the equivalence between mann ishikawa iterations and multistep iteration abstract we show that the convergence of mann ishikawa iterations are equivalent to the convergence of a multistep iteration for various classes of operators

a bifurcation theory for a class of discrete time markovian stochastic systems we present a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities the local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities this dependence ratio is a geometric invariant of the system by introducing an equivalence relation defined on these dependence ratios we arrive at a bifurcation theory for which in the compact case the set of stable i e non bifurcating systems is open and dense the theory is illustrated with some simple examples

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

a local solution adapted tri tree multigrid generator and iterative equation solver for mixed finite element formulation of the navier stokes equations abstract a multigrid algorithm for solving the navier stokes equations is described the generation of multigrids are based on the tri tree unstructured grid generator the tri tree grid generation algorithm include local refinements the spatial location of refinements are based on the solution calculated for the previous grid the convection and diffusion terms are calculated for each tri tree element and tri tree elements with large convection diffusion ratio refined the finite element equation system is solved for subsequent locally refined grids local grid adaption to the solution correspond to linearization of the non linear equation system at each grid level the newton formulation of the finite element equation system is solved by an iterative incomplete coupled node fill preconditioned equation solver the solution adapted multigrid method is compared with the original newton method for a single grid at each grid level the solution from the coarser grid is interpolated and projected to the new nodes on the finer grid and used as start vector the finite element matrices are integrated analytically the test problems considered are constricted tube flow and driven cavity flow

the prolonged adaptive multigrid method for finite element navier stokes equations abstract a tri tree method for generating finite element grids in two and three dimensions is developed the method is based on a new tree search structure the search tree is built upon triangles in two dimensions and tetrahedra in three dimensions the density of elements can be varied throughout the computational domain a coupled node fill in ilu preconditioner for implicit and itererative solution of the navier stokes equations has been designed the properties of preconditioner which is based on new fill in rules permit fill in and preconditioning of the pressure in the equation system the tri tree grid generation algorithm includes local refinements the spatial location of refinements are based on the solution calculated for the previous grid the element reynolds number is calculated for each tri tree element and tri tree elements with reynolds number above a predefined limit are recursively refined elements with reynolds number below this limit are recoarsed a prolonged adaptive multigrid method which is based on the tri tree grid generation method and the coupled node fill in preconditioner is developed the prolonged adaptive multigrid method reveals advantages in both robustness and efficiency especially for nonlinear and nonpositive definite equation systems in order to reduce the overall work the element matrices are integrated analytically

approximating solutions for the systems of strongly accretive operator equations the purpose of this paper is to study a strong convergence of multi step iterative scheme to a common fixed point for a finite family of strongly pseudo contractive mappings as a consequence the strong convergence theorems for the modified multi step iterative sequence to a solution of systems of strongly accretive operator equations are obtained in real uniformly smooth banach spaces the results presented in this paper improve and extend the corresponding xu y xu ishikawa and mann iterative processes with errors for nonlinear strongly accretive operator equations j math anal appl 224 1998 91 101 and noor rassias and huang m a noor t m rassias z huang three step iterations for nonlinear accretive operator equations j math anal appl 274 2002 59 68 and many others

adaptive finite element simulations of the surface currents in the north sea abstract the navier stokes equations are solved for the pressure and flow of the surface currents in the north sea the solution algorithm applied is the nodal adaptive mesh and adaptive time method the navier stokes equations are split in four equations which are solved sequentially the first equation which is solved implicitly is the diffusion equation the second equation which is solved explicitly is the convection equation the third equation which is solved implicitly is the pressure correction equation and the fourth equation which is solved explicitly is the velocity correction equation the two equations the diffusion and the pressure correction equation which are solved implicitly are symmetric linear and positive definite the implicit equations are therefore solved by a symmetric conjugate gradient algorithm the symmetric conjugate gradient algorithm is performed node by node without storage of the equation matrix the nodal solution algorithm therefore permits the solution of larger problems as compared to algorithms which apply an assembled and stored equation matrix the coefficients in the equation matrix in the nodal algorithm are generated whenever needed in the matrix vector multiplication in the conjugate algorithm the initial finite element grid is obtained by adapting the grid to the coastline a solution is first obtained at a low reynolds number the solution is then scaled and used as a start vector for the computation at a higher reynolds number at several time steps in the iteration process the reynolds number and the courant number are computed for each element an element is refined if the element reynolds number is greater than 1 and an element is recoarsed if the element reynolds number is much less than 1 the time step is adjusted simultaneously with adapting the grid to the solution the time step is computed to ensure that the largest element courant number is less than 0 5 the simulation results demonstrate that vortices may develop at the coasts outside england and outside germany

