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

error bars and quadratically convergent methods for the numerical simulation of the hartree fock equations this paper presents an a posteriori error analysis of the discretization methods used in computational quantum chemistry on the hartree fock equations upper and lower bounds for the energy are obtained from any discrete approximation strategy of the solution and the estimator proposed is shown to possess further approximation virtues

holomorphic kernels and commuting operators necessary and sufficient conditions in terms of operator polynomials are obtained for an m tuple t tl t l of commuting bounded linear operators on a separable hilbert space x to extend to an m tuple s s1 sm of operators on some hilbert space x where each s is realized as a representation of the adjoint of a multiplication operator on the tensor product of a special type of functional hilbert spaces a1so necessary and sufficient conditions in terms of operator polynomials are obtained for t to have a commuting normal extension o introduction in this paper some results in 1 and 2 for a single bounded linear operator t on a separable hilbert space x are generalized to m commuting operators on x in 1 agler introduces a special class of functional hilbert spaces and describes conditions under which an operator t on x extends to m where m denotes the multiplication operator on x and m denotes the countable direct sum of m with itself special cases of spaces x of which the classical hardy space is the prototype are considered in 1 2 the relevant conditions for the kind of extension of t referred to above are expressed in terms of the positivity of certain operator polynomials involving t and t these conditions are closely related to the reproducing kernel associated with the space s in 2 necessary and sufficient conditions are also given for a contraction t to be subnormal these conditions are really the requirement that a certain sequence of polynomials in t and t be positive it is natural to seek generalizations of these results to m commuting operators t1 tm on x an appropriate model for this generalization is obtained by constructi ng a finite tensor product of spaces s and exploiting the well known fact that the reproducing kernel of such a tensor product is the product of the reproducing kernels of the illdividual spaces suitable modifications of the reproducing kernels which render them holomorphic on the unit polydisc can be used to describe analogous extension results for t t1 tm the question of t having a commuting normal extension n n1 nm turns out to have a direct link with the multi dimensional hausdorff moment problem from the theory of probability received by the editors march 21 1986 1980 mathematics subject classification primary 47a20 secondary 47b20 47b38

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

local and pollution error estimation for finite element approximations of elliptic boundary value problems this paper addresses the issue of local elementwise error estimation of finite element approximations of elliptic boundary value problems the characterization of element error as local and pollution error component is presented and the relationship between energy norms of local errors and that predicted by means of a posteriori error estimators is investigated in addition techniques for calculating element indicators of local error pollution error and other error indicators are presented

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

a splitting step algorithm for reflected stochastic differential equations in r 1 a new algorithm which is based on the splitting step idea and the penalization method for reflected stochastic differential equation rsde in the upper half space r 1 is presented in this paper after some important estimates about rsdes and penalization odes are obtained the local pathwise error of the suggested algorithm are considered and an estimate of local error in the 2pth moment is proved under the assumptions of lipschitz continuity and linear growth some numerical experiments based on matlab programs are given to support the suggested algorithm

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985

real algebraic geometry for matrices over commutative rings abstract we define and study preorderings and orderings on rings of the form m n r where r is a commutative unital ring we extend the artin lang theorem and krivine stengle stellensatze both abstract and geometric from r to m n r this problem has been open since the seventies when hilbert s 17th problem was extended from usual to matrix polynomials while the orderings of m n r are in one to one correspondence with the orderings of r this is not true for preorderings therefore our theory is not morita equivalent to the classical real algebraic geometry

nonlinear neumann boundary conditions for quasilinear degenerate elliptic equations and applications abstract we prove comparison results between viscosity sub and supersolutions of degenerate elliptic and parabolic equations associated to possibly nonlinear neumann boundary conditions these results are obtained under more general assumptions on the equation in particular the dependence in the gradient of the solution and they allow applications to quasilinear possibly singular elliptic or parabolic equations one of the main applications is the extension of the so called level set approach for equations set in bounded domains with nonlinear neumann boundary conditions in such a framework the level set approach provides a weak notion for the motion of hypersurfaces with curvature dependent velocities and a prescribed contact angle at the boundary

nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain abstract we establish comparison and existence theorems of viscosity solutions of the initial boundary value problem for some singular degenerate parabolic partial differential equations with nonlinear oblique derivative boundary conditions the theorems cover the capillary problem for the mean curvature flow equation and apply to more general neumann type boundary problems for parabolic equations in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and curvature

