effective postprocessing for equilibration a posteriori error estimators guaranteed error control via fully discrete a posteriori error estimators is possible with typical overestimation between 1 25 and 2 in simple computer benchmarks the equilibration techniques due to braess and that due to luce wohlmuth are efficient tools with an accuracy limited by the hyper circle threshold this motivates postprocessing strategies and the analysis of successive improvements of guaranteed upper error bounds with a few pcg iterations result in reduced overestimation factors between 1 and 1 25 numerical simulations for three classes of applications illustrate the efficiency for the poisson model problem with and without jumping coefficients or a simple obstacle problem

brief paper a revisited tsypkin criterion for discrete time nonlinear lur e systems with monotonic sector restrictions this paper revisits a well known tsypkin criterion for stability analysis of discrete time nonlinear luroe systems when nonlinearities are monotonic and sector restricted by 0 1 where 1 is positive de nite it is shown by kapila and haddad that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k ng z is strictly positive real wherek is nonnegative diagonal and g z represents a transfer function of the linear part of the luroe system with invertible or identically zero g 0 this paper extends this criterion when 1 is positive diagonal by choosing a new lyapunov function to obtain an lmi criterion from a frequency domain interpretation of this lmi criterion another su cient criterion is generated which establishes that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k 1 z k n g z is strictly positive real wherek andk are nonnegative diagonal and orthogonal to each other 1998 elsevier science ltd all rights reserved

an optimal adaptive finite element method for the stokes problem a new adaptive finite element method for solving the stokes equations is developed which is shown to converge with the best possible rate the method consists of 3 nested loops the outermost loop consists of an adaptive finite element method for solving the pressure from the elliptic schur complement system that arises by eliminating the velocity each of the arising finite element problems is a stokes type problem with the pressure being sought in the current trial space and the divergence free constraint being reduced to orthogonality of the divergence to this trial space such a problem is still continuous in the velocity field in the middle loop its solution is approximated using the uzawa scheme in the innermost loop the solution of the elliptic system for the velocity field that has to be solved in each uzawa iteration is approximated by an adaptive finite element method

goal oriented adaptivity for gmsfem in this paper we develop two goal oriented adaptive strategies for a posteriori error estimation within the generalized multiscale finite element framework in this methodology one seeks to determine the number of multiscale basis functions adaptively for each coarse region to efficiently reduce the error in the goal functional our first error estimator uses a residual based strategy where local indicators on each coarse neighborhood are the product of local indicators for the primal and dual problems respectively in the second approach viewed as the multiscale extension of the dual weighted residual method dwr the error indicators are computed as the pairing of the local h 1 residual of the primal problem weighed by a projection into the primal space of the h 0 1 dual solution from an enriched space over each coarse neighborhood in both of these strategies the goal oriented indicators are then used in place of a standard residual based indicator to mark coarse neighborhoods of the mesh for further enrichment in the form of additional multiscale basis functions the method is demonstrated on high contrast problems with heterogeneous multiscale coefficients and is seen to outperform the standard residual based strategy with respect to efficient reduction of error in the goal function

goal oriented adaptivity using unconventional error representations for the multidimensional helmholtz equation in goal oriented adaptivity goa the error in the quantity of interest qoi is represented using the error functions of the direct and adjoint problems this error representation is subsequently bounded above by element wise error indicators that are used to drive optimal refinements in this work we propose to replace in the error representation the adjoint problem by an alternative operator the main advantage of the proposed approach is that when judiciously selecting such alternative operator the corresponding upper bound of the error representation becomes sharper leading to a more efficient goa while the method can be applied to a variety of problems we focus here on two and threedimensional 2d and 3d helmholtz problems we show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p adaptive process we also provide guidelines for finding operators delivering sharp error representation upper bounds we further extend the results to a convectiondominated diffusion problem as well as to problems with discontinuous material coefficients finally we consider a sonic logging while drilling lwd problem to illustrate the applicability of the proposed method

a goal oriented adaptive finite element method with convergence rates an adaptive finite element method is analyzed for approximating functionals of the solution of symmetric elliptic second order boundary value problems we show that the method converges and derive a favorable upper bound for its convergence rate and computational complexity we illustrate our theoretical findings with numerical results

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

adaptive fem with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems we analyze adaptive mesh refining algorithms for conforming finite element discretizations of certain nonlinear second order partial differential equations we allow continuous polynomials of arbitrary but fixed polynomial order the adaptivity is driven by the residual error estimator we prove convergence even with optimal algebraic convergence rates in particular our analysis covers general linear second order elliptic operators unlike prior works for linear nonsymmetric operators our analysis avoids the interior node property for the refinement and the differential operator has to satisfy a g rarding inequality only if the differential operator is uniformly elliptic no additional assumption on the initial mesh is posed

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

a machine learning minimal residual ml mres framework for goal oriented finite element discretizations abstract we introduce the concept of machine learning minimal residual ml mres finite element discretizations of partial differential equations pdes which resolve quantities of interest with striking accuracy regardless of the underlying mesh size the methods are obtained within a machine learning framework during which the parameters defining the method are tuned against available training data in particular we use a provably stable parametric petrov galerkin method that is equivalent to a minimal residual formulation using a weighted norm while the trial space is a standard finite element space the test space has parameters that are tuned in an off line stage finding the optimal test space therefore amounts to obtaining a goal oriented discretization that is completely tailored towards the quantity of interest we use an artificial neural network to define the parametric family of test spaces using numerical examples for the laplacian and advection equation in one and two dimensions we demonstrate that the ml mres finite element method has superior approximation of quantities of interest even on very coarse meshes

goal oriented adaptive finite element methods for elliptic problems revisited a goal oriented a posteriori error estimation of an output functional for elliptic problems is presented continuous finite element approximations are used in quadrilateral and triangular meshes the algorithm is similar to the classical dual weighted error estimation however the dual weight contains solutions of the proposed patch problems the patch problems are introduced to apply clement and scott zhang type interpolation operators to estimate point values with the finite element polynomials the algorithm is shown to be reliable efficient and convergent

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

quasi optimality of adaptive finite element methods for controlling local energy errors a rich theory demonstrating convergence and optimality of adaptive finite element methods afem has been developed in recent years in this work we prove optimality of afem which are designed to control local energy errors in elliptic partial differential equations because errors propagate globally in fem controlling local errors requires controlling both local energy solution properties and global error contributions pollution errors which may be measured in a weaker norm such as the l 2 l2 norm we define adaptive methods which control both of these error components and prove that they converge with the best possible rate over all possible refinements of the initial mesh these results are valid for poisson s problem on convex polyhedral domains in arbitrary space dimension our theory establishes afem optimality for several adaptive marking strategies which rigorously control pollution effects we also present numerical examples that illustrate our theory and confirm that local energy afem without pollution control can fail to yield optimal meshes

a less conservative lmi condition for stability of discrete time systems with slope restricted nonlinearities many conditions have been found for the absolute stability of discrete time lur e systems in the literature it is advantageous to find convex searches via lmis where possible in this technical note we construct two less conservative lmi conditions for discrete time systems with slope restricted nonlinearities the first condition is derived via lyapunov theory while the second is derived via the theory of integral quadratic constraints iqcs and noncausal zames falb multipliers both conditions are related to the jury lee criterion most appropriate for systems with such nonlinearities and the second generalizes it numerical examples demonstrate a significant reduction in conservatism over competing approaches

edge residuals dominate a posteriori error estimates for low order finite element methods we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in h1 and l2 norms we present two proofs one uses the standard l2 projection and the other relies on a new weighted clement type interpolation operator

a multivariable extension of the tsypkin criterion using a lyapunov function approach for analyzing the stability of discrete time systems containing a feedback nonlinearity the tsypkin criterion is the closest analog to the popov criterion which is used for analyzing such systems in continuous time traditionally the proof of this criterion is based upon input output properties and function analytic methods in this paper we extend the tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector bounded memoryless time invariant nonlinearities along with providing a lyapunov function proof for this classical result

worst case multi objective error estimation and adaptivity this paper introduces a new computational methodology for determining a posteriori multi objective error estimates for finite element approximations and for constructing corresponding quasi optimal adaptive refinements of finite element spaces as opposed to the classical goal oriented approaches which consider only a single objective functional the presented methodology applies to general closed convex subsets of the dual space and constructs a worst case error estimate of the finite element approximation error this worst case multi objective error estimate conforms to a dual weighted residual in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error we regard both standard approximation errors and data incompatibility errors associated with incompatibility of boundary data with the trace of the finite element space numerical experiments are presented to demonstrate the efficacy of applying the proposed worst case multi objective error estimate in adaptive refinement procedures

