolev norm of thek th derivatives is bounded bycd k l k l k l m 1 c andd are independent ofk in addition if the data is piecewise analytic then in fact the exact solution belongs to the spacesb m 1 problems of this type are related obviously to many engineering problems such as problems of plates and shells and are also important in connection with well known locking problems

eigenvalue convergence in the finite element method in static force deflection applications of the finite element method convergence rates for the p version in which the polynomial degree of element interpolation functions is increased while the mesh remains fixed are superior to those for the h version in which the element degree remains fixed while the mesh is refined so that element size approaches zero in structural dynamics applications one does not seek to approximate a single solution as in static applications but seeks estimates for a number of the lower system eigenvalues this paper identifies factors responsible for poorer accuracy in higher computed eigenvalues in addition it explains why the p version of the finite element method can be expected to exhibit significantly better eigenvalue convergence than the h version numerical examples demonstrate the superiority of the p version over the h version they also show the effects of various mechanisms limiting eigenvalue convergence

a conforming mixed finite element method for the pure traction problem of linear elasticity a conforming mixed finite element method using the method of reduced integration is developed for the two dimensional pure traction boundary value problem of linear elasticity the convergence is uniform as the material becomes nearly incompressible

a comment on the effectiveness of uniform finite element refinements on presente des resultats numeriques sur l utilisation d approches de raffinement h pour la resolution de problemes d elasticite avec singularites et on les compare avec des resultats recents bases sur l approche de raffinement p

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

non uniform dependence for the periodic higher dimensional camassa holm equations abstract in this paper we investigate the dependence on initial data of solutions to the higher dimensional camassa holm equations with periodic boundary condition in besov spaces we show that when s 1 d 2 d 2 and 1 r the solution map is not uniformly continuous from b 2 r s t d into c 0 t b 2 r s t d for r or from b 2 s t d into l 0 t b 2 s t d for r

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

ill posedness of the camassa holm and related equations in the critical space abstract we prove norm inflation and hence ill posedness for a class of shallow water wave equations such as the camassa holm equation degasperis procesi equation and novikov equation etc in the critical sobolev space h 3 2 and even in the besov space b p r 1 1 p for p 1 r 1 our results cover both real line and torus cases only real line case for novikov solving an open problem left in the previous works 5 14 16

first order system least squares for linear elasticity numerical results two first order system least squares fosls methods based on l2 norms are applied to various boundary value problems of planar linear elasticity both use finite element discretization and multigrid solution methods they are two stage algorithms that solve first for the displacement flux variable the gradient of displacement which easily yields the deformation and stress variables then for the displacement variable itself as a complement to a companion theoretical paper this paper focuses on numerical results including finite element accuracy and multigrid convergence estimates that confirm uniform optimal performance even as the material tends to the incompressible limit

a posteriori error indicators for the p version of the finite element method the existence of local a posteriori error indicators for the p version of the finite element method is demonstrated through numerical examples it is shown that it is possible to construct reliable error indicators from the residuums and tractions which have the same rate of convergence as the strain energy the error indicators contribution of an element can be estimated by considering only the element and its immediate neighbors the optimal sequence of p distributions can be closely followed by the indicators even when there are only a very few elements the elements vary greatly in size the polynomial orders vary the mesh grading is poor and the smoothness indices and relative errors vary between wide limits

a variationally consistent mesh adaptation method for triangular elements in explicit lagrangian dynamics in this paper a variational formulation for mesh adaptation procedures involving local mesh changes for triangular meshes is presented such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense they also greatly simplify the projection of field variables from the old to the new meshes crucially the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained int j numer meth engng 2000 49 1295 1325 several examples in 2 d showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed copyright 2009 john wiley sons ltd