a multi level computational model for multi scale damage analysis in composite and porous materials abstract an adaptive multi level methodology is developed in this paper to create a hierarchy of computational sub domains with varying resolution for multiple scale problems it is intended to concurrently predict evolution of variables at the structural and microstructural scales as well as to track the incidence and propagation of microstructural damage in composite and porous materials the microstructural analysis is conducted with the voronoi cell finite element model vcfem while a conventional displacement based fem code executes the macroscopic analysis the model introduces three levels in the computational domain which include macro macro micro and microscopic analysis it differentiates between non critical and critical regions and ranges from macroscopic computations using continuum constitutive relations to zooming in at hotspots for pure microscopic simulations coupling between the scales in regions of periodic microstructure is accomplished through asymptotic homogenization an adaptive process significantly increases the efficiency while retaining appropriate level of accuracy for each region numerical examples are conducted for composite and porous materials with a variety of microscopic architectures to demonstrate the potential of the model

convex duality and the skorokhod problem ii in this paper we consider skorokhod problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain and with a corresponding cone of constraint directions at the intersection of faces in part one of this paper we used convex duality to develop new methods for the construction of solutions to such skorokhod problems and for proving lipschitz continuity of the associated skorokhod maps the main alternative approach to skorokhod problems of this type is the reflection mapping technique introduced by harrison and reiman 8 in this part of the paper we apply the theory developed in part one to show that the reflection mapping technique of 8 is restricted to a slight generalization of the class of problems originally considered in 8 we further illustrate the power of the duality approach by applying it to two other classes of skorokhod problems those with normal directions of constraint and a new class that arises from a model of processor sharing in communication networks in particular we prove existence of solutions to and lipschitz continuity of the skorokhod maps associated with each of these skorokhod problems

a posteriori pointwise error computation for 2 d transport equations based on the variational multiscale method abstract this article presents a general framework to estimate the pointwise error of linear partial differential equations the error estimator is based on the variational multiscale theory in which the error is decomposed in two components according to the nature of the residuals element interior residuals and inter element jumps the relationship between the residuals coarse scales and the error components fine scales is established yielding to a very simple model in particular the pointwise error is modeled as a linear combination of bubble functions and green s functions if residual free bubbles and the classical green s function are employed the technology leads to an exact explicit method for the pointwise error if bubble functions and free space green s functions are employed then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary as a consequence this gives a model for the so called fine scale green s functions the numerical error is studied for the standard galerkin and supg methods with application to the heat equation the reaction diffusion equation and the convection diffusion equation numerical results show that stabilized methods minimize the propagation of pollution errors which stay mostly locally

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

sdes with oblique reflection on nonsmooth domains in this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique the main results are strong existence and uniqueness for solutions of such equations a key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection two cases are treated in the paper in the first case the direction of reflection is single valued and varies smoothly and the main new feature is that the boundary of the domain may be nonsmooth in the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary and at the resulting corner points more than one oblique direction is allowed

a real nullstellensatz for free modules abstract let a be the algebra of all n n matrices with entries from r x 1 x d and let g 1 g m f a we will show that f a v 0 for every a r d and v r n such that g i a v 0 for all i if and only if f belongs to the smallest real left ideal of a which contains g 1 g m here a left ideal j of a is real if for every h 1 h k a such that h 1 t h 1 h k t h k j j t we have that h 1 h k j we call this result the one sided real nullstellensatz for matrix polynomials we first prove by induction on n that it holds when g 1 g m f have zeros everywhere except in the first row this auxiliary result can be formulated as a real nullstellensatz for the free module r x 1 x d n

probabilistic approach for nonlinear partial differential equations and stochastic partial differential equations with neumann boundary conditions abstract in this study we consider nonlinear partial differential equations pdes and stochastic pdes spdes with neumann boundary conditions on domains that satisfy the lions sznitmann saisho conditions using the convergence result based on the penalization approximation for stochastic differential equations with normal reflections on nonsmooth and nonconvex domains we establish the existence and comparison principle for the viscosity solutions of nonlinear pdes with the neumann conditions associated with the optimal control problem we also obtain for nonlinear spdes and backward spdes with neumann condition the representations of the stochastic viscosity solutions

