reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

a posteriori error estimation based on numerical realization of the variational multiscale method this paper presents a numerical realization of the variational multiscale method with the objective of providing a reliable and easy to implement local error estimation technique the variational multiscale framework provides a systematic approach for solution scale decomposition into coarse scales captured by the mesh and fine or subgrid scales in the proposed work the coarse scale errors in the finite element solution are neglected in comparison to the fine scale errors the fine scale variational equation is then localized using a general localization function over an element or a patch of elements to develop a local error estimation technique based on the proposed framework a consistent formulation of a new subdomain error estimator for elliptic problems is derived without the necessity of introducing an error locality assumption the new subdomain error estimator is evaluated numerically within a mesh adaptivity algorithm and it is shown to produce very sharp error estimates that outperform the element residual method estimates

an asymptotic justification of a model of isothermal catalyst an asymptotic model of isothermal catalyst is obtained rigorously from a well known model of porous catalyst for appropriate limiting values of some of the nondimensional parameters large lewis number small prater and thermal biot numbers and bounded damkohler number

a model of porous catalyst accounting for incipiently non isothermal effects abstract an approximate model accounting for incipiently non isothermal effects is derived from a well known model of porous catalyst for appropriate realistic limiting values of the parameters in this limit the original model is a singularly perturbed m d reaction diffusion system and the approximate model is given by the m d heat equation with nonlinear boundary condition coupled with infinitely many ifm 2 1 d semilinear parabolic equations one for each point of the boundary of the spatial domain some limiting cases are still considered in the approximate model that lead to further simplifications

the adaptive nonconforming fem for the pure displacement problem in linear elasticity is optimal and robust this paper presents a natural nonconforming adaptive finite element algorithm and proves its quasi optimal complexity for the pure displacement navier lame equations the convergence rates are robust with respect to the lame parameter lambda to infty in the sense that all constants in the quasi optimal convergence rate stay bounded for almost incompressible materials and so the stokes equations are covered by our analysis in the limit lambda infty

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

an adaptive uzawa fem for the stokes problem convergence without the inf sup condition we introduce and study an adaptive finite element method fem for the stokes system based on an uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity we show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree k for velocity whereas for pressure the elements can be either discontinuous of degree k 1 or continuous of degree k 1 and k the popular taylor hood family is the sole example of stable elements included in the theory which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf sup condition we discuss the realization and complexity of the elliptic adaptive inner solver and provide consistent computational evidence that the resulting meshes are quasi optimal

convergence analysis of an adaptive nonconforming finite element method an adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations the analysis is carried out for the lowest order crouzeix raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels important tools in the convergence proof are a discrete local efficiency and a quasi orthogonality property the proof does neither require regularity of the solution nor uses duality arguments as a consequence on the data control no particular mesh design has to be monitored

a convergent adaptive method for elliptic eigenvalue problems we prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second order symmetric elliptic partial differential operators the weak form is assumed to yield a bilinear form which is bounded and coercive in h 1 each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation the error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem because of this nonlinearity the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

fully reliable localized error control in the fem if the first task in numerical analysis is the calculation of an approximate solution the second is to provide a guaranteed error bound and is often of equal importance the standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some saturation property for coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds the aim of this paper is to provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well the main argument is a localization via a partition of unity which leads to problems on small domains two fully reliable estimates are established the sharper one solves an analytical interface problem with residuals following babuska and rheinboldt siam j numer anal 15 1978 pp 736 754 the second estimate is a modification of the standard residual based a posteriori estimate with explicit constants from local analytical eigenvalue problems for some class of triangulations we show that the efficiency constant is smaller than 2 5 according to our numerical experience the overestimation of our computable estimates proved to be reasonably small with an overestimation by a factor between 2 5 and 4 only

an improved a priori error analysis of nitsche s method for robin boundary conditions in a previous paper juntunen and stenberg in math comput 78 1353 1374 2009 we have extended nitsche s method in abhandlungen aus dem mathematischen seminar der universitat hamburg 1970 71 for the poisson equation with general robin boundary conditions the analysis required that the solution is in h s with s 3 2 here we give an improved error analysis using a technique proposed by gudi math comput 79 2169 2189 2010

