simple efficient solvers for certain ill conditioned systems of linear equations including h div problems abstract simple accurate and efficient iterative methods for the solution of a nearly singular variational problem are described the systems considered arise e g when seeking to determine the flux of second order elliptic partial differential equations each outer iteration uses a robust and efficient solver which may be a direct or iterative method and only a few outer iterations are required to obtain an approximate solution reduced rank extrapolation may be applied to speed up the convergence

pointwise lipschitzian mappings in uniformly convex and uniformly smooth banach spaces abstract let x be a uniformly convex and uniformly smooth banach space let t c c be an asymptotic pointwise nonexpansive mapping where c is a bounded closed and convex subset of x in this paper we investigate conditions sufficient for the weak convergence of the generalized mann and ishikawa processes to a fixed point of t

fixed point iterations for asymptotically nonexpansive mappings in banach spaces in this paper we suggest and analyze a three step iterative scheme for asymptotically nonexpansive mappings in banach spaces the new iterative scheme includes ishikawa type and mann type interations as special cases the results obtained in this paper represent an extension as well as refinement of previous known results

iterative construction of fixed points of asymptotically nonexpansive mappings abstract let t be a completely continuous and asymptotically nonexpansive self mapping of a nonempty closed bounded and convex subset of a hilbert space then under certain conditions the sequence defined by xn 1 ntn xn 1 n x n converges strongly to some fixed point of t a similar result is obtained for asymptotically pseudocontractive mappings whose iterates are lipschitz continuous with respect to some common lipschitz constant

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

iteration processes for nonlinear mappings in convex metric spaces on considere le schema iteratif d ishikawa pour construire des points fixes d applications quasi contractives quasi contractives generalisees et quasi non dilatante dans des espaces metriques convexes

convergence of iterates with errors of asymptotically quasi nonexpansive mappings and applications in this paper we consider a generalized iterative process with errors to approximate the common fixed points of two asymptotically quasi nonexpansive mappings a convergence theorem has been obtained which generalizes a known result this theorem has then been used to prove another convergence theorem which in turn generalizes a number of results

on uniformly lipschitzian multivalued mappings in banach and metric spaces abstract let x d be a metric space a mapping t x x is said to be uniformly lipschitzian if there exists a constant k such that d t n x t n y k d x y for all x y x and n 1 it is known that such mappings always have fixed points in certain metric spaces for k 1 provided k is sufficiently near 1 these spaces include uniformly convex metric and banach spaces as well as metric spaces having lifsic characteristic greater than 1 a uniformly lipschitzian concept for multivalued mappings is introduced in this paper and multivalued analogues of these results are obtained

approximating common fixed points of asymptotically quasi nonexpansive mappings by a k 1 step iterative scheme with error terms in this paper a k 1 step iterative scheme with error terms involving k 1 asymptotically quasi nonexpansive mappings is studied in usual banach spaces some sufficient and necessary conditions are given for the iterative scheme to approximate a common fixed point in uniformly convex banach spaces power equicontinuity for a mapping is introduced and a series of new convergence theorems are established several known results in the current literature are extended and refined

on a fem bem formulation for an exterior quasilinear problem in the plane we use a version of the fem bem method introduced by costabel boundary elements ix vol 1 c a brebbia et al eds springer verlag 1987 and han j comput math 8 1990 pp 223 232 to discretize an exterior quasilinear problem we provide error estimates for the galerkin method and propose a fully discrete scheme based on simple quadrature formulas furthermore we show that these numerical integration schemes preserve the optimal rates of convergence finally we present results of numerical experiments involving our discretization method

common fixed points noor iteration for a finite family of asymptotically quasi nonexpansive mappings in banach spaces in this paper we introduce a general iteration scheme for a finite family of asymptotically quasi nonexpansive mappings the new iterative scheme includes the modified mann and ishikawa iterations three step iterative scheme of xu and noor and khan and takahashi scheme as special cases our results are generalizations as well as refinement of several known results in the current literature

on δ convergence theorems in cat 0 spaces by using the concept of convergence introduced by lim t c lim remarks on some fixed point theorems proc amer math soc 60 1976 179 182 we are able to give the cat 0 space analogs of results on weak convergence of the picard mann and ishikawa iterates proved in uniformly convex banach spaces by opial z opial weak convergence of the sequence of successive approximations for nonexpansive mappings bull amer math soc 73 1967 591 597 ishikawa s ishikawa fixed points and iteration of a nonexpansive mapping in a banach space proc amer math soc 59 1976 65 71 and tan and xu k k tan h k xu approximating fixed points of nonexpansive mappings by the ishikawa iteration process j math anal appl 178 1993 301 308