numerical p version refinement studies for the regularized stress bem in the development of the boundary element method bem and the finite element method fem researchers have typically selected similar basis functions that is both methods typically employ low order interpolations such as piece wise linear or piece wise quadratic and rely on h version refinement to increase accuracy as required in the case of the fem the decision to use low order elements is made for computational efficiency as an attractive compromise between local modeling accuracy and sparseness of the resulting linear system however in many bem formulations low order elements may be the only practical choice given the complexity of using analytic integration formulae in conjunction with special integral interpretations unlike their efficient use in the fem fine meshes of low order elements in the bem are highly inefficient from a computational standpoint given the dense nature of bem systems moreover owing to singularities in the kernel functions the bem should be expected to benefit more so than the fem from very high levels of local accuracy through the use of regularized algorithms which only require numerical integration p version refinement in the bem is easily extended to include any set of basis functions with no significant increase in programming complexity numerical results show that by using interpolations as high as 12th and 16th order one can expect reductions in error by as many as five orders of magnitude over comparable algorithms based on similar system size for two dimensional problems it is also shown that for a given level of error one can expect reductions in system size by an order of magnitude thus leading to a reduction in computational expense for conventional algorithms by three orders of magnitude copyright 2003 john wiley sons ltd

computational error estimates and adaptive processes for some nonlinear structural problems abstract in earlier papers the authors introduced and analyzed the calculation of reliable a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them while most of this work concerned linear problems this paper is intended to show that the same approaches remain also highly effective in the nonlinear case as a model problem a planar elastic rod is considered involving both material as well as geometrical nonlinearities but as far as possible the discussion is kept general for nonlinear problems of this type interest centers on an analysis of the shape and features of the equilibrium surface after some general remarks about such surfaces and the related computational problems a general continuation process is summarized and some of its extensions for determining limit points and tracing stability paths are discussed then estimators are introduced for the error along the solution paths and of the computed critical values experimental results for the model problem show the effectivity of these estimators and of the adaptive procedures based upon them then some aspects relating to the occurrence and identification of spurious solutions are discussed in order to illustrate desirable generalizations of these results the paper ends with an outline of some results for linear problems of the type that should be achievable also in the nonlinear case

a method of grid optimization for finite element methods a computationally useful criterion for grid optimization is derived based on a measure of the interpolation error associated with the finite element model the result is intended to be used to improve the quality of finite element solutions by changing the location of the nodes within a fixed number of degrees of freedom examples of the application of the criterion are provided the finite element method has become an important tool of engineering analysis and design much of the popularity of the method is due to the freedom that it allows in the construction of the discretized model it has long been realized however that this freedom must be carefully exercised since the quality of the finite element solution greatly depends on how the discretization is performed several methods have been proposed to improve the discretized model in an iterative manner one such method is presented in this paper a natural way of improving the quality of finite element solutions is to increase the number of degrees of freedom the process is normally performed after an initial solution is already available several schemes have been devised to introduce the new degrees of freedom in a selective manner in order to produce the greatest possible improvement of the previous solution this calls for the definition of criteria to identify the regions of the domain where the finite element approximation is poorer the new degrees of freedom are added in these regions by either increasing the order of polynomial approximation inside elements the so called p method or by subdivision of elements or h method the process is continued until a specified accuracy is achieved the quality of the finite element solution may be improved also by optimizing the disposition of the nodes analysts often rely on their experience to construct grids that make an efficient use of the available degrees of freedom it is also possible to improve the quality of existing meshes iteratively using predefined guidelines for the redistribution of the nodes this paper considers the development of such guidelines

on the rates of convergence of the finite element method abstract the rate of convergence of the finite element method is a function of the strategy by which the number of degrees of freedom are increased alternative strategies are examined in the light of recent theoretical results and computational experience author

an h p multigrid method for finite element analysis the finite element method generates solutions to partial differential equations by minimizing a strain energy based functional strain energy based techniques for adaptive mesh refinements are not always effective however the adaptive refinement technique proposed in this paper uses strain energy but also incorporates advantages from the h and p finite element methods the multigrid method and a delaunay based mesh generation method the refinement technique converged rapidly and was numerically efficient when applied to determining stress concentrations around the circular hole of a thick plate under tension