partition of unity for localization in implicit a posteriori finite element error control for linear elasticity the partition of unity for localization in adaptive finite element method fem for elliptic partial differential equations has been proposed in carstensen and funken siam j sci comput 2000 21 1465 1484 and is applied therein to the laplace problem a direct adaptation to linear elasticity in this paper yields a first estimator l based on patch oriented local weighted interface problems the global korn inequality with a constant c korn yields reliability u u h korn l for any finite element approximation u h to the exact displacement u in order to localize this inequality further and so to involve the global constant c korn directly in the local computations we deduce a new error estimator l the latter estimator is based on local weighted interface problems with rigid body motions rbm as a kernel and so leads to effective estimates only if rbm are included in the local fe test functions therefore the excluded first order fem has to be enlarged by rbm which leads to a partition of unit method pum with rbm called p 1 rbm or to second order fems called p 2 fem for p 1 rbm and p 2 fem or even higher order schemes one obtains the sharper reliability estimate u u h l efficiency holds in the strict sense of l u u h the local weighted interface problems behind the implicit error estimators l and l are usually not exactly solvable and are rather approximated by some fem on a refined mesh and or with a higher order fem the computable approximations l l and l are shown to be reliable in the sense of u u h min l l osc the oscillations osc are known functions of the given data and higher order terms if the data are smooth for first order fem the mathematical proofs are based on weighted korn inequalities and inverse estimates combined with standard arguments the numerical experiments for uniform and adapted fem on benchmarks such as an l shape problem cook s membrane or a slit problem validate the theoretical estimates and also concern numerical bounds for c korn and the locking phenomena

convergence of natural adaptive least squares finite element methods the first order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional in this paper we prove q linear convergence of the associated adaptive mesh refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right hand sides in a poisson model problem the proof relies on some modification of known supercloseness results to non convex polygonal domains plus the flux representation formula the analysis is carried out for the lowest order case in two dimensions for the simplicity of the presentation

convergence and optimal complexity of adaptive finite element eigenvalue computations in this paper an adaptive finite element method for elliptic eigenvalue problems is studied both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved the analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

an optimally convergent adaptive mixed finite element method we prove convergence and optimal complexity of an adaptive mixed finite element algorithm based on the lowest order raviart thomas finite element space in each step of the algorithm the local refinement is either performed using simple edge residuals or a data oscillation term depending on an adaptive marking strategy the inexact solution of the discrete system is controlled by an adaptive stopping criterion related to the estimator

a posteriori error analysis for finite element solution of one dimensional elliptic differential equations using equidistributing meshes the paper is concerned with the adaptive linear finite element solution of linear one dimensional elliptic differential equations using equidistributing meshes a strategy is developed for defining such meshes based on a residual based a posteriori error estimate the mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation i e the mesh equation an iterative algorithm is proposed for solving this system for the mesh and the finite element solution the existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh which is also established in the current work error bounds for the finite element solution are obtained for the equidistributing and quasi equidistributing meshes they show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes the departure from the equidistributing meshes has only a mild effect on the finite element error numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

convergence and quasi optimality of adaptive fem with inhomogeneous dirichlet data we consider the solution of a second order elliptic pde with inhomogeneous dirichlet data by means of adaptive lowest order fem as is usually done in practice the given dirichlet data are discretized by nodal interpolation as model example serves the poisson equation with mixed dirichlet neumann boundary conditions for error estimation we use an edge based residual error estimator which replaces the volume residual contributions by edge oscillations for 2d we prove convergence of the adaptive algorithm even with optimal convergence rate for 2d and 3d we show convergence if the nodal interpolation operator is replaced by the l 2 projection or the scott zhang quasi interpolation operator as a byproduct of the proof we show that the scott zhang operator converges pointwise to a limiting operator as the mesh is locally refined this property might be of independent interest besides the current application finally numerical experiments conclude the work

the asymptotic justification of a nonlocal 1 d model arising in porous catalyst theory an asymptotic model of isothermal catalyst is obtained from a well known model of porous catalyst for appropriate realistic limiting values of some nondimensional parameters in this limit the original model is a singularly perturbedm d reaction diffusion system the asymptotic model consists of an ordinary differential equation coupled with a semilinear parabolic equation on a semi infinite one dimensional interval

