a posteriori error estimators for the stokes and oseen equations the problem of obtaining posteriori estimates of the discretization error when one uses finite element methods to approximate problems with an incompressibility constraint is discussed a general approach to the treatment of the constraint condition and to the possible non self adjointness of the associated momentum equations is presented a posteriori error estimates are derived for adaptive h p and h p type finite element schemes key features are that the local error residual problems are not subject to an incompressibility constraint thereby avoiding the need for special finite element schemes and that the analysis is valid for essentially any discretization scheme including continuous and discontinuous pressure spaces the estimator bounds the actual error measured in an energy like norm

analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

a residual based a posteriori error estimator for the stokes darcy coupled problem in this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow the flows are governed by the stokes and darcy equations respectively and the transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the finite element subspaces consider bernardi raugel and raviart thomas elements for the velocities piecewise constants for the pressures and continuous piecewise linear elements for a lagrange multiplier defined on the interface we derive a reliable and efficient residual based a posteriori error estimator for this coupled problem the proof of reliability makes use of suitable auxiliary problems diverse continuous inf sup conditions satisfied by the bilinear forms involved and local approximation properties of the clement interpolant and raviart thomas operator on the other hand helmholtz decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are the main tools for proving the efficiency of the estimator up to minor modifications our analysis can be extended to other finite element subspaces yielding a stable galerkin scheme

a mixed finite element method for nonlinear elasticity two fold saddle point approach and a posteriori error estimate we extend the applicability of stable mixed finite elements for linear plane elasticity such as peers to a mixed variational formulation of hyperelasticity the present approach is based on the introduction of the strain tensor as a further unknown which yields a two fold saddle point nonlinear operator equation for the corresponding weak formulation we provide the uniqueness of solution for the continuous and discrete schemes and derive the usual cea estimate for the associated error finally a reliable a posteriori error estimate based on the solution of local dirichlet problems and well suited for adaptive computations is also given

a posteriori error estimates of stabilized finite element method for the steady navier stokes problem in this paper we develop a posteriori error estimates for the steady navier stokes equations based on the lowest equal order mixed finite element pair residual type a posteriori error estimates are derived by means of general framework established by verfurth for the nonlinear equations furthermore a simple error estimator in l 2 norm is also presented by using the duality argument numerical experiments using adaptive computations are presented to demonstrate the effectiveness of these error estimates for three examples the first example is a singular problem with known solution the second example is a physical model of lid driven cavity and the last one is a backward facing step problem

a posteriori error estimation for a pde constrained optimization problem involving the generalized oseen equations we derive globally reliable a posteriori error estimators for a linear quadratic optimal control problem involving the generalized oseen equations as state equations control constraints are also considered the corresponding local error indicators are locally efficient the assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods when stabilized methods are considered no a priori relation between the stabilization terms for the state and adjoint equations is required if a lower bound for the inf sup constant is available a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained we illustrate the theory with numerical examples

a posteriori error estimates for the stokes problem an a posteriors error estimate is derived and analyzed for the mini element discretization of the stokes equations the estimate is based on the solution of a local stokes problem in each element of the finite element mesh using spaces of quadratic bump functions for both velocity and pressure errors this results in solving a 9 times 9 system which reduces to two easily invertible 3 times 3 systems comparisons with other estimates based on a petrov galerkin solution are used in this analysis which shows that it provides a reasonable approximation of the actual discretization error numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures

a posteriori error estimation and dynamic adaptivity for symmetric discontinuous galerkin approximations of reactive transport problems a symmetric discontinuous galerkin scheme is applied for solving reactive transport problems an explicit a posteriori error estimator in l 2 l 2 is derived for the semi discrete scheme applied to transport with general kinetic reactions explicit a posteriori error estimators in terms of linear functionals and negative norms are also obtained for transport problems with simple reactions an adaptive approach based on the a posteriori error estimators is proposed to modify the mesh dynamically numerical examples demonstrate the efficiency and effectivity of the estimators moreover it is shown in this study that the proposed adaptivity eliminates the need of slope limiters 2005 elsevier b v all rights reserved