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

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

non uniform continuous dependence on initial data of solutions to the euler poincare system in this paper we investigate the continuous dependence on initial data of solutions to the euler poincare system by constructing a sequence of approximate solutions and calculating the error terms we show that the data to solution map is not uniformly continuous in sobolev space hs rd for s 1 d2

a full 3d finite element analysis using adaptive refinement and pcg solver with back interpolation in this paper adaptive refinement finite element analyses were carried out for full 3d problems in order to achieve an optimal computation cost and to eliminate the effects of singular points the adaptive refinement procedure was applied in conjunction with a back interpolation for the construction of a good initial guess for the solution of the linear equations system the combined use of the adaptive refinement procedure the back interpolation scheme and the preconditioned conjugate gradient pcg solver lead to a significant reduction in the operations for the solution of the simultaneous linear equations in the adaptive refinement analysis in many cases the number of iterations needed by the pcg solver to reach a converged solution is independent of the number of equations in the global system the numerical results obtained indicated that for a series of carefully designed adaptive meshes the computational cost required to solve the linear system could be made only proportional to the number of degrees of freedom in the mesh to our knowledge this is the first time that the global stiffness equations in 3d stress analysis have been solved with this efficiency the same set of numerical results also showed that in general the total computational cost of the adaptive refinement procedure can usually be minimized by gradually reducing the target relative error of the solution during successive refinements rather than employing a constant target relative error throughout the whole adaptive analysis

hierarchical models of thin elastic structures overview and recent advances in error estimation and adaptivity abstract in the numerical approximation of hierarchical models of thin bodies there are two separate contributions to the total error a modeling error is the error due to dimensional reduction and b discretization error is the error due to the numerical approximation traditionally estimation of these errors has been performed in terms of abstract quantities like the global energy norm which does not give a clear representation of the error in the local features of the solution recently a new class of finite element error estimation techniques has emerged wherein errors in a simulation are measured not in norms but in quantities of interest to the analyst following a brief overview of the literature we focus on extending the theory of a posteriori estimation of the modeling error to local quantities of interest that have physical meaning independent estimates are obtained for the modeling and discretization errors in quantities of interest these local estimates are used to develop goal oriented adaptive modeling strategies in which the model is automatically adapted by controlling the two errors independently and effectively to obtain local quantities of interest to within a preset level of accuracy numerical examples are presented to demonstrate the accuracy of the error estimates and the adaptive strategy

analytical and computational assessment of locking in the hp finite element method locking is the phenomenon by which the numerical approximation of parameter dependent problems deteriorates for values of the parameter close to a limiting value in this paper we give a definition of locking and develop precise computable and analytic ways to quantify it using the example of nearly incompressible elasticity we show by means of computational and theoretical results the difference between the h version and php version in combatting locking our results establish the superiority of high order elements both h p and hp when the standard variational form is used we also discuss other issues such as curved elements mixed methods and locking phenomena for problems over anisotropic materials and over thin domains

remarks on the well posedness of camassa holm type equations in besov spaces abstract in this paper we prove the solution map of the cauchy problem of camassa holm type equations depends continuously on the initial data in nonhomogeneous besov spaces in the sense of hadamard by using the littlewood paley theory and the method introduced by kato 37 and danchin 21

adaptive hp finite element for transient analysis abstract even though a tremendous effort has been devoted to the area of transient analysis many of these models fall short of achieving the overall desired objective due to oversights and rigid simplifying assumptions some of those shortfalls are summarized as follows 1 i the grid used is too fine or too coarse and is usually fixed in size 2 ii no error estimates are being used to trigger a grid size change 3 iii a fixed polynomial degree is being used to interpolate the approximate solution of the differential equation at hand finite element approach 4 iv an approximate differential equation is used to model the physical phenomena rather than the exact one finite difference approach in our study a self adaptive grid refinement technique coupled with the p version of the finite element method is investigated this type of approach is called the hp version of the finite element in the modeling process the approximate solution to the exact differential equation achieves convergence by applying two distinct solution enrichment strategies one strategy is to solve the problem using a very coarse grid and to enrich the quality of the approximation by increasing the degree p of the interpolating polynomial shape functions for those elements where it is needed the other strategy involves local grid h refinements for those elements where it is needed both strategies would interact synergistically to produce an optimal computational model that produces an accurate simulation of the transient phenomena addressed with minimal computational effort the triggering for the increase or decrease in polynomial degree p and placement or removal of local mesh h refinements is based on the calculation of a reliable a posteriori error estimate over each element at the end of each time step acase study of particular interest is the simulation of wave propagation in a semi infinite media

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

closed form stiffness matrices and error estimators for plane hierarchic triangular elements closed form expressions for the stiffness matrix and a simple error estimator and error indicator are derived for plane straight sided triangular finite elements in elasticity problems the calculation of the error estimator is performed on an element by element basis and is found to be very accurate and efficient in general the solutions for benchmark problems using the error indicators for selective refinement of the regions show accelerated convergence when compared to the convergence rate of solutions using uniform mesh refinement evaluation of the stiffness matrices and error estimators using explicit formulations is found to be several times faster than numerical integration

mixed hp finite element methods for problems in elasticity and stokes flow we consider the mixed formulation for the elasticity problem and the limiting stokes problem in bbb r d d 2 3 we derive a set of sufficient conditions under which families of mixed finite element spaces are simultaneously stable with respect to the mesh size h and subject to a maximum loss of o k frac d 1 2 with respect to the polynomial degree k we obtain asymptotic rates of convergence that are optimal up to o k epsilon in the displacement velocity and up to o k frac d 1 2 epsilon in the pressure with epsilon 0 arbitrary both rates being optimal with respect to h several choices of elements are discussed with reference to properties desirable in the context of the hp version

valuing vulnerable geometric asian options an option on a defaultable instrument is prone to default risk arising from the probability that the instrument s issuer will not honor its contractual obligations under this financial circumstance we consider vulnerable asian options in this paper we obtain parabolic partial differential equations with time dependent coefficients on the vulnerable asian option model and derive a closed formula of the pricing of the vulnerable geometric asian option using the pricing formula of the vulnerable european option with time dependent coefficients

higher order finite element modelling of laminated composite plates p version finite elements based on higher order theory are developed for the two dimensional modelling of general bending and cylindrical bending of thin to thick laminated composite plates in the case of general laminated plate elements three displacement fields are used in the special case of cylindrically bent laminated plate elements two displacement fields are needed in each case the displacement is expressed as the product of two functions one in terms of out of plane co ordinates alone and the other in terms of in plane co ordinates the shape functions used to build the displacement fields are based on integral of legendre polynomials the quality and performance of the elements are evaluated in terms of convergence characteristics of displacements and stresses the predicted response quantities are compared with those available in the published literature based on analytical as well as conventional finite element models

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

mid node admissible spaces for quadratic triangular arbitrarily curved 2d finite elements this paper presents the extension of the mid node admissible space mas concept for two dimensional quadratic triangular element 2dqte to the case when more than one edge is curved the mathematical background for the mas for 2dqte with more than one edge curved is developed a metric based on the determination of the jacobian distribution whose evaluation and implementation is aided through the use of the mas is developed the evaluation of the quality of an element based on this metric and procedures to repair an element to produce better results are also presented copyright 2001 john wiley sons ltd

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

on modified mellin transforms gauss laguerre quadrature and the valuation of american call options we extend a framework based on mellin transforms and show how to modify the approach to value american call options on dividend paying stocks we present a new integral equation to determine the price of an american call option and its free boundary using modified mellin transforms we also show how to derive the pricing formula for perpetual american call options using the new framework a result due to kim 1990 24 regarding the optimal exercise price at expiry is also recovered finally we apply gauss laguerre quadrature for the purpose of an efficient and accurate numerical valuation

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

criteria to achieve nearly optimal meshes in theh adaptive finite element method five adaptive modification strategies amss are studied as means to obtain nearly optimal meshes using an adaptive finite element modelling system based on the h version of the fem these strategies include the method traditionally seem in the literature as well as four additional methods the five strategies are tested over five numerical examples one of them producing convergence oscillations in one example the principal characteristic of our system is that it permits the user to control certain parameters of the adaptive process which contributes to an improvement in the results

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

on the accuracy of p version elements for the reissner mindlin plate problem this paper addresses the question of accuracy of p version finite element formulations for reissner mindlin plate problems three model problems a circular arc a rhombic plate and a geometrically complex structure are investigated whereas displacements and bending moments turn out to be very accurate without any post processing even for very coarse meshes the quality of shear forces computed from constitutive equations is poor it is shown that significantly improved results can be obtained if shear forces are computed from equilibrium equations instead a consistent computation of second derivatives of the shape functions is derived 1998 john wiley sons ltd

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

on the euler poincare equation with non zero dispersion we consider the euler poincare equation on mathbb r d d geqq 2 for a large class of smooth initial data we prove that the corresponding solution blows up in finite time this settles an open problem raised by chae and liu commun math phys 314 671 687 2012 our analysis exhibits some new concentration mechanisms and hidden monotonicity formulas associated with the euler poincare flow in particular we show an abundance of blowups emanating from smooth initial data with certain sign properties no size restrictions are imposed on the data we also showcase a class of initial data for which the corresponding solution exists globally in time

multigrid methods for the pure traction problem of linear elasticity mixed formulation a multigrid method using conforming p 1 finite elements is developed for the two dimensional pure traction boundary value problem of linear elasticity the convergence is uniform even as the material becomes nearly incompressible a heuristic argument for acceleration of the multigrid method is discussed as well numerical results with and without this acceleration as well as performance estimates on a parallel computer are included

a posteriori error estimates and adaptivity for finite element solutions in finite elasticity methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems in contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context in the paper a brief outline of continuum theory of finite elasticity is first given using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally the proposed error estimator is physically interpreted in the energy sense we then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences

the pricing of vulnerable options with double mellin transforms abstract many options traded in the over the counter markets are subject to default risks resulting from the probability that the option writer could not honor its contractual obligations there have been growing concerns about financial derivatives subject to default risks in particular since the global financial crisis and eurozone crisis this paper uses double mellin transforms to study european vulnerable options under constant as well as stochastic the hull white interest rates we obtain explicitly an analytic closed form pricing formula in each interest rate case so that the pricing of the options can be computed both accurately and efficiently

blow up zero α limit and the liouville type theorem for the euler poincare equations in this paper we study the euler poincare equations in mathbb r n we prove local existence of weak solutions in w 2 p mathbb r n p n and local existence of unique classical solutions in h k mathbb r n k n 2 3 as well as a blow up criterion for the zero dispersion equation 0 we prove a finite time blow up of the classical solution we also prove that as the dispersion parameter vanishes the weak solution converges to a solution of the zero dispersion equation with sharp rate as 0 provided that the limiting solution belongs to c 0 t h k mathbb r n with k n 2 3 for the stationary weak solutions of the euler poincare equations we prove a liouville type theorem namely for 0 any weak solution mathbf u in h 1 mathbb r n is u 0 for 0 any weak solution mathbf u in l 2 mathbb r n is u 0

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

the orthonormalized generalized finite element method ogfem efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions abstract an extension of the generalized finite element method gfem is proposed with which we efficiently reduce approximation errors the new method constructs a stiffness matrix with a conditioning that is significantly better than the stable generalized finite element method sgfem and the finite element method fem accordingly the risk of a severe loss of accuracy in the computed solution which burdens the gfem is prevented furthermore the computational cost of the inversion of the associated stiffness matrix is significantly reduced the gfem employs a set of enriched basis functions which is chosen to improve the rate at which the approximation converges to the exact solution the stiffness matrix constructed from these basis functions is often ill conditioned and the accuracy of the solution cannot be guaranteed we prevent this by orthonormalizing the basis functions and refer to the method as the orthonormalized generalized finite element method ogfem because the ogfem has the flexibility to orthonormalize either a part or all of the basis functions the method can be considered as a generalization of the gfem the method is applicable with single or multiple global and or local enrichment functions problems in blending elements are avoided by a modification of the enrichment functions the method is demonstrated for the one dimensional modified helmholtz and poisson equations and compared with the fem gfem and sgfem

on using degenerated solid shell elements in adaptive refinement analysis three different degenerated shell elements are studied in an adaptive refinement procedure for the solution of shell problems the stress recovery procedure expressed in a convective patch co ordinate system is used for the construction of continuous smoothed stress fields for the a posteriori error estimation the performance of the stress recovery procedure the error estimator and the adaptive refinement strategy are tested by solving three benchmark shell problems it is found that when adaptive refinement is used the adverse effects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates it is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations in addition if complete lagrangian polynomial terms are used the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed copyright 1999 john wiley sons ltd

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