on the techniques for constructing admissible stress fields in model verification performances on engineering examples robust global goal oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation in the context of computational mechanics the construction of admissible stress fields ie stress tensors which verify the equilibrium equations is required to set up strict and guaranteed error bounds using residual based error estimators and plays an important role in the quality of the error estimates this work focuses on the different procedures used in the calculation of admissible stress fields which is a crucial and technically complicated point the three main techniques that currently exist called the element equilibration technique eet the star patch equilibration technique spet and the element equilibration star patch technique eespt are investigated and compared with respect to three different criteria namely the quality of associated error estimators computational cost and easiness of practical implementation into commercial finite element codes the numerical results which are presented focus on industrial problems they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators which have the same convergence rate as the exact global error is consistent two and three dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches the analysis of the results reveals that the spet is more accurate than eet and eespt methods but the corresponding computational cost is higher overall the numerical tests prove the interest of the hybrid method eespt and show that it is a correct compromise between quality of the error estimate practical implementation and computational cost furthermore the influence of the cost function involved in the eet and the eespt is studied in order to optimize the estimators

an optimal adaptive mixed finite element method various applications in fluid dynamics and computational continuum mechanics motivate the development of reliable and efficient adaptive algorithms for mixed finite element methods in order to save degrees of freedom not all but just some selected set of finite element domains are refined hence the fundamental question of convergence as well as the question of optimality require new mathematical arguments the presented adaptive algorithm for raviart thomas mixed finite element methods solves the poisson model problem with optimal convergence rate chen holst and xu presented convergence and optimality of adaptive mixed finite element methods 2008 following arguments of rob stevenson for the conforming finite element method their algorithm reduces oscillations separately before approximating the solution by some adaptive algorithm in the spirit of w dorfler 1996 the algorithm proposed here appears more natural in switching to either reduction of the edge error estimator or of the oscillations

adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations by a well known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned cg iterations using a multilevel preconditioner in the spirit of bramble pasciak and xu designed for standard nonconforming approximations local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results the performance of the preconditioner and the error estimator is illustrated by several numerical examples

a convergent adaptive finite element method for the primal problem of elastoplasticity summary the boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional this paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and up to higher order terms the r linear convergence of the stresses with respect to the number of loops applications include several plasticity models linear isotropic kinematic hardening linear kinematic hardening and multisurface plasticity as model for nonlinear hardening laws for perfect plasticity the adaptive algorithm yields strong convergence of the stresses numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule copyright q 2006 john wiley sons ltd

convergence analysis of an adaptive interior penalty discontinuous galerkin method we study the convergence of an adaptive interior penalty discontinuous galerkin ipdg method for a two dimensional model second order elliptic boundary value problem based on a residual type a posteriori error estimator we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh dependent energy norm associated with the ipdg method in contrast to recent work on adaptive ipdg methods o karakashian and f pascal convergence of adaptive discontinuous galerkin approximations of second order elliptic problems preprint university of tennessee knoxville tn 2007 the convergence analysis does not require multiple interior nodes for refined elements of the triangulation in fact it will be shown that bisection of the elements is sufficient the main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator a bulk criterion that takes care of a proper selection of edges and elements for refinement and a perturbed galerkin orthogonality property with respect to the energy inner product the results of numerical experiments are given to illustrate the performance of the adaptive method

convergence and quasi optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems recently an adaptive nonconforming finite element method anfem has been developed by carstensen and hoppe in numer math 103 251 266 2006 in this paper we extend the result to some nonsymmetric and indefinite problems the main tools in our analysis are a posteriori error estimators and a quasi orthogonality property in this case we need to overcome two main difficulties one stems from the nonconformity of the finite element space the other is how to handle the effect of a nonsymmetric and indefinite bilinear form an appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order crouzeix raviart element it is shown that the anfem is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops moreover quasi optimality in the sense of quasi optimal algorithmic complexity can be shown for the anfem the results of numerical experiments confirm the theoretical findings