a posteriori error estimators for stabilized p1 nonconforming approximation of the stokes problem in this paper we derive and analyze some a posteriori error estimators for the stabilized p1 nonconforming approximation of the stokes problem involving the strain tensor this will be done by decomposing the numerical error in a proper way into conforming and nonconforming contributions the error estimator for the nonconforming error is obtained in the standard way and the implicit error estimator for the conforming error is derived by applying the equilibrated residual method a crucial part of this work is construction of approximate normal stresses on interelement boundaries which will serve as equilibrated neumann data for local stokes problems it turns out that such normal stresses can be simply computed by local weak residuals of the discrete system plus jumps of the velocity solution and that a stronger equilibration condition is satisfied to ensure solvability of local stokes problems we also derive a simple explicit error estimator based on the nonsymmetric tensor recovery of the normal stress error numerical results are provided to illustrate the performance of our error estimators

a mixed finite element method with lagrange multipliers for nonlinear exterior transmission problems we apply a mixed finite element method to numerically solve a class of nonlinear exterior transmission problems in r 2 with inhomogeneous interface conditions besides the usual unknowns required for the dual mixed method which include the gradient of the temperature in this nonlinear case our approach makes use of the trace of the outer solution on the transmission boundary as a suitable lagrange multiplier in addition we use a boundary integral operator to reduce the original transmission problem on the unbounded region into a nonlocal one on a bounded domain in this way we are lead to a two fold saddle point operator equation as the resulting variational formulation we prove that the continuous formulation and the associated galerkin scheme defined with raviart thomas spaces are well posed and derive the a priori estimates and the corresponding rate of convergence then we introduce suitable local problems and deduce first an implicit reliable and quasi efficient a posteriori error estimate and then a fully explicit reliable one finally several numerical results illustrate the effectivity of the explicit estimate for the adaptive computation of the discrete solutions

adaptive inexact iterative algorithms based on polynomial degree robust a posteriori estimates for the stokes problem in this paper we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear stokes problem we base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components namely the discretization error component the inner algebraic solver error component and possibly the outer algebraic solver error component for algorithms of the uzawa type we prove that our estimate gives a guaranteed upper bound on the total error as well as a polynomial degree robust local efficiency and this on each step of the employed iterative algorithm our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error the developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver implementation into the freefem programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies as example we choose here the unpreconditioned and preconditioned uzawa algorithm and the preconditioned minimum residual minres algorithm in combination with the taylor hood discretization

numerical investigation of effectivity indices of space time error indicators for navier stokes equations abstract in this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady navier stokes equations we define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error moreover we verify the reliability of the estimations through numerical experiments and we propose an effective space time adaptive strategy for the unsteady navier stokes equations such technique is based on two residual based error indicators that suitably drive the mesh and the timestep length modifications adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way

an a posteriori finite element error analysis for the stokes equations abstract conforming and nonconforming error estimators are presented and analyzed for the mixed finite element approximation of the stokes problem the estimators are obtained by solving local poisson type problems that do not involve boundary conditions compatibility and balancing conditions incompressibility constraint or flux jumps across inter element boundaries the estimators are bounded from above and below by constant multiples of the actual error in an energy like norm and can be used in adaptive h p and hp computations

variational multiscale error estimators for solid mechanics adaptive simulations an orthogonal subgrid scale approach abstract in this work we present a general error estimator for the finite element solution of solid mechanics problems based on the variational multiscale method the main idea is to consider a rich model for the subgrid scales as an error estimator the subscales are considered to belong to a space orthogonal to the finite element space orthogonal subgrid scales and we take into account their contribution both in the element interiors and on the element boundaries subscales on the element boundaries a simple analysis shows that the upper bound for the obtained error estimator is sharper than in other error estimators based on the variational multiscale method numerical examples show that the proposed error estimator is an accurate approximation for the energy norm error and can be used both in simple linear constitutive models and in more complex non linear cases

a posteriori error estimators for the stokes equations ii non conforming discretizations we present an a posteriori error estimator for the non conforming crouzeix raviart discretization of the stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation the error estimator yields global upper and local lower bounds for the error of the finite element solution it can easily be generalized to the stationary incompressible navier stokes equations and to other non conforming finite element methods numerical examples show the efficiency of the proposed error estimator

a fast efficient technique for the estimation of frequency interpretation and generalisation in quinn fernandes 1991 an iterative technique was presented for the estimation of the frequency of a sinusoid in the presence of noise the technique is more robust to poor initial estimates than is the periodogram maximiser a generalisation to the case of multiple sinusoids is presented in this paper and it is shown that the original estimator may be interpreted as the maximiser of a smoothed version of the periodogram

specification and towers in shift spaces we show that a shift space on a finite alphabet with a non uniform specification property can be modeled by a strongly positive recurrent countable state markov shift to which every equilibrium state lifts in addition to uniqueness of the equilibrium state this gives strong statistical properties including the bernoulli property exponential decay of correlations central limit theorem and analyticity of pressure which are new even for uniform specification we give applications to shifts of quasi finite type synchronised and coded shifts and factors of beta shifts and s gap shifts

