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

explicit and averaging a posteriori error estimates for adaptive finite volume methods local mesh refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady state convection diffusion reaction equations the paper establishes residual type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global h1 and l2 norms reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh refining algorithms over uniform mesh refining a discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper

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

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

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

adaptive h and h r methods for symm s integral equation in this paper we present the adaptive h r and h r methods for the galerkin approximation of symm s integral equation the a posteriori error estimate depends on a localized a priori error estimate and local finite differences of the computed solution the optimal mesh for these three methods is obtained by using a mesh grading function the numerical results for a polygonal problem and a comparison between our error estimate and the residual error estimate are also presented moreover we briefly describe how to apply our method to a two dimensional elasticity problem

an adaptive finite element procedure for compressible high speed flows abstract a two step explicit finite element procedure for obtaining time accurate solutions to the compressible euler equations is described the method is coupled with an adaptive mesh refinement process which enables steady state solutions of improved quality to be obtained

adaptive recovery of near optimal meshes in the finite element method for parameter dependent problems abstract a method for adaptive construction of nearly optimal finite element meshes is presented the method is based on the notion of mesh density for parametrized problems optimal meshes for new parameter values can be predicted from previous meshes the method is of the one shot type normally giving a sufficiently good approximation to the optimal mesh at the first attempt but when the first shot is unsuccessful the method can be iterated for parametrized problems no coarse mesh solutions are required except for the very first problem and whenever the method fails in providing an acceptable mesh and iteration becomes necessary

spline based computational technique for linear singularly perturbed boundary value problems a fourth order difference scheme via cubic spline in compression for singularly perturbed two point boundary value problems of the form ey p x y q x y r x y a a y b bis presented the convergence analysis is given finally computational efficiency of the method is verified through numerical examples

the theory of density representation of finite element meshes examples of density operators with quadrilateral elements in the mapped domain abstract a method to represent finite element meshes in any dimension by functions called mesh density functions is described the method is based on a generalization of the notion of mesh grading the idea of the mesh density function is to represent a mesh by the density of elements in all points of the domain the density representation is part of a methodology for adaptive solution of linear and nonlinear parameter dependent problems the general theory is given for any space dimension and examples are presented for one and two dimensional problems with point singularities

an energy minimizing mesh for the schro dinger equation abstract a formula is derived which gives the optimum placement of mesh points in the sense of minimizing the error in energy for a given number of degrees of freedom the wave function is assumed to be given in a finite difference or finite element representation with polynomial completeness to an arbitrary but fixed degree the result depends explicitly on the wave function the kinetic energy operator and the degree of polynomial completeness for the representation but does not depend explicitly on the potential even in the presence of a coulomb singularity the optimum mesh predicted here for the hydrogen atom is compared to the widely used herman skillman mesh a 1 dimensional example is given in which the calculated error in energy displays a sharp minimum at the predicted optimal mesh density the critical role of reproducing the analytic structure of the solution is illustrated with an additional example in one dimension the hydrogen is wave function is considered as a 3 dimensional problem and an optimal mesh density is calculated a singular mesh density is required to account for the cusp while retaining the convergence properties of the basis set a few percent of the available degrees of freedom are devoted to the description of the wave function cusp in the optimal mesh

on adaptive finite element methods for fredholm integral equations of the second kind a posteriors and a priori error estimates are derived for a finite element discretization of a fredholm integral equation of the second kind a reliable and efficient adaptive algorithm is then designed for a specific computational goal with applications to potential problems the reliability of the algorithm is guaranteed by the a posteriors error estimate and the efficiency follows from the a priori error estimate which shows that the a posteriors error bound is sharp

two dimensional mesh generation adaptive remeshing and refinement this paper discusses the generation of unstructured triangular meshes by the advancing front technique we also employ the technique to discretize surfaces a simple and effective algorithm for the specification of element sizes is described a similar idea is then applied to adaptive remeshing where a completely new mesh is created on the basis of the isolines of a certain important variable the isolines are obtained from the fe calculation based on the previous mesh in this paper attention is focused on the poisson s equation the optimal a priori error estimate for the problem is used to estimate the number of isolines needed for the remeshing when the estimated error exceeds the tolerance in only a small part of the domain the adaptive procedure is switched automatically to an adaptive refinement

goal oriented local a posteriori error estimators for h div least squares finite element methods we propose a goal oriented local a posteriori error estimator for h div least squares ls finite element methods our main interest is to develop an a posteriori error estimator for the flux approximation in a preassigned region of interest d subset omega the estimator is obtained from the ls functional by scaling residuals with proper weight coefficients the weight coefficients are given in terms of local mesh size h t and a function omega d depending on the distance to d this new error estimator measures the pollution effect from the outside region of d and provides a basis for local refinement in order to efficiently approximate the solution in d numerical experiments show superior performances of our goal oriented a posteriori estimators over the standard ls functional and global error estimators

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

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

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

a finite element formulation of compressible flows using various sets of independent variables abstract this paper presents a finite element method for the simulation of compressible flows the navier stokes and euler equations are solved in the conservation form using various sets of independent variables a variational formulation is developed based upon a variant of the petrov galerkin method and uses a shock capturing operator an adaptive algorithm based on a particular residual norm is proposed several numerical examples are presented to demonstrate the performances of each set of variables in solving compressible high speed flows

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 posteriori error estimation in least squares stabilized finite element schemes abstract we introduce a new approach to a posteriori error estimation in finite element galerkin schemes with least squares stabilization typical examples are the stabilization of equal order approximation in mixed finite element schemes and the streamline diffusion method for convection dominated problems the underlying framework is the general optimal control approach recently proposed for error control in finite element models in this method the approximate solutions of certain dual problems are used as weighting factors in residual based a posteriori bounds for arbitrary functionals of the error on this basis almost optimally economical meshes as well as reliable and efficient error bounds can be developed in a feed back process on successively adapted meshes the new feature in this method when applied for least squares finite element methods is the use of mesh dependent duality arguments to guarantee optimal order estimation of the stabilization terms

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

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

a second order finite difference error indicator for adaptive transonic flow computations an adaptive finite element method for the calculation of transonic potential flows was developed an error indicator based on first order finite differences of gradients is introduced as a local error estimator it measures second order distributional derivatives estimates involving this error estimator a residual and the error are given the error estimator can be used as a criterion for mesh refinement we also give some computational results

an adaptive finite element method for two phase stefan problems in two space dimensions i stability and error estimates a simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two phase stefan problems in 2d a typical triangula tion is coarse away from the discrete interface where discretization parameters satisfy a parabolic relation whereas it is locally refined in the vicinity of the dis crete interface so that the relation becomes hyperbolic several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations mesh selection is based upon equidistributing pointwise interpolation errors be tween consecutive meshes and imposing that discrete interfaces belong to the so called refined region consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated a general theory for interpolation between noncompatible meshes is set up in lp based norms the resulting scheme is stable in various sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods 3 2 2 on quasi uniform meshes namely 0 r as opposed to 0 x to achieve the same global asymptotic accuracy here r 0 is the uniform time step 112 a rate of convergence of essentially 0 x is derived in the natural energy spaces provided the total number of mesh changes is restricted to 0 x which in turn is compatible with the mesh selection procedure an auxiliary quasi optimal pointwise error estimate for the laplace operator is proved as well numerical results illustrate the scheme s efficiency in approximating both solutions and interfaces

the multiscale approach to error estimation and adaptivity abstract a new explicit a posteriori error estimator is investigated which emanates from the variational multiscale theory the error estimator uses an approximation of the green s function which propagates the error according to the dual problem the technique predicts the error in the l2 norm from element residuals both interior and inter element jumps therefore leading to a very economical procedure finally the methodology is applied to fluid flow transport showing that for convection dominated flows the efficiency index is independent of the diffusion coefficient

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

a posteriori error estimation for the semidiscrete finite element method of parabolic differential equations in this paper three a posteriori error estimators of the error in the semidiscrete finite element solution discrete in space and continuous in time of parabolic partial differential equations are analyzed this approach is based on a posteriori error estimators for the elliptic pdes it is proven that guaranteed resp asymptotically exact a posteriori error estimator for the elliptic problem yields the guaranteed resp asymptotically exact estimator for the parabolic problem

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

a posteriori error estimates for the primary and dual variables for the div first order least squares finite element method abstract we propose a posteriori error estimators for first order div least squares ls finite element method for linear elasticity stokes equations and general second order scalar elliptic problems our main interest is obtaining a posteriori error estimators for the dual variables fluxes strains stress etc which are main quantity of interest in many applications we also provide a posteriori error estimators for the primary variable these estimators are obtained from the local least squares functional by assigning weight coefficients scaling the respective residuals the weight coefficients are given in terms of local meshsize h k we establish the global upper bounds and local lower bounds for the estimators the estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution the indicators and estimators are identical numerical experiments show a superior performance of our a posteriori estimators for user specific norm over the standard ls functional

interior estimates for nonconforming finite element methods interior error estimates are derived for a wide class of nonconforming finite element methods for second order scalar elliptic boundary value problems it is shown that the error in an interior domain can be estimated by three terms the first one measures the local approximability of the finite element space to the exact solution the second one measures the degree of continuity of the finite element space the consistency error and the last one expresses the global effect through the error in an arbitrarily weak sobolev norm over a slightly larger domain as an application interior superconvergences of some difference quotients of the finite element solution are obtained for the derivatives of the exact solution when the mesh satisfies some translation invariant condition

a hybridizable discontinuous galerkin method for elliptic interface problems in the formulation of boundary integral equations abstract a novel numerical method is presented to solve the general elliptic interface problem with piecewise constant diffusion coefficients to overcome the difficulties caused by piecewise constant diffusion coefficients a combination of the kernel free boundary integral kfbi method and the unfitted mesh based hybridizable discontinuous galerkin hdg method is considered to this end we first reformulate the elliptic interface problem into a fredholm boundary integral equation of the second kind and further solve the well conditioned discrete integral equation iteratively with the gmres method in each iteration evaluation of boundary and volume integrals is made by interpolating numerical solutions to equivalent but simple interface problems which is efficiently calculated with a hdg method on unfitted meshes the advantage of using the unfitted hdg method to treat the equivalent but simple interface problem is that the interpolation along boundary integrals on the interface can be easily evaluated via a linear lagrange polynomial element by element which is an important improvement from ying and henriquez 2007 numerical experiments are given to verify the effectiveness of the novel method which has second order accuracy even in the case of high contrast diffusion coefficients

goal oriented error estimation for fluid structure interaction problems in this work we present an adaptive finite element method for the numerical simulation of stationary fluid structure interaction problems the coupled system is given in a variational and monolithic arbitrary lagrangian eulerian framework we derive methods for goal oriented error estimation and mesh adaptation with the dual weighted residual method key to applying this error estimator is the underlying canonic variational formulation of the fluid structure interaction problem by mapping the flow problem to ale coordinates the developed method is applied to two and three dimensional stationary benchmark problems coupling the incompressible navier stokes equations with a nonlinear hyper elastic material law

adaptive techniques for time dependent problems abstract both a priori and a posteriori error estimation techniques have played an important role in the adaptivity of numerical methods for static problems with important localized phenomena timedependent problems with localized temporal effects offer an even greater challenge to effective adaptivity the need for flexible data structures and solution algorithms to accommodate moving patch refinement to follow moving sharp fronts in fluid flow applications is discussed in parabolic problems often transient temporal effects are generated by spatially fixed but dynamic source terms local time stepping methods are presented to treat these transient phenomena stability and accuracy of the methods are considered results of numerical experiments are presented that indicate the relative roles of the local temporal and spatial discretizations in the performance of the methods

explicit error bounds in a conforming finite element method the goal of this paper is to define a procedure for bounding the error in a conforming finite element method the new point is that this upper bound is fully explicit and can be computed locally numerical tests prove the efficiency of the method it is presented here for the case of the poisson equation and a first order finite element approximation

an adaptive finite element strategy for the three dimensional time dependent navier stokes equations abstract an adaptive strategy for three dimensional time dependent problems in the context of the fem is presented the basic tools are a mechanism for local refinement and coarsening of simplicial meshes and an unexpensive error estimator the algorithm for local grid modification is based on bisecting tetrahedra the method is applied to the navier stokes equations

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

an adaptive finite element algorithm for a two dimensional stationary stefan like problem an efficient adaptive algorithm is presented for a stationary regularized stefan problem in 2d the adaptive criteria relies upon a posteriori estimates based on the residual equation since the problem we are studying is a non linear diffusion convection problem these error estimates are relatively crude in the sense that the constant relating the true error to the error indicator is unknown thus instead of trying to build a mesh such that the true error is below a given tolerance the local error indicator is equidistributed in a way such that the final number of triangles is close to a desired value at each iteration of the adaptive algorithm the new triangulation is obtained from the previous one by adding or deleting vertices according to the error indicator and a global mesh regeneration is performed using a delaunay mesh generator numerical experiments for two practical situations show the efficiency and the robustness of our approach

adaptive wavelet enriched hierarchical finite element model for polycrystalline microstructures abstract this paper proposes an adaptive wavelet enriched hierarchical finite element model fem to enhance computational efficiency and solution accuracy for heterogeneous domains of anisotropic elastic materials it is motivated by the need to overcome shortcomings of conventional methods like the standard fem or fast fourier transformation fft based method for analyzing deformation in large polycrystalline microstructures the multi resolution wavelet functions are advantageous for selection of an optimal set of functions that can adaptively enrich the solution space with a prescribed level of accuracy the proposed method introduces a discretization space that conforms to the profile of the solution sought the method engages a second generation family of wavelets with a lifting scheme to generate hierarchical interpolation functions an iterative algorithm efficiently calculates estimates of the solution from the previous iterate using a modified jacobi method the adaptive fe method performs significantly better than uniformly refined fem in validation tests with focus on the convergence rate and error mitigation a polycrystalline microstructure with elastically anisotropic grains is simulated by the adaptive wavelet enriched hierarchical method showing high convergence rates

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part ii the piecewise linear case we extend results from part i about estimating gradient errors elementwise a posteriori given there for quadratic and higher elements to the piecewise linear case the key to our new result is to consider certain technical estimates for differences in the error e x 1 e x 2 rather than for e x itself we also give a posteriori estimators for second derivatives on each element

on the adaptive coupling of fem and bem in 2 d elasticity this paper concerns the combination of the finite element method fem and the boundary element method bem using the symmetric coupling as a model problem in two dimensions we consider the hencky material a certain nonlinear elastic material in a bounded domain with navier lame differential equation in the unbounded complementary domain using some boundary integral operators the problem is rewritten such that the galerkin procedure leads to a fem bem coupling and quasi optimally convergent discrete solutions beside this a priori information we derive an a posteriori error estimate which allows up to a constant factor the error control in the energy norm since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh refinement we state adaptive algorithms for the h version of the fem bem coupling illustrating numerical results are included

a priori estimates for rough pdes with application to rough conservation laws we introduce a general framework to study pdes driven by rough paths we develop new a priori estimates based on a rough gronwall lemma argument for weak solutions to rough pdes this will allow us to follow standard pde strategies to obtain existence and uniqueness results in particular our approach does not rely on any sort of transformation formula flow transformation feynman kac representation formula etc and is therefore rather flexible as an application we study conservation laws driven by rough paths establishing well posedness for the corresponding kinetic formulation

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

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

a posteriori error estimates for boundary element methods this paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples namely symm s integral equation an integral equation with a hypersingular operator and a boundary integral equation for a transmission problem based on these estimates an analog of eriksson and johnson s adaptive finite element method is proposed for the h version of the galerkin boundary element method for integral equations of the first kind the efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities the construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations and we refer to the pioneering work of babuska and miller 3 and eriksson and johnson 10 11 whereas the main features ofadaptivity for finite element methods now seem to be visible and the door is open to implementation 15 comparably little is known for boundary element methods for integral equations see e g 1 13 18 19 24 in this paper a new adaptive version of the galerkin discretization for the boundary element method is presented based on a posteriori error estimates a general framework for these a posteriori error estimates is derived in 2 and three examples are discussed in 3 5 involving the dirichlet problem the neumann problem for a closed and an open surface and a transmission problem for the laplacian leading to integral equations with strongly elliptic pseudodifferential operators even for smooth data the lack of regularity of the solution near corners of a polygonal domain q leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used in practical problems such information is missing e g when we have singular or nearly singular data and the main problem is how to balance a graded mesh refinement towards singularities and a global

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

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

on an h type mesh refinement strategy based on minimization of interpolation errors abstract an adaptive finite element method is proposed which involves an automatic mesh refinement in areas of the mesh where local errors are determined to exceed a pre assigned limit the estimation of local errors is based on interpolation error bounds and extraction formulas for highly accurate estimates of second derivatives applications to several two dimensional model problems are discussed the results indicate that the method can be very effective for both regular problems and problems with strong singularities

goal oriented r adaptivity based on variational arguments in the physical and material spaces abstract the error in a local quantity of interest depends on the error in the corresponding dual solution or generalized green s function this fact is used in dual weighted based goal oriented error estimation techniques and h or p adaptive algorithms this contribution is concerned with a novel approach for goal oriented r adaptivity based on energy minimization principles for the primal and the dual problems the minimization of the energy of the primal problem as well as the minimization of the energy of the dual problem with respect to a design function lead to the primal and dual material residuals respectively these residuals are error indicators for the discretization error on the current mesh we propose a goal oriented mesh optimization algorithm which ends in an optimized mesh with respect to a chosen quantity of interest by means of selected computational examples we demonstrate the capability of the proposed framework

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

adaptive boundary element methods for some first kind integral equations in this paper we present an adaptive boundary element method for the boundary integral equations of the first kind concerning the dirichlet problem and the neumann problem for the laplacian in a two dimensional lipschitz domain for the h version of the finite element galerkin discretization of the single layer potential and the hypersingular operator we derive a posteriori error estimates which guarantee a given bound for the error in the energy norm up to a multiplicative constant following eriksson and johnson this yields adaptive algorithms steering the mesh refinement numerical examples confirm that our adaptive algorithms yield automatically the expected convergence rate

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

stabilized element residual method serm a posteriori error estimation for the advection diffusion equation residual based a posteriori error estimation techniques have been developed for linear elliptic symmetric positive definite problems one asymptotically exact error estimator for the elliptic laplacian operator relies on solving local neumann problems in each element this technique is extended to the unsymmetric and positive semi definite advection diffusion ad operator here a novel approach the stabilized element residual method serm is presented in this method the unsymmetric advection terms are retained in the formulation of the local error problem through the use of stabilized methods the selection of the optimum stabilization parameter is discussed

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

h adaptive finite element computation of time harmonic exterior acoustics problems in two dimensions abstract in this paper we carry out adaptive finite element computations of the helmholtz equation in two dimensions in the context of time harmonic exterior acoustics the purpose is to demonstrate potentially significant cost savings engendered through adaptivity for propagating solutions at moderate wave numbers ka 2 4 the computations are performed on meshes of linear triangles and are adapted to the solution by locally changing element sizes h refinement the adaptive procedure involves applying an explicit residual based a posteriori error estimator and an h adaptive strategy which were derived in a previous paper the adaptive meshes are then obtained through global mesh regeneration using an advancing front mesh generator two different finite element formulations are considered the galerkin and galerkin least squares gls methods the infinite physical domain is truncated by an artificial exterior boundary on which a fully coupled dirichlet to neumann dtn boundary condition is applied two different problems are computed plane wave scattering by a rigid infinite surface circular and square cross sections and nonuniform radiation from an infinite circular cylinder detailed cost studies with respect to an active column direct solver are performed for the nonuniform radiation problem the adaptive mesh is twenty times more cost effective than a uniform mesh for galerkin when coupled with the gls formulation the adaptive mesh is forty times more efficient than galerkin computations on a uniform mesh

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

error controlled adaptive goal oriented modeling and finite element approximations in elasticity abstract in our earlier work e stein s ohnimus coupled model and solution adaptivity in the finite element method comput methods appl mech engrg 150 1997 327 350 e stein s ohnimus anisotropic discretization and model error estimation in solid mechanics by local neumann problems comput methods appl mech engrg 176 1999 363 385 e stein m ruter s ohnimus adaptive finite element analysis and modelling of solids and structures findings problems and trends int j numer methods engrg 60 2004 103 138 residual based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of neumann problems on the element level in particular these estimators control the errors obtained while expanding the model hierarchically from 1d to 2d and from 2d to 3d elastic theories in the global energy norm in this paper we extend both types of estimates to goal oriented a posteriori error estimation within the framework of nonlinear structural mechanics structural model adaptivity is characterized by the fact that the dimensions in general the constitutive equations and especially the related subdomains and thus the test and solution spaces of the process dependent models considered are different this has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model m to those of the refined model m 1 are necessary therefore such prolongation operators are discussed in detail furthermore duality techniques are used for the model and discretization error estimates using system and load dependent quantities of interest in this paper we also consider different finite element meshes for the primal and the dual problem finally some numerical examples demonstrate the expected effects realizing verification and even validation in structural mechanics i e approaching a new paradigm in computational mechanics for achieving reliability and efficiency

mathematical foundation of adaptive boundary element methods abstract in this paper we review some mathematical aspects of adaptive boundary element methods in brief and give some a posteriori local error estimates for the h version and the p version

explicit error bounds for a nonconforming finite element method let u be the solution of the following model left begin array l mbox find u in h 1 0 omega such that 3pt delta u f quad mbox in omega end array right where f is a given function in l2 omega and omega a bounded open set in mathbb r 2 with a polygonal boundary let uh be a finite element approximation of u the goal of this paper is to suggest an explicit bound of the error u uh furthermore this bound is obtained without complex computations the basic idea is to define locally an admissible vector field for the dual model