equivalence theorems of the convergence between ishikawa and mann iterations with errors for generalized strongly successively φ pseudocontractive mappings without lipschitzian assumptions abstract in this paper the equivalence of the strong convergence between the modified mann and ishikawa iterations with errors in two different schemes by xu y g xu ishikawa and mann iteration process with errors for nonlinear strongly accretive operator equations j math anal appl 224 1998 91 101 and liu l s liu ishikawa and mann iterative process with errors for nonlinear strongly accretive mappings in banach spaces j math anal appl 194 1995 114 125 respectively is proven for the generalized strongly successively pseudocontractive mappings without lipschitzian assumption our results generalize the recent results of the papers zhenyu huang f bu the equivalence between the convergence of ishikawa and mann iterations with errors for strongly successively pseudocontractive mappings without lipschitzian assumption j math anal appl 325 1 2007 586 594 b e rhoades s m soltuz the equivalence between the convergences of ishikawa and mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps j math anal appl 289 2004 266 278 b e rhoades s m soltuz the equivalence between mann ishikawa iterations and multi step iteration nonlinear anal 58 2004 219 228 by extending to the most general class of the generalized strongly successively pseudocontractive mappings and hence improve the corresponding results of all the references given in this paper by providing the equivalence of convergence between all of these iteration schemes for any initial points u 1 x 1 in uniformly smooth banach spaces

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

a dynamic adaption algorithm for grid time and satellite processors for nodal finite element formulations of navier stokes equations abstract two parallel nodal finite element algorithms for solving the navier stokes equations are developed the algorithms are based on operator splitting of the original navier stokes equations linear basis functions are applied in the finite element formulation of the equation system the finite element grid is generated by the tri tree grid generation algorithm the grid is adapted to the solution according to the element reynolds number the length of each time step is computed from the courant number the linearized equation system is solved iteratively by conjugate gradient algorithms the most time consuming part of the conjugate gradient algorithm the generation of the right hand side of the equation system and the matrix vector multiplication is distributed to satellite processors the number of processors which can be used by the parallel algorithm is only limited by the number of finite elements or the number of processors available the parallel algorithm can either select a fixed number of satellite processors or the number of satellite processors can be adapted to the amount of work performed during the computations

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 tri tree multigrid recoarsement algorithm for the finite element formulation of the navier stokes equations abstract in the present paper local recoarsements of finite element grids for solving the navier stokes equations are investigated the recoarsement algorithm is based on the hierarchic tri tree multigrid generation algorithm and computes a local tri tree element reynolds number previous investigations show that the element reynolds number is critical for convergence of the linear cgstab equation solver a necessary condition for convergence is found to be that the element reynolds number is less than 10 the tri tree multigrid solution algorithm starts by solving the equations for a low reynolds number for a coarse grid solutions for higher reynolds numbers are obtained by gradually increasing the velocity boundary conditions while refining the grid correspondingly to satisfy the element reynolds criterion as the global reynolds number is increasing the non linear convection moves in space the recoarsement algorithm is then making the grid coarser at locations where the previous fine grid resolution is no longer required

transformation invariant stochastic catastrophe theory catastrophe theory is a mathematical formalism for modeling nonlinear systems whose discontinuous behavior is determined by smooth changes in a small number of driving parameters fitting a catastrophe model to noisy data constitutes a serious challenge however because catastrophe theory was formulated specifically for deterministic systems loren cobb addressed this challenge by developing a stochastic counterpart of catastrophe theory sct based on ito stochastic differential equations in sct the stable and unstable equilibrium states of the system correspond to the modes and the antimodes of the empirical probability density function respectively unfortunately sct is not invariant under smooth and invertible transformations of variables this is an important limitation since invariance to diffeomorphic transformations is essential in deterministic catastrophe theory from the ito transformation rules we derive a generalized version of sct that does remain invariant under transformation and can include cobb s sct as a special case we show that an invariant function is obtained by multiplying the probability density function with the diffusion function of the stochastic process this invariant function can be estimated by a straightforward time series analysis based on level crossings we illustrate the invariance problem and its solution with two applications

