error estimates of mixed finite element methods for quadratic optimal control problems in this paper we investigate the error estimates for the solutions of optimal control problems by mixed finite element methods the state and costate are approximated by raviart thomas mixed finite element spaces of order k and the control is approximated by piecewise polynomials of order k under the special constraint set we will show that the control variable can be smooth in the whole domain we derive error estimates of optimal order both for the state variables and the control variable

invariant and attractor sets for fuzzy dynamical systems abstract in this work we establish conditions for the existence and stability of invariant sets for dynamical systems defined on metric space of fuzzy subsets of r n we show that the results presented in this work generalize some conditions of existence and stability of equilibrium points and periodicity of fuzzy differential equations solutions

continuous dependence and convergence results for brinkman and forchheimer models with variable viscosity the equations for convective fluid motion in a porous medium of brinkman or forchheimer type are analysed when the viscosity varies with either temperature or a salt concentration mundane situations such as salinization require models which incorporate strong viscosity variation therefore we establish rigorous a priori bounds with coefficients which depend only on boundary data initial data and the geometry of the problem which demonstrate continuous dependence of the solution on changes in the viscosity a convergence result is established for the darcy equations when the variable viscosity is allowed to tend to a constant viscosity

fuzzy differential equations and the extension principle we study the cauchy problem for differential equations considering its parameters and or initial conditions given by fuzzy sets these fuzzy differential equations are approached in two different ways a by using a family of differential inclusions and b the zadeh extension principle for the solution of the model we conclude that the solutions of the cauchy problem obtained by both are the same we also provide some illustrative examples

generic simplicity of the spectrum and stabilization for a plate equation in this work we prove the generic simplicity of the spectrum of the clamped plate equation in a bounded regular domain of mathbb r d that is given omega subset mathbb r d we show that there exists an arbitrarily small deformation of the domain u such that all the eigenvalues of the plate system in the deformed domain omega u are simple to prove this result we first prove a nonstandard unique continuation property for this system that also holds generically with respect to the perturbations of the domain both the proof of this generic uniqueness result and the generic simplicity of the spectrum use baire s lemma and shape differentiation finally we show an application of this unique continuation property to a result of generic stabilization for a plate system with one dissipative boundary condition

alternating direction iterative technique for mixed finite element methods for stokes equations alternating direction iterative technique for solving the algebraic systems associated with mixed finite element methods for two and three dimensional stokes equations and the navier stokes equations are formulated this method due to brown brown alternating direction iterative schemes for mixed finite element methods for second order elliptic problems thesis university of chicago 1982 is related to the uzawa iterative method for saddle point problems stabilization analysis for the uzawa iterative method is given

a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

a framework for the adaptive finite element solution of large scale inverse problems since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging efficient methods are indispensable in this paper we will introduce a framework for the efficient solution of such problems this comprises the use of adaptive finite element schemes solvers for the large linear systems arising from discretization and methods to treat additional information in the form of inequality constraints on the parameter to be recovered the methods to be developed will be based on an all at once approach in which the inverse problem is solved through a lagrangian formulation the main feature of the paper is the use of a continuous function space setting to formulate algorithms in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed this entails that steps such as the description of a newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions on the other hand this approach avoids the dependence of finite dimensional norms on the mesh size making individual steps of the algorithm comparable even if they used differently refined meshes numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10 000 parameters

ciarlet raviart mixed finite element approximation for an optimal control problem governed by the first bi harmonic equation the ciarlet raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi harmonic equation the optimality conditions consisting of the state and the co state equations is derived also the a priori error estimates are analyzed in the analysis of the a priori error estimates the improved convergent rate of the higher order than existed results is proved some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

error analysis for optimal control problem governed by convection diffusion equations dg method in the current paper we study the convergence properties of the dgfe approximation of optimal control problem governed by convection diffusion equations we derive a posteriori error estimates and a priori error estimates for both the states ad joint and the control variable approximation for the optimal control problem these estimates are apparently not available in the literature