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

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

analysis finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity abstract in this paper we present constitutive models for nearly incompressible transversely isotropic materials in finite hyperelasticity particularly for reinforced rubber like materials which are of essential engineering interest the theory is developed using a convected curvilinear coordinate system based on a mixed two field displacement pressure energy functional furthermore an a posteriori error estimator without multiplicative constants is derived for non linear anisotropic problems which measures the discretization error in the first piola kirchhoff stresses in the l 2 norm by solving local neumann problems with equilibrated tractions illustrative numerical examples demonstrate the anisotropic material behaviour of reinforced materials and the efficiency of using adaptive finite element methods

residual a posteriori error estimator for a three field model of a non linear generalized stokes problem in this article we propose and analyze an a posteriori error estimator for a three field model of a generalized stokes problem the components of the a posteriori error estimator are defined via a non linear projection of the residues of the variational equations both upper and lower bounds for the approximation error are derived in terms of the components of the a posteriori error estimator the non linear projections do not need to be explicitly computed to construct the a posteriori error estimates

a posteriori error estimation for stabilized mixed approximations of the stokes equations the practical implementation of the lowest order pl p0 linear velocity constant pressure finite element method for the steady state incompressible navier stokes equations is addressed in this work three different types of a posteriori error indicator are introduced and each is shown to be equivalent to the discretization error our numerical results show that these indicators can be used to drive an adaptive refinement process which is specially tailored to create grids which conform to the requirements of the local stabilization it is also shown that the indicators provide an effective method for detecting local singularities in the flow solution

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 estimations for mixed finite element approximations to the navier stokes equations based on newton type linearization abstract in this paper we derive a posteriori error estimates for inf sup stable mixed finite element approximations to the evolutionary navier stokes equations we reduce the problem of getting a posteriori error estimations of a non linear evolutionary problem to that of getting a posteriori estimations of a linear steady problem the main idea is based on the fact that the solutions of some newton type linearized navier stokes equations around the plain galerkin approximations approach the solution of the original navier stokes equations with a bigger rate of convergence than the plain galerkin method as a consequence the difference between the galerkin approximations and the solution of the linearized problem is an estimator of the galerkin error moreover since the galerkin approximation of the evolutionary navier stokes equations is also the galerkin approximation of the linearized equations any a posteriori estimator of the error in the newton type linearized navier stokes equations is also an a posteriori error estimator of the full navier stokes equations

adaptive variational multiscale methods for incompressible flow based on two local gauss integrations we consider variational multiscale vms methods with h adaptive technique for the stationary incompressible navier stokes equations the natural combination of vms with adaptive strategy retains the best features of both methods and overcomes many of their deficits a reliable a posteriori projection error estimator is derived which can be computed by two local gauss integrations at the element level finally some numerical tests are presented to illustrate the method s efficiency

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

stabilized finite element approximations for a generalized boussinesq problem a posteriori error analysis abstract the purpose of this work is the design and analysis of a posteriori error estimators for low order stabilized finite element approximations of a generalized boussinesq problem we consider standard stabilization procedures over conforming finite element spaces and a nonconforming one that delivers a divergence free discrete velocity field the analysis that is valid for two and three dimensional domains relies on a smallness assumption on the solution and is based on a technique that involves the ritz projection of the residuals the devised a posteriori error estimators are proven to be globally reliable and locally efficient three dimensional numerical experiments reveal a competitive performance of adaptive procedures driven by the designed a posteriori error estimators

a posteriori error estimates for a mixed fem formulation of a non linear elliptic problem abstract we consider the numerical solution via the mixed finite element method of a non linear elliptic partial differential equation in divergence form with dirichlet boundary conditions besides the temperature u and the flux we introduce u as a further unknown which yields a variational formulation with a twofold saddle point structure we derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem which does not need any equilibrium property for its solvability in addition for specific finite element subspaces of raviart thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems our approach does not need the exact finite element solution but any reasonable approximation of it such as for instance the one obtained with a fully discrete galerkin scheme in particular we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi linear forms involved finally several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions

variational multiscale a posteriori error estimation for systems the euler and navier stokes equations this paper extends explicit a posteriori error estimators based on the variational multiscale theory to systems of equations in particular the emphasis is placed on flow problems the euler and navier stokes equations three error estimators are proposed the standard the naive and the upper bound numerical results show that with a very economical algorithm the attained global and local efficiencies for the naive approach are reasonably close to unity whereas the standard and upper bound approaches give respectively approximate lower and higher error estimates

a posteriori error estimate for the h d i v conforming mixed finite element for the coupled darcy stokes system an h d i v conforming mixed finite element method has been proposed for the coupled darcy stokes flow in kanschat and riviere 2010 12 which imposes normal continuity on the velocity field strongly across the darcy stokes interface here we develop an a posteriori error estimator for this h d i v conforming mixed method and prove its global reliability and efficiency due to the strong coupling on the interface special techniques need to be employed in the proof this is the main difference between this paper and babuska and gatica s 2010 work 38 in which they analyzed an a posteriori error estimator for the mixed formulation using weakly coupled interface conditions

an adaptive stabilized finite element method for the generalized stokes problem in this work we present an adaptive strategy based on an a posteriori error estimator for a stabilized finite element method for the stokes problem with and without a reaction term the hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble like functions an equivalence result between the norm of the finite element error and the estimator is given where the dependence of the constants on the physics of the problem is explicited several numerical results confirming both the theoretical results and the good performance of the estimator are given

a posteriori error estimates of stabilization of low order mixed finite elements for incompressible flow in this paper we derive a posteriori error estimates for the stabilization of low order mixed finite element methods for the stokes equations by defining different projection estimators we prove that up to higher order perturbation terms the estimators yield global upper and lower bounds on the error of stabilized finite element methods in numerical tests each error estimator is shown to be equivalent to the discretization error it is also shown that the adaptive strategy based on both projection estimators is efficient to detect local singularities in the flow problems

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

a posteriori error analysis of an augmented fully mixed formulation for the stationary boussinesq model abstract in this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the boussinesq approximation setting our approach incorporates as additional unknowns a modified pseudostress tensor field and an auxiliary vector field in the fluid and heat equations respectively which possibilitates the elimination of the pressure this unknown however can be easily recovered by a postprocessing formula in turn redundant galerkin terms are included into the weak formulation to ensure well posedness in this way the resulting variational formulation is a four field augmented scheme whose galerkin discretization allows a raviart thomas approximation for the auxiliary unknowns and a lagrange approximation for the velocity and the temperature in the present work we propose a reliable and efficient fully local and computable residual based a posteriori error estimator in two and three dimensions for the aforementioned method standard arguments based on duality techniques stable helmholtz decompositions and well known results from previous works are the main underlying tools used in our methodology several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm

a posteriori error estimates for spectral element solutions to viscoelastic flow problems abstract in this paper we develop an a posteriori error estimate for viscoelastic flow calculations using spectral elements the error estimate is compared with the exact error obtained by computing the difference between the numerical solution and a reference calculation for the benchmark problem of viscoelastic flow past a rigid sphere in a tube

augmented mixed finite element method for the oseen problem a priori and a posteriori error analyses abstract we propose a new augmented dual mixed method for the oseen problem based on the pseudostress velocity formulation the stabilized formulation is obtained by adding to the dual mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations we prove that for appropriate values of the stabilization parameters the new variational formulation and the corresponding galerkin scheme are well posed and a cea estimate holds for any finite element subspaces we also provide the rate of convergence when each row of the pseudostress is approximated by raviart thomas or brezzi douglas marini elements and the velocity is approximated by continuous piecewise polynomials moreover we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient finally we include several numerical experiments that support the theoretical results

anisotropic finite elements for the stokes problem a posteriori error estimator and adaptive mesh we propose an a posteriori error estimator for the stokes problem using the crouzeix raviart p0 pair its efficiency and reliability on highly stretched meshes are investigated the analysis is based on hierarchical space splitting whose main ingredients are the strengthened cauchy schwarz inequality and the saturation assumption we give a theoretical proof of a method to enrich the crouzeix raviart element so that the strengthened cauchy constant is always bounded away from unity independently of the aspect ratio an anisotropic self adaptive mesh refinement approach for which the saturation assumption is valid will be described our theory is confirmed by corroborative numerical tests which include an internal layer a boundary layer a re entrant corner and a crack simulation a comparison of the exact error and the a posteriori one with respect to the aspect ratio will be demonstrated

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