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 the convergence of adaptive nonconforming finite element methods for a class of convex variational problems we formulate and analyze an adaptive nonconforming finite element method for the solution of convex variational problems the class of minimization problems we admit includes highly singular problems for which no euler lagrange equation or inequality is available as a consequence our arguments only use the structure of the energy functional we are nevertheless able to prove convergence of an adaptive algorithm using even refinement indicators that are not reliable error indicators

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

guaranteed computable bounds for the exact error in the finite element solution part ii bounds for the energy norm of the error in two dimensions this paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution these bounds are constructed in terms of the solutions of local residual problems with equilibrated residual loads and are rather sharp even for coarse meshes he sharpness of the bounds can be further improved by employing few iterations of a relatively inexpensive iterative scheme he main result is that the bounds are guaranteed for the nergy norm of the exact error unlike the bounds which ave been proposed in 13 14 which are guaranteed only for the nergy norm of the error with respect to an enriched truth esh finite element solution copyright 2000 john wiley sons ltd

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

convergence of an adaptive mixed finite element method for kirchhoff plate bending problems some reliable and efficient a posteriori error estimators are produced for a mixed finite element method the hellan herrmann johnson h h j method for kirchhoff plate bending problems cf k hellan acta polytech scand civil engrg ser 46 1967 pp 1 28 l herrmann j eng mech div asce 93 1967 pp 13 26 c johnson numer math 21 1973 pp 43 62 based on these results with k 0 1 where k denotes the polynomial order of the discrete moment field space an adaptive mixed finite element method amfem is set up and its convergence and complexity are studied thoroughly the key points of the theoretical analysis include achieving a discrete helmholtz decomposition and a discrete inf sup condition which serve as the main tools in deducing the quasi orthogonality of the moment field and the discrete reliability of the estimator it is shown that the amfem is a contraction for the sum of the moment field error in an energy norm and the scaled error estimator between two consecutive adaptive loops moreover an estimate for the amfem s complexity via the number of elements is developed

shout options a framework for pricing contracts which can be modified by the investor a shout option may be broadly defined as a financial contract which can be modified by the holder according to specified rules in a simple example the holder could have the right to set the strike of an option equal to the current value of the underlying asset in such a case the holder effectively has the right to select when to take ownership of an at the money option more generally the holder could have multiple rights along these lines in some cases with a limit placed on the number of rights which may be exercised within a given time period e g four times per year the value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems this paper describes a general framework for the valuation of complex types of shout options numerical issues related to interpolation and choice of timestepping method are considered in detail some illustrative examples are provided

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

convergence analysis of a conforming adaptive finite element method for an obstacle problem the adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual based a posteriori error estimator each cycle of the adaptive loop consists of the steps solve estimate mark and refine the techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of galerkin orthogonality we establish r linear convergence of the part of the energy above its minimal value if there is appropriate control of the data oscillations surprisingly the adaptive mesh refinement algorithm is the same as in the unconstrained case of a linear pde in fact there is no modification near the discrete free boundary necessary for r linear convergence the arguments are presented for a model obstacle problem with an affine obstacle and homogeneous dirichlet boundary conditions the proof of the discrete local efficiency is more involved than in the unconstrained case numerical results are given to illustrate the performance of the error estimator

a multilevel correction type of adaptive finite element method for eigenvalue problems an adaptive finite element method for eigenvalue problems is proposed based on the multilevel correction scheme different from the standard adaptive finite element method which requires solving ei

a parabolic variational inequality arising from the valuation of strike reset options abstract a strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder the pricing model of the option can be formulated as a one dimensional parabolic variational inequality or equivalently a free boundary problem where the free boundary just corresponds to the optimal reset strategy adopted by the holder of the option this paper is concerned with the theoretical analysis of the model the existence and uniqueness of the solution are established furthermore we study properties of the free boundary the monotonicity and c smoothness of the free boundary are proven in some situations

error reduction and convergence for an adaptive mixed finite element method an adaptive mixed finite element method amfem is designed to guarantee an error reduction also known as saturation property after each refinement step the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms this error reduction property is established here for the raviart thomas finite element method with a reduction factor p 1 uniformly for the l 2 norm of the flux errors our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels the adaptive algorithm surprisingly does not require any particular mesh design unlike the conforming finite element method the new arguments are a discrete local efficiency and a quasi orthogonality estimate the proof does not rely on duality or on regularity