concurrent multi scale analysis of elastic composites by a multi level computational model abstract in this paper an adaptive multi level computational model is developed to comprehensively address issues of discretization and physically based modeling error in structures consisting of heterogeneous micro structures the model consists of three essential levels i a purely macroscopic domain level 0 with homogenized material parameters ii a macro micro domain level 1 with the micro domain represented by the periodic repetition of a representative volume element rve and iii purely microscopic domain level 2 where the rve ceases to exist and extended micro structural regions need to be modeled all micro structural computations of arbitrary heterogeneous domains are conducted using the adaptive voronoi cell finite element model an intermediate layer is used between the macroscopic and microscopic regions for smooth transition the level 1 domain is utilized to estimate criteria for switching from macroscopic to microscopic calculations physically based error indicators are developed for transition from macroscopic to microscopic domain statistical functions are used to assess the extent of the rve domains for non uniform micro structures for the macroscopic domains h and hp adaptations are executed using conventional error indicators based on the energy norm the numerical simulations are compared with results from pure micro mechanics and other multi level models in the literature the model is also tested for a realistic problem with macroscopic stress risers

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

positivstellensatze for algebras of matrices abstract noncommutative positivstellensatze express positive elements of algebras in terms of sums of squares here positive elements can be defined by means of representations point evaluations or abstract orderings squares are elements of the form a a we prove various types of noncommutative positivstellensatze for matrix algebras m n a where a is an algebra with involution and subalgebras of m n a such as path algebras crossed product algebras and cyclic algebras the notion of a noncommutative sum of squares is proposed and new versions of positivstellensatze are proved

guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains this is the pre peer reviewed version of the following article pares n santos h diez p guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains international journal for numerical methods in engineering 23 marc 2009 vol 79 num 10 p 1203 1244 which has been published in final form at http www3 interscience wiley com journal 122269073 abstract

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

identification and a posteriori estimation of transported errors in finite element analysis a systematic approach for developing reliable and efficient a posteriori error estimators for quite general classes of problems is presented a major difficulty to be overcome is the definition and identification of the transported error for certain classes of problem there is no significant transported error and the only issue is how to develop appropriate local estimators for the error for other types of problem there is no alternative but to estimate the transported error a general framework is presented for achieving this goal the theory is reminiscent of the additive schwarz methods that have proved successful in the development of effective preconditioners the general theory is illustrated by applying it a singularly perturbed reaction diffusion problem to derive estimators similar to those recently developed and to derive new estimators for the radial part of the laplace operator in cylindrical polar coordinates that are reliable and efficient the results obtained indicate the theory gives sharp and practical estimates

a posteriori estimation of the error in the error estimate in this paper we address the problem of a posteriori estimation of the error in the error estimate we consider the case of estimates for the error in the derivatives the strains or the stresses which are constructed in terms of locally computed element error indicators of the element residual or the least squares recovery type the estimates of the error in the error estimate have the same structure as the original error estimates and are determined by locally averaging recycling the original error indicators the most accurate indicators of the error in the error indicators are obtained by employing a harmonic basis in the recycling of the indicators namely a basis which locally satisfies the partial differential equation and the boundary conditions

distribution of the discretization and algebraic error in numerical solution of partial differential equations abstract in the adaptive numerical solution of partial differential equations local mesh refinement is used together with a posteriori error analysis in order to equilibrate the discretization error distribution over the domain since the discretized algebraic problems are not solved exactly a natural question is whether the spatial distribution of the algebraic error is analogous to the spatial distribution of the discretization error the main goal of this paper is to illustrate using standard boundary value model problems that this may not hold on the contrary the algebraic error can have large local components which can significantly dominate the total error in some parts of the domain the illustrated phenomenon is of general significance and it is not restricted to some particular problems or dimensions to our knowledge the discrepancy between the spatial distribution of the discretization and algebraic errors has not been studied in detail elsewhere

a stochastic differential game in the orthrant we study a zero sum stochastic differential game in the nonnegative orthrant the state of the system is governed by controlled reflecting diffusions in the nonnegative orthrant we consider discounted and average payoff evaluation criteria we prove the existence of values and optimal strategies for both payoff criteria

a posteriori estimation and adaptive control of the pollution error in the h version of the finite element method abstract we studied the pollution error in the h version of the finite element method and its effect on the quality of the local error indicators resp the quality of the derivatives recovered by local postprocessing in the interior of the mesh here we show that it is possible to construct a posteriori estimates of the pollution error in a patch of elements by employing the local error indicators over the entire mesh we also give an adaptive algorithm for the local control of the pollution error in a patch of elements of interest

a posteriori error estimators for second order elliptic systems part 2 an optimal order process for calculating self equilibrating fluxes abstract we present full details of an algorithm which can be used to obtain self equilibrating fluxes for arbitrary h p finite element approximations all of the calculations are performed on a local scale yielding an optimal order process o n where n is the number of elements in the mesh owing to the local nature of the process the algorithm is perfectly suited to parallel implementation the algorithm can be applied in both two and three dimensions for unstructured k irregular meshes and elements of non uniform degree finally it is shown that the implementation of the flux equilibration algorithm can take advantage of calculations already performed in obtaining the finite element approximation itself

a posteriori error estimators for second order elliptic systems part 1 theoretical foundations and a posteriori error analysis abstract this is the first in a series of two papers dealing with a posteriori error estimation for h p finite element approximation of second order elliptic systems in this paper we shall present the fundamental ideas behind the error estimator and the theoretical foundations for the method the second paper deals with the algorithmical details

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

finsler s lemma for matrix polynomials abstract finsler s lemma characterizes all pairs of symmetric n n real matrices a and b which satisfy the property that v t a v 0 for every nonzero v r n such that v t b v 0 we extend this characterization to all symmetric matrices of real multivariate polynomials but we need an additional assumption that b is negative semidefinite outside some ball we also give two applications of this result to noncommutative real algebraic geometry which for n 1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets

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

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

a procedure for a posteriori error estimation for h p finite element methods abstract a new approach to a posteriori error estimation is outlined which is applicable to general h p finite element approximations of general classes of boundary value problems the approach makes use of duality arguments and is based on the element residual method erm important aspects of the method are that it provides a systematic approach toward deriving element boundary conditions for the erm it leads to an upper bound for the global error in an appropriate energy norm and it is valid for non uniform and irregular h p meshes in the present exposition a brief outline of the theoretical foundations of the method is given together with the results of its application to several representative problems these results show that the approach is applicable to general linearly elliptic systems including unsymmetrical operators and that the method is valid for broad classes of linear and non linear problems

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples

barrier option hedging under constraints a viscosity approach we study the problem of finding the minimal initial capital needed in order to hedge without risk a barrier option when the vector of proportions of wealth invested in each risky asset is constrained to lie in a closed convex domain in the context of a brownian diffusion model we provide a pde characterization of the superhedging price this extends the result of broadie cvitanicg and soner rev financial stud 11 1998 pp 59 79 and cvitanicg pham and touzi j appl probab 36 1999 pp 523 545 which was obtained for plain vanilla options and provides a natural numerical procedure for computing the corresponding superhedging price as a by product we obtain a comparison theorem for a class of parabolic pdes with relaxed dirichlet conditions involving a constraint on the gradient

a defect corrected finite element approach for the accurate evaluation of magnetic fields on unstructured grids in electromagnetic simulations of magnets and machines one is often interested in a highly accurate and local evaluation of the magnetic field uniformity based on local post processing of the solution a defect correction scheme is proposed as an easy to realize alternative to higher order finite element or hybrid approaches radial basis functions rbfs are key for the generality of the method which in particular can handle unstructured grids also contrary to conventional finite element basis functions higher derivatives of the solution can be evaluated as required e g for deflection magnets defect correction is applied to obtain a solution with improved accuracy and adjoint techniques are used to estimate the remaining error for a specific quantity of interest significantly improved local convergence orders are obtained the scheme is also applied to the simulation of a sterngerlach magnet currently in operation

a new equilibrated residual method improving accuracy and efficiency of flux free error estimates this paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction diffusion equation the new approach revisits the ideas in pares et al 2009 6 4 with the goal of substantially reducing the computational cost of the flux free method while retaining the good quality of the bounds the new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies the zeroth order equilibration conditions are imposed using an alternative less restrictive form of the first order equilibration conditions along with a new efficient minimization criterion this new equilibration strategy provides much more accurate upper bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved

a posteriori estimation and adaptive control of the error in the quantity of interest part i a posteriori estimation of the error in the von mises stress and the stress intensity factor in this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution as an example we consider the plane elasticity problem with the von mises stress and the stress intensity factor as the quantities of interest the estimates of the error in the von mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point the local and the pollution errors and by constructing separate estimates for each component the estimates of the error in the stress intensity factors are constructed by employing an extraction method we demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees in part ii we will address the problem of the adaptive control of the error in the quantity of interest the goal of the computation and the construction of goal adaptive meshes for one or multiple goals

the self equilibration of residuals and complementary a posteriori error estimates in the finite element method it is demonstrated that the residual in a compatible displacement finite element solution can be partitioned into local self equilibrating systems on each element an a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved

pollution error in the h version of the finite element method and the local quality of the recovered derivatives abstract in this paper we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes as an example we consider the zienkiewicz zhu superconvergent patch recovery scheme the zz spr scheme and we study its accuracy in the interior of the mesh for finite element approximations of solutions of laplace s equation in polygonal domains we will demonstrate the following 1 in general the accuracy of the derivatives recovered by the zz spr or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution 2 if the mesh is globally adaptive i e it is nearly equilibrated in the energy norm then we can practically always gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution 3 it is possible to guarantee that the recovered solution derivatives have higher accuracy than the derivatives computed directly from the finite element solution in any patch of elements of interest by employing a mesh which is adaptive only with respect to the patch of interest i e it is nearly equilibrated in a weighted energy norm 4 in practice we are often interested in obtaining highly accurate derivatives or heat fluxes stresses etc only in a few critical regions which are identified by a preliminary analysis a grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions

a variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity this paper presents an error estimation framework for a mixed displacement pressure finite element method for nearly incompressible elasticity the proposed method is based on variational multiscale vms concepts wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh variational projection of fine scales onto the coarse scale space via variational embedding of the fine scale solution into the coarse scale formulation leads to the stabilized method with two major attributes first it is free of volumetric locking and second it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields this vms based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error specifically two error estimators are explored the first method employs an element based strategy and a representation of error via a fine scale error equation defined over element interiors which is evaluated by a direct post solution evaluation this quantity when combined with the global pollution error results in a simple explicit error estimator the second method involves solving the fine scale error equation through localization to overlapping patches spread across the domain thereby leading to an implicit calculation of the local error this implicit calculation when combined with the global pollution error results in an implicit error estimator the performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes the sharpness and robustness of the estimators is shown to be consistent across the test cases employed

a multidimensional process involving local time on presente une approche probabiliste a un probleme pose par mckean 1967 pour la construction d un systeme de n particules sur r avec un generateur formel

optimal reflection of diffusions and barrier options pricing under constraints we introduce a new class of control problems in which the gain depends on the solution of a stochastic differential equation sde reflected at the boundary of a bounded domain along directions which are controlled by a bounded variation process we provide a pde characterization of the associated value function this study is motivated by applications in mathematical finance where such equations are related to the pricing of barrier options under portfolio constraints

adaptivity and convergence in the voronoi cell finite element model for analyzing heterogeneous materials in this paper an adaptive voronoi cell finite element model is presented for analyzing micromechanical response of composites and porous materials both elastic and elastic plastic materials are considered two error measures viz a traction reciprocity error and an error in the kinematic relation are formulated as indicators of the quality of vcfem solutions based on a posteriori evaluation of these error measures element adaptation is executed in two consecutive stages in the first stage displacement function adaptations on the element boundaries and matrix inclusion void interfaces are carried out to minimize the corresponding traction reciprocity errors this is accomplished through a sequence of h refinement and spectral p enrichment strategy in an optimal displacement direction following this an enrichment of matrix and inclusion stress functions or enrp adaptation is conducted to reduce the error in kinematic relations the complete process improves convergence characteristics of the vcfem solution numerical analysis is conducted to examine the potential of the resulting vcfem code in analyzing microstructures with different distributions sizes and shapes of heterogeneities the method is seen to perform very well for the wide variety of problems solved

on the sharpness of certain local estimates for projections into finite element spaces influence of a re entrant corner in a plane polygonal domain with a reentrant corner consider a homogeneous dirichlet problem for poisson s equation au f with f smooth and the corresponding galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi uniform uniformly regular triangulations with the diameter of each element compara