local bisection refinement for n simplicial grids generated by reflection a simple local bisection refinement algorithm for the adaptive refinement of n simplicial grids is presented the algorithm requires that the vertices of each simplex be ordered in a special way re

an optimal adaptive finite element method although existing adaptive finite element methods for solving second order elliptic equations often perform better in practical computations than nonadaptive ones usually they are not even proven to converge only recently in the work of dorfler siam j numer anal 33 1996 pp 1106 1124 and that of morin nochetto and siebert siam j numer anal 38 2000 pp 466 488 adaptive methods were constructed for which convergence could be demonstrated however convergence alone does not imply that the method is more efficient than its nonadaptive counterpart in numer math 97 2004 pp 219 268 binev dahmen and devore added a coarsening step to the routine of morin nochetto and siebert and proved that the resulting method is quasi optimal in the following sense if the solution is such that for some s 0 the error in energy norm of the best continuous piecewise linear approximations subordinate to any partition with n triangles is mathcal o n s then given an eps 0 the adaptive method produces an approximation with an error less than eps subordinate to a partition with mathcal o eps 1 s triangles in only mathcal o eps 1 s operations in this paper employing a different type of adaptive partition we develop an adaptive method with properties similar to those of binev dahmen and devore s method but unlike their method our coarsening routine will be based on a transformation to a wavelet basis and we expect it to have better quantitative properties furthermore all our results are valid uniformly in the size of possible jumps of the diffusion coefficients since the algorithm uses solely approximations of the right hand side we can even allow right hand sides in h 1 omega that lie outside l 2 omega at least when they can be sufficiently well approximated by piecewise constants in our final adaptive algorithm all tolerances depend on an a posteriori estimate of the current error instead of an a priori one this can be expected to provide quantitative advantages

parameter dependent lyapunov functions and the discrete time popov criterion for robust analysis abstract parameter dependent lyapunov functions are developed for discrete time systems with constant real parameter uncertainty this construction of a family of lyapunov functions is then used to analyze robust stability with h 2 performance bounds for state space systems a special case of these results generalizes the classical discrete time popov criterion

weighted marking for goal oriented adaptive finite element methods we propose a new goal oriented adaptive finite element method which is based on the weighting of the residuals from the primal and dual problem at each step of the iteration our main result is the quasi optimality of the resulting algorithm numerical experiments are reported showing the convergence behavior of the algorithm

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

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

goal oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers abstract we derive a unified framework for goal oriented a posteriori estimation covering in particular higher order conforming nonconforming and discontinuous galerkin finite element methods as well as the finite volume method the considered problem is a model linear second order elliptic equation with inhomogeneous dirichlet and neumann boundary conditions and the quantity of interest is given by an arbitrary functional composed of a volumetric weighted mean value source term and a surface weighted mean dirichlet boundary flux term we specifically do not request the primal and dual discrete problems to be resolved exactly allowing for inexact solves our estimates are based on h div conforming flux reconstructions and h 1 conforming potential reconstructions and provide a guaranteed upper bound on the goal error the overall estimator is split into components corresponding to the primal and dual discretization and algebraic errors which are then used to prescribe efficient stopping criteria for the employed iterative algebraic solvers numerical experiments are performed for the finite volume method applied to the darcy porous media flow problem in two and three space dimensions they show excellent effectivity indices even in presence of primal and dual algebraic errors and enable to spare a large percentage of unnecessary algebraic iterations

goal oriented error estimation and adaptivity for free boundary problems the domain map linearization approach in free boundary problems the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation we develop duality based a posteriori error estimates for functional outputs of solutions of free boundary problems that include both sources of error the derivation of an appropriate dual problem linearized adjoint is however nonobvious for free boundary problems to derive an appropriate dual problem we present the domain map linearization approach in this approach the free boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map we show for a bernoulli type free boundary problem that this dual problem corresponds to a poisson problem with a nonlocal robin type boundary condition furthermore we present numerical experiments that demonstrate the effectivity of the dual based error estimate and its usefulness in goal oriented adaptive mesh refinement

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

adaptive boundary element methods for optimal convergence of point errors one particular strength of the boundary element method is that it allows for a high order pointwise approximation of the solution of the related partial differential equation via the representation formula however the high order convergence and hence accuracy usually suffers from singularities of the cauchy data we propose two adaptive mesh refining algorithms and prove their quasi optimal convergence behavior with respect to an a posteriori computable bound for the point error in the representation formula numerical examples for the weakly singular integral equations for the 2d and 3d laplacian underline our theoretical findings