multiscale asymptotic analysis and computation of optimal control for elliptic systems with constraints this paper reports the multiscale analysis and numerical algorithms for a class of distributed elliptic optimal control problems with constraints multiscale asymptotic expansions are presented and an explicit rate of convergence is derived a multiscale numerical method is developed numerical experiments are carried out to validate the predicted convergence results

on the stability of fuzzy dynamical systems abstract in this work we study the asymptotic behavior of fuzzy solutions obtained by using zadeh s extension at the deterministic solutions of initial value problems we obtain some result regarding the existence of fuzzy equilibrium points that generalize the already known results as show earlier the membership function of the fuzzy equilibrium points may be obtained in a relatively simple way also we study the projection of the fuzzy solution on the appropriate subspaces of the phase space several examples are given to illustrate the obtained results

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

a priori and a posteriori error estimates of h 1 galerkin mixed finite element methods for elliptic optimal control problems in this paper we investigate numerical approximations of h 1 galerkin mixed finite element methods for elliptic optimal control problems the presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables moreover the matching relation i e lbb condition between the mixed element spaces v h and w h is not necessary thus we can choose the approximation spaces more flexibly the state and co state are approximated by the lowest order raviart thomas mixed finite element spaces and the standard finite element spaces the control variable is approximated by piecewise constant functions we derive a priori and a posteriori error estimates for the control variable the state variables and the adjoint state variables finally some numerical examples are given to demonstrate the theoretical results about a priori error estimates

performance of a parallel mixed finite element implementation for fourth order clamped anisotropic plate bending problems in distributed memory environments the paper deals with a parallel implementation of a mixed finite element method of solution of a fourth order elliptic partial differential equation which in the particular case of aniso ortho isotropic plate bending problems gives a simultaneous approximation to the bending moment tensor field j j i j 1 i j 2 and the displacement field u the parallel implementation has been done in silicon graphics origin 3800 a 4 processor intel xeon symmetric multiprocessor and a beowulf cluster of four intel pentium iii pcs the indefinite linear system obtained after discretization using mixed finite element method is solved in parallel using direct schur s complement approach and iterative method using argonne national laboratory s petsc package the numerical results obtained and the time speed up comparisons for the direct method and iterative methods for a few interesting examples are also given

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

adaptive finite element approximation for a class of parameter estimation problems in this paper we investigate the finite element approximation of a class of parameter estimation problems which are the form of performance as the optimal control problems governed by bilinear parabolic equations we propose to use adaptive multi meshes in developing efficient algorithms for the estimation problems we derive a posteriori error estimators for both the control and state approximations which particularly suit an adaptive multi meshes finite element scheme the error estimators are then implemented and tested with promising numerical results

a mixed finite element method for fourth order elliptic equations with variable coefficients abstract a new mixed finite element method for the dirichlet problem of fourth order elliptic equations with variable coefficients on convex polygonal domain has been developed in this paper the biharmonic problem and bending problems of elastic plates being particular cases of the general problem considered in the paper for bending problems this method gives a simultaneous approximation to the displacement u and the actual bending and twisting moment ij i j 1 2 error estimate for the mixed finite element solution has been given

finite element approximation of parabolic time optimal control problems the numerical approximation of a parabolic time optimal control problem via piecewise linear splines is considered at each stage of the approximation a bang bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem the well posedness of the approximation scheme is shown and the rate of convergence to the exact solution investigated numerical results for some one and two dimensional parabolic control problems are given

goal oriented adaptivity in pointwise state constrained optimal control of partial differential equations we derive primal dual weighted goal oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations the constraints give rise to a primal dual weighted error term representing the mismatch in the complementarity system due to discretization in the case of sufficiently regular active or coincidence sets and problem data a further decomposition of the multiplier into a regular l 2 part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error the paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points

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 moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

optimal control problem of the two dimensional modified anomalous subdiffusion equation with discontinuous galerkin approximation abstract in this paper we consider an optimal control problem ocp governed by a modified anomalous subdiffusion equation with box constraints on the control we apply space time discontinuous galerkin method to discretize the problem specifically symmetric interior penalty galerkin method and piecewise discontinuous constant temporal approximations are used then we discuss the equivalence of optimize then discretize and discretize then optimize approaches in a general setting we derive stability and convergence analysis for the fully discrete problem at the end we present some numerical results to measure the numerical rate of convergence

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

fast intersections on nested tetrahedrons fint an algorithm for adaptive finite element based distributed parameter estimation a variety of biomedical imaging techniques such as optical and fluorescence tomography electrical impedance tomography and ultrasound imaging can be cast as inverse problems wherein image reconstruction involves the estimation of spatially distributed parameter s of the pde system describing the physics of the imaging process finite element discretization of imaged domain with tetrahedral elements is a popular way of solving the forward and inverse imaging problems on complicated geometries a dual adaptive mesh based approach wherein one mesh is used for solving the forward imaging problem and the other mesh used for iteratively estimating the unknown distributed parameter can result in high resolution image reconstruction at minimum computation effort if both the meshes are allowed to adapt independently till date no efficient method has been reported to identify and resolve intersection between tetrahedrons in independently refined or coarsened dual meshes herein we report a fast and robust algorithm to identify and resolve intersection of tetrahedrons within nested dual meshes generated by 8 similar subtetrahedron subdivision scheme the algorithm exploits finite element weight functions and gives rise to a set of weight functions on each vertex of disjoint tetrahedron pieces that completely cover up the intersection region of two tetrahedrons the procedure enables fully adaptive tetrahedral finite elements by supporting independent refinement and coarsening of each individual mesh while preserving fast identification and resolution of intersection the computational efficiency of the algorithm is demonstrated by diffuse photon density wave solutions obtained from a single and a dual mesh and by reconstructing a fluorescent inclusion in simulated phantom from boundary frequency domain fluorescence measurements

analysis of new augmented lagrangian formulations for mixed finite element schemes two new augmented lagrangian formulations for mixed finite element schemes are presented the methods lead in some cases to an improvement in the order of the approximation an error analysis is provided together with some interesting examples of applications

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

a posteriori error estimates for discontinuous galerkin time stepping method for optimal control problems governed by parabolic equations in this paper we examine the discontinuous galerkin dg finite element approximation to convex distributed optimal control problems governed by linear parabolic equations where the discontinuous finite element method is used for the time discretization and the conforming finite element method is used for the space discretization we derive a posteriori error estimates for both the state and the control approximation assuming only that the underlying mesh in space is nondegenerate for problems with control constraints of obstacle type which are the kind most frequently met in applications further improved error estimates are obtained

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

fuzzy differential equations for universal oscillators abstract this study considers universal oscillator fuzzy differential equations the differential inclusion method is applied to this problem by considering the universal oscillator fuzzy differential equations as universal oscillator uncertain dynamical systems three types of damped oscillators are considered comprising underdamped critically damped and overdamped oscillators the existence and uniqueness of the solutions to each case are discussed and proofs are given under some certain conditions for universal oscillator uncertain dynamical systems with an undamped forcing function the existence and uniqueness of the solutions and big solutions are also verified while the scopes of the trajectories to the solutions are obtained based on the inclusion relationship between solutions and big solutions we also prove the existence and uniqueness of solutions to universal oscillator uncertain dynamical systems with a damped forcing function by using other more direct conditions

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

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

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

solving a class of linear projection equations many interesting and important constrained optimization problems in mathematical programming can be translated into an equivalent linear projection equation u p omega u mu q here p omega cdot is the orthogonal projection on some convex set omega e g omega bbb r n and m is a positive semidefinite matrix this paper presents some new methods for solving a class of linear projection equations the search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization based on the idea of a projection and contraction method 7 we get a simple step length rule and are able to obtain global linear convergence

a discontinuous galerkin method for optimal control problems governed by a system of convection diffusion pdes with nonlinear reaction terms in this paper we study the numerical solution of optimal control problems governed by a system of convection diffusion pdes with nonlinear reaction terms arising from chemical processes the symmetric interior penalty galerkin sipg method with upwinding for the convection term is used as a discretization method we use a residual based error estimator for the state and the adjoint variables an adaptive mesh refinement indicated by a posteriori error estimates is applied the arising saddle point system is solved using a suitable preconditioner numerical results are presented to illustrate the performance of the proposed error estimator

some properties of solutions for a class of semi linear uncertain dynamical systems in this paper some properties of solutions are obtained for semi linear fuzzy differential equations understood as differential inclusions i e semi linear uncertain dynamical systems the definitions of the solution to the semi linear uncertain dynamical system and its corresponding big solution are given the existence of the two kinds of solutions is established and the inclusion relation between them is also obtained

adaptive finite element method for elliptic optimal control problems convergence and optimality in this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints we use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable based on the well established convergence theory of afem for elliptic boundary value problems we rigorously prove the convergence and quasi optimality of afem for optimal control problems with respect to the state and adjoint state variables by using the so called perturbation argument numerical experiments confirm our theoretical analysis

conjugate gradient boundary element solution for distributed elliptic optimal control problems an optimality system of equations for the optimal control problem governed by helmholtz type equations is derived by the associated first order necessary optimality condition we obtain the conjugate gradient method cgm in the continuous case introducing the sequence of higher order fundamental solutions we propose an iterative algorithm based on the conjugate gradient boundary element method using the multiple reciprocity method cgm mrbem for solving the discrete control input this algorithm has an advantage over that of the existing literatures because the main attribute the reduced dimensionality of the boundary element method is fully utilized finally the local error estimates for this scheme are obtained and a test problem is given to illustrate the efficiency of the proposed method

a mixed finite element method for fourth order eigenvalue problems a mixed finite element method for approximating eigenpairs of iv order elliptic eigenvalue problems with dirichlet boundary conditions has been given the method can be applied to the vibration analysis of anisotropic orthotropic isotropic biharmonic plates computer implementation procedures for this mixed method are given along with the results of numerical experiments

adaptive optimal control approximation for solving a fourth order elliptic variational inequality an optimal control approach is proposed to solve the fourth order elliptic variational inequality with curvature obstacle it is proved that the variational inequality is equivalent to the constrained optimal control problem the finite element approximation of the optimal control problem is constructed and the a priori error estimates and the equivalent a posteriori error estimators are derived some numerical experiments are performed to confirm a priori error estimates and demonstrate the effectiveness of the a posteriori estimators

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

continuous dependence and decay for the forchheimer equations the forchheimer equations for non slow flow in a saturated porous medium are studied we demonstrate that the solution depends continuously on changes in the forchheimer coefficients in addition upper and lower bounds are derived for the energy decay rate these establish the energy decays exponentially but not faster than this

superconvergence of mixed finite element methods for optimal control problems t in this paper we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods the state and co state variables are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique we derive l 2 superconvergence properties for the flux functions along the gauss lines and for the scalar functions at the gauss points via mixed projections moreover global l 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique thus based on these superconvergence estimates some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

weak series bv convergence of a moving finite element method for singular axisymmetric harmonic maps we prove the convergence of a moving finite element method for the minimization of a relaxed dirichlet energy among axisymmetric maps from the disc into the sphere the optimal mesh and the discrete minimizer exist for a discrete relaxed energy similarly to the true solution the discrete minimizer has a boundary layer because of the consistency error introduced by the discretization of the energy this discrete minimizer is nonconforming we show that it converges to the solution of the continuous problem in an appropriate bv space the proof is done for three different piecewise linear finite element discretizations

an interior penalty method for distributed optimal control problems governed by the biharmonic operator in this paper a c 0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator the state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations a priori and a posteriori error estimates are derived for the state adjoint and control variables under minimal regularity assumptions numerical results justify the theoretical results obtained the a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement

mixed finite element and boundary element approximation in 3d magnetostatics for computation of the magnetic induction we describe a mixed method for the computation of the magnetic induction in three dimensional magnetostatics this method uses mixed finite elements in association with boundary elements we present numerical results and make a comparison between these results and those obtained from a classical method with magnetic scalar potential we conclude that our method leads to better numerical accuracy

a posteriori error estimates for hp finite element solutions of convex optimal control problems in this paper we present a posteriori error analysis for hp finite element approximation of convex optimal control problems we derive a new quasi interpolation operator of clement type and a new quasi interpolation operator of scott zhang type that preserves homogeneous boundary condition the scott zhang type quasi interpolation is suitable for an application in bounding the errors in l 2 norm then hp a posteriori error estimators are obtained for the coupled state and control approximations such estimators can be used to construct reliable adaptive finite elements for the control problems

a parallel mixed finite element implementation of fourth order plate bending problems in distributed memory environments the paper deals with a parallel implementation of a mixed finite element method of solution of general fourth order elliptic partial differential equations which in the case of aniso ortho isotropic plate bending problems simultaneously approximates the bending moment tensor field j j i j 1 i j 2 and the displacement field u the experiments on the performance of this method have been done in two parallel environments namely silicon graphics origin 3800 and cray t3e the details of the parallel computer implementation procedures for this mixed method with time speed up comparisons for a few illustrative examples are given

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

superconvergence of quadratic optimal control problems by triangular mixed finite element methods the aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method the state and co state are approximated by the order k 1 raviart thomas mixed finite element spaces and the control is approximated by piecewise constant functions we construct an interpolation of the exact control and a projection of the discrete scalar co state to be the approximated solution of the control variable for the continuous optimal control problem as a result it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h 3 2 for k 0 and of order h 2 for k 1 moreover only for the order k 1 raviart thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h 2 finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems a new procedure for accelerating the convergence of mixed finite element approximations of the eigenpairs and of the biharmonic operator is proposed it is based on a postprocessing technique that involves an additional solution of a source problem on an augmented finite element space this space could be obtained either by substantially refining the grid the two grid method or by using the same grid but increasing the order of polynomials by one the two space method the numerical results presented and discussed in the paper illustrate the efficiency of the postprocessing method

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

three dimensional magnetostatic problem the paper is devoted to an approximation of the solution of maxwell s equations in three dimensional sapce we present two methods which couple a finite element method inside the magnetic materials with a boundary integral method which uses poincare steklov operator to describe the exterior domain

a mixed multiscale finite element method for convex optimal control problems with oscillating coefficients abstract we study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients since the objective functional contains flux we approximate the problems using the mixed finite element methods we first analyze the standard mixed finite element approximation schemes then motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media we use a multiscale mixed finite element method to solve the state equations the multiscale finite element bases are constructed by locally solving neumann boundary value problems the analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic which allows us to use homogenization theory to obtain the asymptotic structure of the solutions although the numerical schemes are designed for the general case

l error estimates of rectangular mixed finite element methods for bilinear optimal control problem in this paper we investigate l error estimates of the bilinear elliptic optimal control problem by rectangular raviartthomas mixed finite element methods the control variable enters the state equation as a coefficient the state and the co state variables are approximated by the raviartthomas mixed finite elements of order k 1 and the control variable is approximated by piecewise linear functions the l error estimates are obtained for the control variable and coupled state variable and the convergence rates of orders o h2 and o h32 lnh 12 are also gained for the control and state variables and the flux of the state and co state variables respectively in addition the performance of the error estimates is assessed by two numerical examples

quasi norm local error estimators for p laplacian in this paper we extend the quasi norm techniques used in a priori error estimation of finite element approximation of degenerate nonlinear systems in order to carry out an improved a posteriori error analysis for the p laplacian we derive quasi norm a posteriori error estimators of residual type which are shown to provide improved upper and lower bounds on the discretization error for sufficiently regular solutions these estimators are further shown to be equivalent on the discretization error in a quasi norm numerical results demonstrating these a posteriori estimators are also presented

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

a priori and a posteriori error analysis of edge stabilization galerkin method for the optimal control problem governed by convection dominated diffusion equation in this paper we study an edge stabilization galerkin approximation scheme for the constrained optimal control problem governed by convection dominated diffusion equation the method uses least square stabilization of the gradient jumps across element edges a priori and a posteriori error estimates are derived for both the state co state and the control the theoretical results are illustrated by two numerical experiments

continuous dependence on the soret coefficient for double diffusive convection in darcy flow in this paper the darcy model is used to describe the double diffusive flow of a fluid containing a solute continuous dependence of the solution on the soret coefficient is established

immersed finite elements for optimal control problems of elliptic pdes with interfaces this paper presents a numerical method and analysis based on the variational discretization concept for optimal control problems governed by elliptic pdes with interfaces the method uses a simple uniform mesh which is independent of the interface due to the jump of the coefficient across the interface the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used therefore the immersed finite element method ifem developed in li et al 20 is used to discretize the state equation required in the variational discretization approach optimal error estimates for the control state and adjoint state are derived numerical examples are provided to confirm the theoretical results

iterative schemes for mixed finite element methods with applications to elasticity and compressible flow problems iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations the first one has applications to elliptic equations and incompressible fluid flow problems while the second has applications to linear elasticity and compressible stokes problems these schemes are constructed through iteratively penalizing the mixed finite element scheme of which iterated penalty method and augmented lagrangian method are special cases convergence theorems are demonstrated in abstract formulations in hilbert spaces and applications to individual physical problems are considered as examples theoretical analysis and computational experiments both show that the proposed schemes have very fast convergence a few iterations are normally enough to reduce the iterative error to a prescribed precision numerical examples with continuous and discontinuous coefficients are presented

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

a posteriori error estimates for some model boundary control problems in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal boundary control problems we derive a posteriori error estimates for both the state and the control approximation on polygonal domains such estimates which are apparently not available in the literature can be used to construct reliable adaptive finite element approximation schemes for the control problems

equivalent a posteriori error estimates for elliptic optimal control problems with l1 control cost abstract an elliptic optimal control problem involving the l 1 norm of the control in the cost functional is considered in this paper we use the full discretization and the variational discretization to approximate the control problem and the efficient and reliable a posteriori error estimates are obtained for the two cases for the variational discretization we also analyze the convergence of adaptive finite element methods in the end some examples are provided to validate our analysis

adaptive finite elements for elliptic optimization problems with control constraints in this paper we develop a posteriori error estimates for finite element discretization of elliptic optimization problems with pointwise inequality constraints on the control variable we derive error estimators for assessing the discretization error with respect to the cost functional as well as with respect to a given quantity of interest these error estimators provide quantitative information about the discretization error and guide an adaptive mesh refinement algorithm allowing for substantial saving in degrees of freedom the behavior of the method is demonstrated on numerical examples

new rectangular mixed finite element method for second order elliptic problems a low order rectangular mixed finite element for second order elliptic problems is constructed stability and optimal order error estimates will be derived

on gradient type optimization method utilizing mixed finite element approximation for optimal boundary control problem governed by bi harmonic equation in this paper a numerical method based on mixed finite element method for optimal boundary control problem governed by bi harmonic equation is presented the system of optimality equations consisting of state and costate function is derived based on the system a gradient type optimization method using a mixed finite approximation for solving the optimal boundary control is developed in our algorithm the trace of vortex function of costate equation on boundary plays a key role finally a local error analysis at every iteration for this gradient type optimization method is given

structural stability for the brinkman equations of flow in double diffusive convection abstract this paper continues the investigation of structural stability for the brinkman equations modeling the double diffusive convection for flow in a porous medium it supplements earlier results of straughan and hutter b straughan k hutter a priori bounds and structural stability for double diffusive convection incorporating the soret effect proc r soc lond ser a 455 1999 767 777

periodic orbits for fuzzy flows abstract periodic solutions are present in many of the various mathematical models that describe physical chemical or biological phenomena in this work we investigate the existence of periodic solutions for fuzzy initial value problem we will show that fuzzy solutions can present periodic points and we will develop tools of qualitative analysis for such solutions some models are presented to illustrate the results obtained in this paper

accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system the finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered stability of the finite difference optimality system and optimal order error estimates in the discrete l2 norm and in the discrete h1 norm under minimum smoothness requirements on the exact solution are proved sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local fourier analysis a multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system