a combined octree delaunay method for fully automatic 3 d mesh generation fully automatic three dimensional mesh generation is a fundamental requirement for automating the numerical solution of partial differential equations two techniques in particular the octree and delaunay approaches have been used towards this end a method that combines both approaches to fully automatic mesh generation is presented here the resulting algorithm provides the linear growth rate and divide and conquer approach of the octree method with the simplicity and optimal properties of the delaunay triangulation

bifurcations of optimal vector fields we study the structure of the solution set of a class of infinite horizon dynamic programming problems with one dimensional state spaces as well as their bifurcations as problem parameters are varied the solutions are represented as the integral curves of a multivalued optimal vector field on state space generically there are three types of integral curves stable points open intervals that are forward asymptotic to a stable point and backward asymptotic to an unstable point and half open intervals that are forward asymptotic to a stable point and backward asymptotic to an indifference point the latter are initial states to multiple optimal trajectories we characterize all bifurcations that occur generically in one and two parameter families most of these are related to global dynamical bifurcations of the state costate system of the problem

the equivalence between the convergences of ishikawa and mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps the convergence of modified mann iteration is equivalent to the convergence of modified ishikawa iterations when t is an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive map

convergence theorems for multivalued φ hemicontractive operators and φ strongly accretive operators abstract suppose e is a uniformly smooth banach space and t e 2e is a multivalued hemicontractive operator with bounded range suppose an bn cn and an bn cn are real sequences in 0 1 satisfying the following conditions i a n b n c n a n b n c n 1 for all n n ii lim n b n lim n b n lim n c n 0 iii n 1 b n iv c n o b n for arbitrary xi u1 v1 e define the sequence xn n 1 by xn 1 anxn bn n cnun n tyn n n yn a nxn b n n c nvn n txn n n where un n 1 vn n 1 are arbitrary bounded sequences in e then xn n 1 converges strongly to the unique fixed point of t related results deal with the iterative solutions of nonlinear multivalued accretive operator equation f tx

the equivalence between the convergence of ishikawa and mann iterations with errors for strongly successively pseudocontractive mappings without lipschitzian assumption abstract in this paper we prove some new equivalences between convergence of the ishikawa and mann iteration sequences with errors in two schemes by xu y g xu ishikawa and mann iteration process with errors for nonlinear strongly accretive operator equations j math anal appl 224 1998 91 101 and liu l s liu ishikawa and mann iterative process with errors for nonlinear strongly accretive mappings in banach spaces j math anal appl 194 1995 114 125 respectively for strongly successively pseudocontractive mappings our main results improve and extend the corresponding results of the all references listed in this article

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

steepest descent approximations for accretive operator equations a necessary and sufficient condition is established for the strong convergence of the steepest descent approximation to a solution of equations involving quasi accretive operators defined on a uniformly smooth banach space miramare trieste march 1993

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

a local predictive convection diffusion refinement indicator for the tri tree adapted finite element multigrid algorithm of the navier stokes equations abstract an element multigrid refinement algorithm based on the magnitude of the convection diffusion ratio is investigated the multigrid is generated by the tri tree algorithm the solution of the navier stokes equations is first found for a low reynolds number for a coarse grid a directional element reynolds number is defined the magnitude of the element reynolds number is investigated for driven cavity flow for different grid resolutions velocity boundary conditions and different cavity sizes the results of these investigations show that cgstab iterative equation solver applied to the newton formulation of the navier stokes equation converges when the element reynolds number is below a limit the element reynolds number limit is independent of element resolution boundary velocity and grid size the element reynolds number is evaluated for all elements in the tri tree grid and the tri tree elements are refined recursively until the element reynolds number for all tri tree elements has been reduced below a predefined limit

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner