on the quadratic finite element approximation to the obstacle problem in this paper we obtain the error bound o left h frac 32 epsilon right for any epsilon 0 for the piecewise quadratic finite element approximation to the obstacle problem without the hypothesis that the free boundary has finite length see 3

an iterative scheme for a class of quasi variational inequalities an iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities it is shown that the approximate solution obtained by the iterative scheme converges strongly in the hilbert space to the exact solution as a special case we obtain the corresponding iterative scheme for variational inequalities

variational formulation of sandpile dynamics principle of stationary action was applied to derive a system of governing equations and boundary conditions describing dynamics of sandpile avalanches the derived general system of equations for the sandpile dynamics is demonstrated to include equations of flow of granular material down a rigid wedge as a particular case it is shown that the variational principle can be readily implemented in a numerical algorithm

convergence analysis of penalty based numerical methods for constrained inequality problems this paper presents a general convergence theory of penalty based numerical methods for elliptic constrained inequality problems including variational inequalities hemivariational inequalities and variational hemivariational inequalities the constraint is relaxed by a penalty formulation and is re stored as the penalty parameter tends to zero the main theoretical result of the paper is the convergence of the penalty based numerical solutions to the solution of the constrained inequality problem as the mesh size and the penalty parameter approach zero independently the convergence of the penalty based numerical methods is first established for a general elliptic variational hemivariational inequality with constraints and then for hemivariational inequalities and variational inequalities as special cases applications to problems in contact mechanics are described

a new domain decomposition method for an hjb equation in this paper we propose a new domain decompostion method for solving a hamilton jacobi bellman equation of second order the basic idea is to solve an equivalent quasivariational inequality instead of the original discretized hjb equation

on an iterative method for variational inequalities a number of numerical solutions are presented as examples of a new iterative method for variational inequalities the iterative method is based on the reduction of variational inequalities to the wiener hopf equations for obstacle problems the convergence is guaranteed inw 1 p spaces forp 2 the examples presented are one and two dimensional obstacle problems in cases when the greens function is known but the method is very general

analysis and approximation of a nonlocal obstacle problem abstract an obstacle problem for a nonlocal operator is considered the operator is a nonlocal integral analogue of the laplacian operator and as a special case reduces to the fractional laplacian in the analysis of classical obstacle problems for the laplacian the obstacle is taken to be a smooth function for the nonlocal obstacle problem considered here obstacles are allowed to have jump discontinuities we cast the nonlocal obstacle problem as a minimization problem wherein the solution is constrained to lie above the obstacle we prove the existence and uniqueness of a solution in an appropriate function space then the well posedness and convergence of finite element approximations are demonstrated the results of numerical experiments are provided that illustrate the theoretical results and the differences between solutions of local i e partial differential equation and nonlocal obstacle problems

error estimates for the numerical approximation of time dependent flow of bingham fluid in cylindrical pipes by the regularization method the flow of a bingham fluid in a cylindrical pipe can give rise to free boundary problems the fluid behaves like a viscous fluid if the shear stress expressed as a linear function of the shear rate exceeds a yield value and like a rigid body otherwise the surfaces dividing fluid and rigid zones are the free boundaries therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods considerable progress has been made in the development of numerical algorithms for bingham fluids 2 20 27 29 30 32 however very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution that is the error estimate of these numerical methods error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters this paper concerns the error estimates of a unsteady bingham fluid modeled as a variational inequality due to duvaut lions 16 and glowinski 30 the difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term a common technique called the regularization method is to replace the non differentiable term by a perturbed differentiable term which depends on a small regularization parameter the regularization method effectively reduces the variational inequality to an equation a regularized problem which is much easier to cope with this paper has achieved the following 1 error estimates are derived for a continuous time galerkin method in suitable norms 2 we give an estimate of the difference between the true solution and the regularized solution in terms of 3 some regularity properties for both regularized solution and the true solution are proved 4 the error estimates for full discretization of the regularized problem using piecewise linear finite elements in space and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t 5 we are able to improve our estimates in the one dimensional case or under stronger regularity assumptions on the true solution the estimates for the one dimensional case are optimal and confirmed by numerical experiment the estimates from 4 and 5 provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h t and in order to control the error see corollary 4 4 the above estimates extend the error bounds derived in glowinski lions and tremolieres 32 chapter 5 pp 348 404 for the stationary bingham fluid to the time dependent one which is the main contribution of this paper

a bi projection method for bingham type flows we propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic bingham medium the main difficulty for both theoretical and numerical approaches is due to the non differentiability of the plastic part of the stress tensor in regions where the rate of strain tensor vanishes this is handled by reformulating the definition of the plastic stress tensor in terms of a projection a new time scheme based on the classical incremental projection method for the newtonian navier stokes equations is proposed the plastic tensor is treated implicitly in the first sub step of the projection scheme and is computed by using a fixed point procedure a pseudo time relaxation is added into the bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm this is a key feature of the bi projection scheme which provides a fast and accurate computation of the plastic tensor stability and error analyses of the numerical scheme are provided the error induced by the pseudo time relaxation term is controlled by a prescribed numerical parameter so that a first order estimate of the time error is derived for the velocity field a second order cell centred finite volume scheme on staggered grids is applied for the spatial discretisation the scheme is assessed against previously published benchmark results for both newtonian and bingham flows in a two dimensional lid driven cavity for reynolds number equals 1000 moreover the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplastic medium in the absence of any energy source term in the equations for a fixed value 100 of the bingham number various numerical simulations for a range of reynolds numbers up to 200000 were performed with the bi projection scheme on a grid with 1024 2 mesh points the effect of this physical parameter on the flow behaviour is discussed

a douglas rachford type primal dual method for solving inclusions with mixtures of composite and parallel sum type monotone operators in this paper we propose two different primal dual splitting algorithms for solving inclusions involving mixtures of composite and parallel sum type monotone operators which rely on an inexact douglas rachford splitting method but applied in different underlying hilbert spaces most importantly the algorithms allow one to process the bounded linear operators and the set valued operators occurring in the formulation of the monotone inclusion problem separately at each iteration the latter being individually accessed via their resolvents the performance of the primal dual algorithms is emphasized via some numerical experiments on location and image denoising problems

solvability of multivalued general mixed variational inequalities in this paper we use the auxiliary principle technique to suggest a new class of predictor corrector algorithms for solving multivalued general mixed variational inequalities the convergence of the proposed method only requires the partially relaxed strong monotonicity of the operator which is weaker than co coercivity as special cases we obtain a number of known and new results for solving various classes of variational inequalities

analysis of the brezzi pitkaranta stabilized galerkin scheme for creeping flows of bingham fluids in this paper we propose and analyze a finite element scheme for a class of variational nonlinear and nondifferentiable mixed inequalities including balance equations governing incompressible creeping flows of bingham fluids for numerical efficiency reasons equal order piecewise linear approximations are used for both velocity and pressure and the numerical scheme is stabilized by a brezzi pitkaranta perturbation term we obtain error estimates of the same order as for stable discretizations namely h1 2 for velocity and pressure solutions in htwod and hun respectively a decomposition coordination algorithm to solve the discrete nonlinear algebraic system is presented together with its convergence properties finally numerical tests are performed the solution of the problem under consideration presents particular regularity properties that are shown to permit convergence order improvement to h log h 1 2 this estimate is confirmed by numerical results

variational principles and convergence of finite element approximations of a holonomic elastic plastic problem it is shown that a boundary value problem based on a holonomic elastic plastic constitutive law may be formulated equivalently as a variational inequality of the second kind a regularised form of the problem is analysed and finite element approximations are considered it is shown that solutions based on finite element approximation of the regularised problem converge

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

surrogate projection methods for finding fixed points of firmly nonexpansive mappings we present methods for finding common fixed points of finitely many firmly nonexpansive mappings on a hilbert space at every iteration an approximation to each mapping generates a halfspace containing its set of fixed points the next iterate is found by projecting the current iterate on a surrogate halfspace formed by taking a convex combination of the halfspace inequalities this acceleration technique extends one for convex feasibility problems cfps since projection operators onto closed convex sets are firmly nonexpansive the resulting methods are block iterative and hence lend themselves to parallel implementation we extend to accelerated methods some recent results of bauschke and borwein siam rev 38 1996 pp 367 426 on the convergence of projection methods

dualization of generalized equations of maximal monotone type this paper develops a simple duality framework for generalized equations defined by set valued mappings from a linear space to another the original problem is related to two auxiliary problems of the similar form corresponding to lagrangian and dual problems in the theory of convex programming as in convex programming the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically in particular dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations also a given problem can often be solved more easily by way of a dual method the strongest results of this paper concern monotone mappings in this context the duality framework yields several new criteria for maximal monotonicity of composite mappings these results are useful theoretically as well as in numerical solution of generalized equations the duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator splitting methods and other special methods can be applied

the probabilistic analysis of a heuristic for the assignment problem we present a heuristic to solve the m times m assignment problem in o m 2 time the assignment problem is formulated as a weighted complete bipartite graph g s t e s t m for convenience we assume that m is even the main procedure in the heuristic is to construct a graph g d s t e d which is a subgraph of g e d 4dn n m 2 such that we can find a perfect matching in g d with probability at least 1 frac 1 3 d n d 2 4d 1 an o s e exact algorithm is used to find a minimum weight matching m in g d any unmatched vertices in g relative to m are then matched by a greedy algorithm the expected value of the total cost of the matching found by the heuristic is shown to be less than six if the costs are independent and identically distributed uniformly in the unit interval further with the above probability the heuristic produces a solution which is at most six times the optimal solution

algorithm of solutions for mixed implicit quasi variational inequalities with fuzzy mappings abstract in this paper by applying the auxiliary variational principle technique some existence theorems of solutions for a class of mixed implicit quasi variational inequalities with fuzzy mappings are proved in hilbert spaces a novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed the convergence criteria is also given as special cases of these results the open problem put forward by noor is answered positively our results are new and generalize a number of known results to mixed implicity quasi variational inequalities with fuzzy mappings

finite element method for stokes equations under leak boundary condition of friction type stokes equations are considered with the leak boundary condition of friction type we propose a discrete variational inequality based on one of the p1 p1 stabilized p1b p1 and p2 p1 finite elements we show that a suitable numerical integration approximation for the nonlinear term preserves in a discrete sense some properties known in continuous problems those properties are essential to justify our error estimate and numerical implementation in a consistent way numerical examples are presented to confirm the theoretical results

asynchronous block iterative primal dual decomposition methods for monotone inclusions we propose new primal dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators the principal innovation in these algorithms is that they are block iterative in the sense that at each iteration only a subset of the monotone operators needs to be processed as opposed to all operators as in established methods flexible strategies are used to select the blocks of operators activated at each iteration in addition we allow lags in operator processing permitting asynchronous implementation the decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators in order to construct a half space containing the kuhn tucker set associated with the system the coordination phase of each iteration involves a projection onto this half space we present two related methods the first method provides weakly convergent primal and dual sequences under general conditions while the second is a variant in which strong convergence is guaranteed without additional assumptions neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal dual projective splitting methods

solving quasi variational inequalities via their kkt conditions we propose to solve a general quasi variational inequality by using its karush kuhn tucker conditions to this end we use a globally convergent algorithm based on a potential reduction approach we establish global convergence results for many interesting instances of quasi variational inequalities vastly broadening the class of problems that can be solved with theoretical guarantees our numerical testings are very promising and show the practical viability of the approach

nonconforming finite element analysis for a plate contact problem in this paper we analyze nonconforming finite element methods for solving a fourth order elliptic variational inequality of the second kind arising in a plate frictional contact problem the variational inequality involves a nondifferentiable term due to the frictional contact optimal order error estimates are derived for both continuous and discontinuous nonconforming finite elements

a new stabilized finite element method for the transient navier stokes equations abstract this paper is concerned with the development and analysis of a new stabilized finite element method based on two local gauss integrations for the two dimensional transient navier stokes equations by using the lowest equal order pair of finite elements this new stabilized finite element method has some prominent features parameter free avoiding higher order derivatives or edge based data structures and stabilization being completely local at the element level an optimal error estimate for approximate velocity and pressure is obtained by applying the technique of the galerkin finite element method under certain regularity assumptions on the solution compared with other stabilized methods using the same pair of mixed finite elements for the two dimensional transient navier stokes equations through a series of numerical experiments it is shown that this new stabilized method has better stability and accuracy results

auxiliary principle for generalized mixed variational like inequalities abstract in this paper we use the auxiliary principle technique to prove the existence of a solution of some new classes of variational inequalities which are called the generalized mixed variational like inequalities this technique is used to analyze an iterative algorithm several special cases which can be obtained from our main result are also discussed

hemivariational like inequalities in this paper we introduce and consider a new class of variational inequalities known as the hemivariational like inequalities it is shown that the hemivariational like inequalities include hemivariational inequalities variational like inequalities and the classical variational inequalities as special cases the auxiliary principle is used to suggest and analyze some iterative methods for solving hemivariational like inequalities under mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

investigations on two kinds of two level stabilized finite element methods for the stationary navier stokes equations abstract this article considers two kinds of two level stabilized finite element methods based on local gauss integral technique for the two dimensional stationary navier stokes equations approximated by the lowest equal order elements which do not satisfy the inf sup condition the two level methods consist of solving a small non linear system on the coarse mesh and then solving a linear system on the fine mesh the error analysis shows that the two level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the navier stokes equations on a fine mesh for a related choice of mesh widths therefore the two level methods are of practical importance in scientific computation finally the performance of two kinds of two level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments the conclusion is that simple two level stabilized method is a viable choice for the lowest equal order approximations of the stationary navier stokes problem