convergence of a general algorithm of asymptotically nonexpansive maps in uniformly convex hyperbolic spaces abstract in this paper we establish convergence theorems for a general algorithm of an asymptotically nonexpansive map in a uniformly convex hyperbolic space our results generalize simultaneously the approximation results of rhoades 1994 18 suantai 2005 20 and xu and noor 2002 26 on a nonlinear domain our results are refinements and generalizations of the corresponding ones in uniformly convex banach spaces and cat 0 spaces

on the coupling of boundary integral and mixed finite element methods we present a numerical method for solving an exterior dirichlet problem in the plane the technique consists in coupling boundary integral and mixed finite element methods an artificial boundary is introduced separating an interior region from an exterior one from an integral representation of the solution in the exterior domain we deduce two integral equations which relate the solution and its normal derivative over the artificial boundary these integral equations are incorporated into the so called mixed formulation of the problem in the interior region and a finite element method is used to approximate the resulting variational problem

convergence theorems for fixed points of uniformly quasi lipschitzian mappings in convex metric spaces abstract in this paper we consider an ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi lipschitzian mappings in convex metric spaces some results have been obtained which generalize and unify many important known results in recent literature

a two grid eulerian lagrangian localized adjoint method to miscible displacement problems with dispersion term abstract in this paper we propose a novel two grid method of eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solutions for nonlinear and coupled miscible displacement problems which describe complex fluid flow processes in porous media an ellam is used to solve the nonlinear convection diffusion equation for concentration and we present a mfem to compute the pressure equation for the pressure and darcy velocity a two grid scheme is investigated to linearize and decouple the mixed method equations it is illustrated that the coarse space can be selected as coarse as h o h 1 2 and the asymptotically optimal approximation as the nonlinear schemes can be still achieved as a result a large class of such nonlinear and coupled problems can be solved efficiently with large time steps

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

a low order mixed finite element method for a class of quasi newtonian stokes flows part i a priori error analysis abstract this is the second part of a work dealing with a low order mixed finite element method for a class of nonlinear stokes models arising in quasi newtonian fluids in the first part we showed that the resulting variational formulation is given by a twofold saddle point operator equation and that the corresponding galerkin scheme becomes well posed with piecewise constant functions and raviart thomas spaces of lowest order as the associated finite element subspaces in this paper we develop a bank weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed finite element solutions several numerical results illustrating the efficiency of the method are also provided

on the coupling of mixed fem and bem for an exterior helmholtz problem in the plane this paper deals with an elliptic boundary value problem posed in the plane with variable coefficients but whose restriction to the exterior of a bounded domain reduces to a helmholtz equation we consider a mixed variational formulation in a bounded domain that contains the heterogeneous medium coupled with a boundary integral method applied to the helmholtz equation in we utilize suitable auxiliary problems duality arguments and fredholm alternative to show that the resulting formulation of the problem is well posed then we define a corresponding galerkin scheme by using rotated raviart thomas subspaces and spectral elements on the interface we show that the discrete problem is uniquely solvable and convergent and prove optimal error estimates finally we illustrate our analysis with some results from computational experiments

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces abstract let m d be a complete 2 uniformly convex metric space let c be a nonempty bounded closed and convex subset of m and let t c c be an asymptotic pointwise nonexpansive mapping in this paper we prove that the modified mann iteration process defined by x n 1 t n t n x n 1 t n x n converges in a weaker sense to a fixed point of t

nonsymmetric coupling of bem and mixed fem on polyhedral interfaces in this paper we propose and analyze some new methods for coupling mixed finite element and boundary element methods for the model problem of the laplace equation in free space or in the exterior of a bounded domain as opposed to the existing methods which use the complete matrix of operators of the calderon projector to obtain a symmetric coupled system we propose methods with only one integral equation the system can be considered as a further generalization of the johnson nedelec coupling of bem fem to the case of mixed formulations in the bounded domain using some recent analytical tools we are able to prove stability and convergence of galerkin methods with very general conditions on the discrete spaces and no restriction relating the finite and boundary element spaces this can be done for general lipschitz interfaces and in particular the coupling boundary can be taken to be a lipschitz polyhedron both the indirect and the direct approaches for the boundary integral formulation are explored

