on the asymptotic exactness of error estimators for linear triangular finite elements this paper deals with a posteriori error estimates for piecewise linear finite element approximations of elliptic problems we analyze two estimators based on recovery operators for the gradient of the approximate solution by using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution one of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation this estimator is equivalent to the error in the energy norm under rather general conditions however we show that for the asymptotic exactness the regularity assumption on the mesh is not merely technical while doing this we analyze the relation between superconvergence and asymptotic exactness for some particular examples

augemented lagrangian techniques for elliptic state constrained optimal control problems we propose augmented lagrangian methods to solve state and control constrained optimal control problems the approach is based on the lagrangian formulation of nonsmooth convex optimization in hilbert spaces developed in k ito and k kunisch augmented lagrangian methods for nonsmooth convex optimization in hilbert spaces preprint 1994 we investigate a linear optimal control problem with a boundary control function as in m bergounioux numer funct anal optim 14 1993 pp 515 543 both the equation and the constraints are augmented the proposed methods are general and can be adapted to a much wider class of problems

multigrid optimization in applications abstract iterative techniques are a key methodology for the numerical solution of optimization problems in differential equations in two practical application problems with different characteristics this paper shows how multigrid methods can be applied efficiently to this problem class problem formulations solution approaches as well as numerical results are presented

elimination of first order errors in shock calculations first order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions based on a matched asymptotic expansion analysis we show how to modify the artificial viscosity and raise the order of accuracy

grid convergence error analysis for mixed order numerical schemes new developments are presented in the area of grid convergence error analysis and error estimation for mixed order numerical schemes a mixed order scheme is defined here as a numerical method where the order of the local truncation error varies either spatially e g at a shock wave or for different terms in the governing equations e g first order convection with second order diffusion the case examined herein is the mach 8 laminar flow of a perfect gas over a sphere cone geometry this flowfield contains a strong bow shock wave where the formally second order numerical scheme is reduced to first order via a flux limiting procedure the mixedorder error analysis method allows for non monotone behavior in the solutions variables as the mesh is refined non monotonicity in the local solution variables is shown to arise from a cancellation of first and second order error terms for the present case the proposed error estimator which is based on the mixed order analysis is shown to provide good estimates of the actual error furthermore this error estimator nearly always provides conservative error estimates in the sense that the actual error is less than the error estimate for the case examined

the convergence rate of finite difference schemes in the presence of shocks finite difference approximations generically have cal o 1 pointwise errors close to a shock we show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher order methods analytic and numerical examples of this form of accuracy are given we also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock

gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes in this paper the gradient recovery type a posteriori error estimators for finite element approximations are proposed for irregular meshes both the global and the local a posteriori error estimates are derived moreover it is shown that the a posteriori error estimates is asymptotically exact on where the mesh is regular enough and the exact solution is smooth

adjoint recovery of superconvergent functionals from pde approximations motivated by applications in computational fluid dynamics a method is presented for obtaining estimates of integral functionals such as lift or drag that have twice the order of accuracy of the computed flow solution on which they are based this is achieved through error analysis that uses an adjoint pde to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy numerical results are presented for the poisson equation in one and two dimensions and for the nonlinear quasi one dimensional euler equations the theory is equally applicable to nonlinear equations in complex multi dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

convergence of finite element approximations to state constrained convex parabolic boundary control problems this paper is concerned with the numerical solution of state constrained parabolic boundary control problems by finite element approximations error estimates for the optimal values and in the coercive case for the optimal solutions are derived these estimates are used to prove convergence results under rather weak assumptions in the noncoercive case a bang bang principle is shown to obtain convergence results for the discrete controls also a discussion of several numerical examples concludes the paper

constrained lqr problems in elliptic distributed control systems with point observations in this paper we study bound constrained lqr problems in distributed control systems governed by the elliptic equation with point observations which are motivated by problems in corrosion engineering and contemporary smart materials several regularity and characterization theorems have been established in particular three decomposition formulas are obtained to characterize the optimal control and optimal layer density and are used to direct the numerical computations these results cannot be obtained by the traditional galerkin variational method in the process several useful lemmas are established which are of independent interest we point out that the classical lagrangian multiplier method lmm may fail to provide a reliable numerical algorithm based on our characterization results and the boundary element method two algorithms are proposed to carry out numerical computations it has been shown by our numerical experiments that both algorithms are efficient and insensitive to the partition number of the boundary an adaptive local refinement scheme has also been designed to handle the rough behavior of the optimal solution around sensor locations

on a posteriori pointwise error estimation using adjoint temperature and lagrange remainder the pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is addressed the truncation error is estimated using a taylor series with the remainder in the lagrange form the contribution of the local error to the total pointwise error is estimated via an adjoint temperature it is demonstrated that the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error correction and that an asymptotic error bound may be found

new elliptic projections and a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations in this paper we discuss a priori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by parabolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions both semidiscrete and fully discrete schemes are considered based on some new elliptic projections we derive a priori error estimates for the control variable the state variables and the adjoint state variables the related a priori error estimates for the new projections error are also established a numerical example is given to demonstrate the theoretical results

a remark on numerical errors downstream of slightly viscous shocks lower order errors downstream of a shock layer have been detected in computations with nonconstant solutions when using higher order shock capturing schemes in one and two dimensions b engquist and b sjogreen siam j numer anal 35 1998 pp 2464 2485 by analyzing the steady state solution of slightly viscous hyperbolic systems of conservation laws we find that the solution can have an cal o h dependence downstream of a shock layer although the viscous term in that region is of cal o h 2 numerical examples illustrate the analysis

feature driven cartesian adaptive mesh refinement for vortex dominated flows we develop locally normalized feature detection methods to guide the adaptive mesh refinement amr process for cartesian grid systems to improve the resolution of vortical features in aerodynamic wakes the methods include the q criterion 1 the l 2 method 2 the l c i method 3 and the l method 4 specific attention is given to automate the feature identification process by applying a local normalization based upon the shear strain rate so that they can be applied to a wide range of flow fields without the need for user intervention to validate the methods we assess tagging efficiency and accuracy using a series of static vortex dominated flow fields and use the methods to drive the amr process for several theoretical and practical simulations we demonstrate that the adaptive solutions provide comparable accuracy to solutions obtained on uniformly refined meshes at a fraction of the computational cost overall the normalized feature detection methods are shown to be effective in driving the amr process in an automated and efficient manner

nonhomogeneous boundary value problems the diffusion equation is solved under stochastic nonhomogeneity using eigen function expansion and the georges method the statistical moments of the solution process are computed through the two previously mentioned techniques and proved to be the same a general solution is obtained under general initial and boundary conditions a random source composed of deterministic and stochastic parts is taken into consideration the stochastic part is then restricted to a generalized gaussian field mainly modulated white noise a special case is considered under constant noise level and constant average noise a numerical case study concerning pollution in a stream is solved and a parametric study is achieved through various figures

adaptive finite element methods for optimal control of partial differential equations basic concept a new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations the lagrangian formalism yields the first order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive galerkin finite element method the mesh design in the resulting reduced models is controlled by residual based a posteriori error estimates these are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error in this case the computed state and costate variables can be used as sensitivity factors multiplying the local cell residuals in the error estimators this results in a generic and simple algorithm for mesh adaptation within the optimization process this method is developed and tested for simple boundary control problems in semiconductor models

adaptive finite element approximation for distributed elliptic optimal control problems in this paper sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems these error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work

a posteriori error estimates for elliptic variational inequalities we derive hierarchical a posteriori error estimates for elliptic variational inequalities the evaluation amounts to the solution of corresponding scalar local subproblems we derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples

a posteriori error estimates for control problems governed by stokes equations in this paper we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the stokes equations we obtain a posteriori error estimators for both the state and the control approximation in the l2 norm and the h1 norm these estimates can be used to construct reliable adaptive finite element approximation for the control problems

error estimates and superconvergence of mixed finite element methods for fourth order hyperbolic control problems abstract in this paper we investigate the error estimates and superconvergence of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear fourth order hyperbolic equations the state and the co state are discretized by the order k raviart thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k k 0 we derive error estimates for both the state and the control approximation moreover we present the superconvergence analysis for mixed finite element approximation of the optimal control problems

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

ritz galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions this paper deals with ritz galerkin approximations of the following two problems i boundary value problems with l 2 boundary data given in the form of dirichlet boundary conditions ii time optimal control problems for parabolic systems with control acting on the boundary in both cases optimal rates of convergence of approximation are obtained also the specialization is considered to approximating with nonconformal elements which are not required to satisfy boundary conditions corresponding rates of convergence are also proved

adjoint correction and bounding of error using lagrange form of truncation term the a posteriori error evaluation based on differential approximation of a finite difference scheme and adjoint equations is addressed the differential approximation is composed of primal equations and a local truncation error determined by a taylor series in lagrange form this approach provides the feasibility of both refining the solution and using the holder inequality for asymptotic bounding of the remaining error

on the structure of lagrange multipliers for state constrained optimal control problems abstract state constrained optimal control problems represent severe analytical and numerical challenges it is shown that under certain conditions on the active set the measure representing the lagrange multiplier associated to the state constraint can be decomposed in a distributed part in l2 with support on the active set and a boundary measure concentrated on the interface between active and inactive sets

a posteriori error estimates for optimal control problems governed by parabolic equations in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation we obtain a posteriori error estimates for both the state and the control approximation such estimates which are apparently not available in the literature are an important step towards developing reliable adaptive finite element approximation schemes for the control problem

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

an adjoint based a posteriori estimation of iterative convergence error the iterative convergence error of goal functional is considered for solving the steady problem by time iterations the functional error is calculated using adjoint parameter and time derivative the numerical tests demonstrate the applicability of this approach for linear and nonlinear heat conduction equations

the numerical solution of nonlinear two dimensional volterra fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis in this paper we present a numerical method for solving two dimensional nonlinear volterra fredholm integral equations of the second kind the method approximates the solution by the discrete collocation method based on radial basis functions rbfs constructed on a set of disordered data the proposed method is meshless since it does not require any background mesh or domain elements error analysis of this method is also investigated numerical examples which compare the proposed method with 2d tfs method 4 approve its supremacy in terms of accuracy and computational cost using various rbfs we have concluded that mq rbf is the best choice for the proposed method

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

grid adaptation for functional outputs application to two dimensional inviscid flows this paper presents an error estimation and grid adaptive strategy for estimating and reducing simulation errors in integral outputs functionals of partial differential equations adaptive criteria are derived using an adjoint based error correction technique that relates the local residual errors of both the primal and adjoint solutions to the global error in the prescribed functional this relationship allows local error contributions to be used as indicators in a grid adaptive strategy designed to produce specially tuned grids for accurately estimating the chosen functional in this paper attention is limited to two dimensional inviscid flows using a standard finite volume discretization although the procedure may be readily applied to other types of multidimensional problems and discretizations numerical examples demonstrate that the proposed adaptive procedure compares favorably in terms of accuracy efficiency and robustness relative to a commonly used gradient based adaptive method

a priori and a posteriori error estimates of h 1 galerkin mixed finite element methods for optimal control problems governed by pseudo hyperbolic integro differential equations abstract in this paper we investigate a priori and a posteriori error estimates of h1 galerkin mixed finite element methods for optimal control problems governed by pseudo hyperbolic integro differential equations the state variables and co state variables are approximated by the lowest order raviart thomas mixed finite element and linear finite element and the control variable is approximated by piecewise constant functions based on two new elliptic projections we derive a priori error estimates both for the control variable the state variable and the co state variable the related a priori error estimates for the new projections error are also established moreover a posteriori error estimates for all variables are derived via energy method such a posteriori error estimates which are apparently not available in the literature are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem

optimal control for variational inequalities consider the problem of maximizing a functional which depends on a control function k and on the solution of an elliptic variational inequality with k appearing in the data the variational problem for k is nondiflerentiable and nonconvex we obtain necessary conditions on a maximizes k 0 and then use them to determine the structure of k 0 in some cases

local a posteriori error estimates for convex boundary control problems in this paper we derive some improved a posteriori error estimates for finite element approximation of neumann boundary control problems we first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems we then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications

central gaussian semigroups of measures with continuous density abstract this paper investigates the existence and properties of symmetric central gaussian semigroups t t 0 which are absolutely continuous and have a continuous density x t x t 0 with respect to haar measure on groups of the form r n k where k is compact connected locally connected and has a countable basis for its topology we prove that there always exists a wealth of such gaussian semigroups on any such group for instance if is any positive function increasing to infinity there exists a symmetric central gaussian semigroup having a continuous density such that log t e log 1 1 t 1 t as t tends to zero among other results of this type we give a necessary and sufficient condition on the structure of k for the existence of symmetric central gaussian semigroups having a continuous density and such that t log t e is bounded above and below by positive constants for t 0 1 and some fixed 0 this condition is independent of these results are proved by splitting any gaussian semigroup in a canonical way into a semisimple part living on the commutator group g and an abelian part living on a g g for symmetric central gaussian semigroups we show that many properties hold for t t 0 if and only if they hold for both the semisimple and the abelian parts this splitting principle is one of the main new tools developed in this paper it leads to a much better understanding of central gaussian semigroups and related objects and lets us answer a number of open questions for instance to any gaussian convolution semigroup are associated a harmonic sheaf h and a quasi distance d on g for symmetric central gaussian semigroups on g r n k we show that h is a brelot sheaf if and only if limt 0 t log t e 0 a sufficient but not necessary condition is that the distance d is continuous together with a celebrated result of bony our results show that the compact group k is a lie group if and only if any bi invariant elliptic bauer harmonic sheaf on g is a brelot sheaf

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

optimal control of flow with discontinuities optimal control of the 1 d riemann problem of euler equations is studied with the initial values for pressure and density as control parameters the least squares type cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window existence of solutions for the optimal control problem is proven smooth and nonsmooth optimization methods employ the numerical gradient respectively a subgradient of the cost functional obtained from the adjoint of the discrete forward model the numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations the algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon except when the observations corresponding to shocks are detected and removed

finite element methods for optimal control problems governed by integral equations and integro differential equations in this paper we analyze finite element galerkin discretizations for a class of constrained optimal control problems that are governed by fredholm integral or integro differential equations the analysis focuses on the derivation of a priori error estimates and a posteriori error estimators for the approximation schemes

some dichotomy results for central gaussian convolution semigroups on compact groups consider a central gaussian convolution semigroup t t 0 on a connected compact semisimple group then either the measure t is singular with respect to haar measure for all t 0 or there exists a time t such that t is absolutely continuous with respect to haar measure and admits a continuous density

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

on accuracy of adaptive grid methods for captured shocks the accuracy of two grid adaptation strategies grid redistribution and local grid refinement is examined by solving the 2 d euler equations for the supersonic steady flow around a cylinder second and fourth order linear finite difference shock capturing schemes based on the lax friedrichs flux splitting are used to discretize the governing equations the grid refinement study shows that for the second order scheme neither grid adaptation strategy improves the numerical solution accuracy compared to that calculated on a uniform grid with the same number of grid points for the fourth order scheme the dominant first order error component is reduced by the grid adaptation while the design order error component drastically increases because of the grid nonuniformity as a result both grid adaptation techniques improve the numerical solution accuracy only on the coarsest mesh or on very fine grids that are seldom found in practical applications because of the computational cost involved similar error behavior has been obtained for the pressure integral across the shock a simple analysis shows that both grid adaptation strategies are not without penalties in the numerical solution accuracy based on these results a new grid adaptation criterion for captured shocks is proposed