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part i low order conforming nonconforming and mixed fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper their reliablility is shown for conforming nonconforming and mixed low order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

a comparison of a posteriori error estimators for mixed finite element discretizations by raviart thomas elements we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations in particular we present and analyze four different kinds of error estimators a residual based estimator a hierarchical one error estimators relying on the solution of local subproblems and on a superconvergence result respectively finally we examine the relationship between the presented error estimators and compare their local components

a convergent nonconforming adaptive finite element method with quasi optimal complexity in this paper we prove convergence and quasi optimal complexity of a simple adaptive nonconforming finite element method in each step of the algorithm the iterative solution of the discrete system is controlled by an adaptive stopping criterion and the local refinement is based on either a simple edge residual or a volume term depending on an adaptive marking strategy we prove that this marking strategy guarantees a strict reduction of the error augmented by the volume term and an additional oscillation term and quasi optimal complexity of the generated sequence of meshes

infinite reload options pricing and analysis infinite reload options allow the user to exercise his reload right as often as he chooses during the lifetime of the contract each time a reload occurs the owner receives new options where the strike price is set to the current stock price we consider a modified version of the infinite reload option contract where the strike price of the new options received by the owner is increased by a certain percentage we refer to this new contract as an increased reload option the pricing problem for this modified contract is characterized as an impulse control problem resulting in a hamilton jacobi bellman equation we use fully implicit timestepping and prove that the discretized equations are monotone stable and consistent implying convergence to the viscosity solution we also derive a globally convergent iterative method for solving the non linear discrete equations numerical examples show that both the exercise policy and the option value are very sensitive to the percentage increase in the reload strike

variational localizations of the dual weighted residual estimator the dual weighted residual method dwr and its localization for mesh adaptivity applied to elliptic partial differential equations are investigated the contribution of this paper is twofold first we introduce a novel localization technique based on the introduction of a partition of unity this new technique is very easy to apply as neither strong residuals nor jumps over element edges are required second we compare and analyze theoretically and numerically different localization techniques used for mesh adaptivity with respect to their effectivity here we focus on localizations in variational formulations that do not require the evaluation of the corresponding differential operator in the classical strong formulation in our mathematical analysis we show for different localization techniques established methods and our new approach that the local error indicators used for mesh adaptivity converge with proper order in the error functional several numerical tests substantiate our theoretical investigations

ten remarks on nonconvex minimisation for phase transition simulations nonconvex minimisation problems are encountered in many applications such as phase transitions in solids or liquids but also in optimal design tasks or micromagnetism in contrast to rubber type elastic materials and many other variational problems in continuum mechanics the minimal energy may be not attained in the sense of sobolev functions the nonrank 1 convex minimisation problem m is ill posed the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures their limit is a measure called gradient young measure gym and describes the effective energy density w qc the quasiconvex envelope of the original energy density w this gives rise to a relaxed minimisation problem r which is well posed in the sense that the minimum is attained the paper compares computational aspects of the two problems m and r neither adaptive finite element methods nor effective solvers may work for discrete versions of m but theoretical and numerical evidence supports that they work for r the drawback of r is that the relaxed energy density w qc is not always given by a known closed form formula instead a numerical relaxation has to be involved which is of the form m and hence yields a computational challenge other semiconvexity notions are requested and give rise to a new category of algorithms the demand for quasiconvexification algorithms is highly motivated by recent models in finite plasticity as the time discretisation in the latter models typically leads to nonrank 1 convex minimisation problems 2004 published by elsevier b v

optimal adaptive nonconforming fem for the stokes problem this paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi optimal complexity for the stokes equations with respect to natural approximation classes the proof does not explicitly involve the pressure variable and follows from a novel discrete helmholtz decomposition of deviatoric functions

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

a penalty method for american options with jump diffusion processes the fair price for an american option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem we develop an implicit discretization method for pricing such american options the jump diffusion correlation integral term is computed using an iterative method coupled with an fft while the american constraint is imposed by using a penalty method we derive sufficient conditions for global convergence of the discrete penalized equations at each timestep finally we present numerical tests which illustrate such convergence

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