a dual dual mixed formulation for nonlinear exterior transmission problems we combine a dual mixed finite element method with a dirichletto neumann mapping derived by the boundary integral equation method to study the solvability and galerkin approximations of a class of exterior nonlinear transmission problems in the plane as a model problem we consider a nonlinear elliptic equation in divergence form coupled with the laplace equation in an unbounded region of the plane our combined approach leads to what we call a dual dual mixed variational formulation since the main operator involved has itself a dual type structure we establish existence and uniqueness of solution for the continuous and discrete formulations and provide the corresponding error analysis by using raviart thomas elements the main tool of our analysis is given by a generalization of the usual babuska brezzi theory to a class of nonlinear variational problems with constraints

comments on smith forms over r z 1 z 2 it is pointed out that a proof of a result in the above paper is incorrect however the result is well known and its correct proof is available in the existing mathematical literature concerning serre s conjecture

the simplified derivation of the completion theorem to a unimodular matrix over r z l z 2 as indicated in a previous paper there is a need for a simple constructive proof of the completion theorem of a 2 d unimodular matrix the proof provided here is based on the concept of coprimeness of 2 d polynomial matrices and the 2 d version of the matrix fraction description mfd of a 2 d transfer function matrix as investigated by morf et al 6 and youla and gnavi 5

analysis of a fem bem model posed on the conducting domain for the time dependent eddy current problem the three dimensional eddy current time dependent problem is considered we formulate it in terms of two variables one lying only on the conducting domain and the other on its boundary we combine finite elements fem and boundary elements bem to obtain a fem bem coupled variational formulation we establish the existence and uniqueness of the solution in the continuous and the fully discrete case finally we investigate the convergence order of the fully discrete scheme

smith forms over r z_ 1 z_ 2 relationships between a polynomial matrix and its smith forms are explored it is shown for example that a matrix over r z 1 z 2 is equivalent to its smith form over r z 1 z 2 if and only if a certain system of linear polynomial equations has a solution

an optimal iterative process for the johnson nedelec method of coupling boundary and finite elements we reformulate the discretization of the johnson nedelec method 11 of coupling boundary elements and finite elements for an exterior bidimensional laplacian this new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix we show that if the parameter of discretization is sufficiently small the fully discrete scheme is well posed and the error estimates remain unaltered the rest of the paper is devoted to the study of an efficient algorithm for solving the resulting discrete linear systems

bezout identities and common minor zeros of 2 d polynomial matrices and applications to time delay systems this paper is concerned with factor zero coprimeness of 2 d polynomial matrices and studies relating bezout identities and some of their properties first we show that a pair of factor coprime matrices satisfies the particular bezout identities which are useful for feedback stabilization secondly we investigate the properties of the common minor zeros common zeros of all the maximum minors of factor coprime factorizations of 2 d transfer function matrices finally we apply the results to time delay systems of neutral or retarded type and provide tests for the spectrally controllability canonicality

a quadratic rate of asymptotic regularity for cat 0 spaces abstract in this paper we obtain a quadratic bound on the rate of asymptotic regularity for the krasnoselski mann iterations of nonexpansive mappings in cat 0 spaces whereas previous results guarantee only exponential bounds the method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of groetsch s theorem obtained by kohlenbach using methods from mathematical logic so called proof mining

a dual dual formulation for the coupling of mixed fem and bem in hyperelasticity we combine a dual mixed finite element method with the boundary integral equation method to study the solvability and galerkin approximations of a linear nonlinear transmission problem arising in plane elastostatics our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain and it leads to what we call a dual dual mixed formulation since the corresponding operator equation becomes of a twofold saddle point structure we derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis in particular we extend the finite element space for plane elasticity given by the peers element and define an explicit finite element boundary element subspace satisfying all the required compatibility conditions most of our analysis makes use of an extension of the classical babuska brezzi theory to a class of nonlinear saddle point problems

least squares methods for the coupling of fem and bem in the present paper we propose least squares formulations for the numerical solution of exterior boundary value problems the partial differential equation is a first order system in a bounded subdomain and the unbounded subdomain is treated by means of boundary integral equations the first order system is derived from a strongly elliptic second order system the analysis of the present least squares formulations is reduced to the analysis of the galerkin method for the coupling of finite element and boundary element methods fem and bem of the second order problem the least squares approach requires no stability condition however it requires the computation of negative as well as of half integer sobolev norms the arising linear systems can be preconditioned to have condition numbers sim 1 the present methods benefit strongly from the use of biorthogonal wavelets on the coupling boundary and the computation of corresponding equivalent norms in sobolev spaces in particular the application of green s formula leads to an efficient discretization of least squares formulations

convergence theorems for nonexpansive semigroups in cat 0 spaces abstract let c be a convex and compact subset of a cat 0 space x in this paper we consider the following iterative scheme for a one parameter nonexpansive semigroup t t t 0 on c x 1 c x n 1 t t n x n 1 x n where 0 1 and t n 0 and we prove that under certain conditions x n converges to a common fixed point of the semigroup t t t 0

strong convergence of a general iteration scheme in cat 0 spaces abstract we introduce and study strong convergence of a general iteration scheme for a finite family of asymptotically quasi nonexpansive maps in convex metric spaces and c a t 0 spaces the new iteration scheme includes modified mann and ishikawa iterations the three step iteration scheme of xu and noor and the scheme of khan domlo and fukhar ud din as special cases in banach spaces our results are refinements and generalizations of several recent results from the current literature

a primal mixed formulation for exterior transmission problems in bf r 2 we combine a primal mixed finite element approach with a dirichlet to neumann mapping arising from the boundary integral equation method to study the weak solvability and galerkin approximations of a class of linear exterior transmission problems in potential theory our results are mainly based on the babuska brezzi theory for variational problems with constraints we establish the uniqueness of solution for the continuous and discrete formulations and show that finite element subspac es of lagrange type satisfy the discrete compatibility conditions in addition we provide the error analysis including polygonal approximations of the domain and prove strong convergence of the galerkin solutions moreover under additional regularity assumptions on the solution of the continuous formulation we obtain the asymptotic rate of convergence o h

grid modification for the wave equation with attenuation the wave equation with attenuation due to a linear friction is approximated by a new mixed finite element method which allows one to use different grids and basis functions at different times when necessary this method enables one to track sharp moving wave fronts more efficiently and accurately error estimates with optimal convergent rates are established unconditional stability is also proved for this method

variational formulations of transmission problems via fem bem and dtn mappings the purpose of this work is to review some of the main results on the weak solvability and galerkin approximations of interior and exterior transmission problems arising in potential theory and elastostatics both linear and nonlinear model problems are included in our presentation we consider variational formulations based on the combination of mixed finite elements with either boundary integral equations or dirichlet to neumann dtn mappings we provide existence and uniqueness of solution for the continuous formulations and give general approximation estimates for the associated galerkin schemes some open questions are also mentioned

coupling of raviart thomas and hybridizable discontinuous galerkin methods with bem we present new a priori error analyses of the coupling of the raviart thomas rt method and bem as well as of the coupling of the hybridizable discontinuous galerkin hdg method with bem the novel features of the analysis of the rt bem coupling are the superconvergence estimates for the scalar approximation of the interior problem and new bounds in weak and strong norms for the boundary variables the analysis of the hdg bem coupling is the first analysis of this coupling and shows that the coupling provides approximations with the same convergence properties as those of the rt bem coupling

coprime fraction computation of 2 d rational matrices this note presents a numerical method of computing a coprime fraction of a two dimensional 2 d rational matrix not necessarily proper it is achieved by searching the primary linearly dependent rows in order from top to bottom of the two generalized resultants the procedure can be extended to the three or higher dimensional case and the result can also be used to compute the greatest common divisor gcd of 2 d polynomial matrices without employing primitive factorizations which does not exist in the three or higher dimensional case

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

δ convergence theorems for multi valued nonexpansive mappings in hyperbolic spaces the purpose of this paper is to introduce the mixed agarwal o regan sahu type iterative scheme agarwal et al 2007 for finding a common fixed point of the multi valued nonexpansive mappings in the setting of hyperbolic spaces under suitable conditions some convergence theorems of the iterative sequence generated by the proposed scheme to approximate a common fixed point of multi valued nonexpansive mappings are proved the results presented in the paper extend and improve some recent results announced in the current literature

an expanded mixed finite element approach via a dual dual formulation and the minimum residual method abstract we apply an expanded mixed finite element method which introduces the gradient as a third explicit unknown to solve a linear second order elliptic equation in divergence form instead of using the standard dual form we show that the corresponding variational formulation can be written as a dual dual operator equation we establish existence and uniqueness of solution for the continuous and discrete formulations and provide the corresponding error analysis by using raviart thomas elements in addition we show that the corresponding dual dual linear system can be efficiently solved by a preconditioned minimum residual method some numerical results illustrating this fact and the rate of convergence of the mixed finite element method are also provided

a mixed finite element method for a strongly nonlinear second order elliptic problem the approximation of the solution of the first boundary value problem for a strongly nonlinear second order elliptic problem in divergence form by the mixed finite element method is considered existence and uniqueness of the approximation are proved and optimal error estimates in l 2 are established for both the scalar and vector functions approximated by the method error estimates are also derived in l q 2 q