average case analysis of a heuristic for the assignment problem our main contribution is an on log n algorithm that determines with high probability a perfect matching in a random 2 out bipartite graph we also show that this algorithm runs in on expected time this algorithm can be used as a subroutine in an on2 heuristic for the assignment problem when the weights in the assignment problem are independently and uniformly distributed in the interval 0 1 we prove that the expected weight of the assignment returned by thus heuristic is bounded above by 3 on for some posrtive constant a

systems of structured monotone inclusions duality algorithms and applications a general primal dual splitting algorithm for solving systems of structured coupled monotone inclusions in hilbert spaces is introduced and its asymptotic behavior is analyzed each inclusion in the primal system features compositions with linear operators parallel sums and lipschitzian operators all the operators involved in this structured model are used separately in the proposed algorithm most steps of which can be executed in parallel this provides a flexible solution method applicable to a variety of problems beyond the reach of the state of the art several applications are discussed to illustrate this point

a two grids projection algorithm for obstacle problems in order to emphasize the possible relation between discontinuous and continuous approximations on different meshes a two grids method for the resolution of parabolic variational inequality problems is presented the numerical methodology combines a time splitting algorithm to decouple a diffusion phenomenon from an obstacle problem the diffusion problem is solved by using finite differences while piecewise linear finite element techniques are used together with a newton method for the obstacle problem projections are used to interpolate the solution from one grid to the other numerical experiments show that the resulting method has good accuracy properties

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

iterative algorithms for solving generalized nonlinear mixed variational inequalities a new concept of g partially relaxed strong monotonicity of mappings is introduced by applying the auxiliary variational inequality technique some new predictor corrector iterative algorithms for solving generalized nonlinear mixed variational inequalities are suggested and analyzed the convergence of the algorithms only need the continuity and the g partially relaxed strongly monotonicity of mappings these algorithms and convergence result are new and generalize some known results in literature

penalty finite element method for stokes problem with nonlinear slip boundary conditions abstract the penalty finite element method for stokes problem with nonlinear slip boundary conditions based on the finite element subspace v h m h which satisfies the discrete inf sup condition is investigated in this paper since this class of nonlinear slip boundary conditions include the subdifferential property the weak variational formulation associated with stokes problem is variational inequality under some regularity assumptions we obtain the optimal h 1 and l 2 error estimates between u and u h and between u and u h e where the error orders are e h for h 1 error and e h 2 for l 2 error

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

warped proximal iterations for monotone inclusions abstract resolvents of set valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions we propose a generalization of this notion called warped resolvent which is constructed with the help of an auxiliary operator the properties of warped resolvents are investigated and connections are made with existing notions abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems

further applications of a splitting algorithm to decomposition in variational inequalities and convex programming a classical method for solving the variational inequality problem is the projection algorithm we show that existing convergence results for this algorithm follow from one given by gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators moreover we extend the projection algorithm to solveany monotone affine variational inequality problem when applied to linear complementarity problems we obtain a matrix splitting algorithm that is simple and for linear quadratic programs massively parallelizable unlike existing matrix splitting algorithms this algorithm converges under no additional assumption on the problem when applied to generalized linear quadratic programs we obtain a decomposition method that unlike existing decomposition methods can simultaneously dualize the linear constraints and diagonalize the cost function this method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets

stress distribution in sandpiles a variational approach two variational closure models are proposed in order to describe stress distribution and arch formation in sandpiles located on a rigid base the maximum strength and the maximum entropy closures are shown to yield the normal stress minimum below the pile apex stress distributions in sandpiles are determined numerically the force uctuations in a model regular granular assembly are studied by monte carlo simulation method a dependence of the level of the uctuations vs the macroscopic stress state is determined c 1998 elsevier science b v all rights reserved

solving coupled composite monotone inclusions by successive fejer approximations of their kuhn tucker set we propose a new class of primal dual fejer monotone algorithms for solving systems of composite monotone inclusions our construction is inspired by a framework used by eckstein and svaiter for the basic problem of finding a zero of the sum of two monotone operators at each iteration points in the graph of the monotone operators present in the model are used to construct a half space containing the kuhn tucker set associated with the system the primal dual update is then obtained via a relaxed projection of the current iterate onto this half space an important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

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

a stabilized finite element method based on two local gauss integrations for the stokes equations this paper considers a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for the stokes equations approximated by the lowest equal order finite element pairs i e the p 1 p 1 and q 1 q 1 pairs this method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf sup condition optimal error estimates are obtained by applying the standard galerkin technique finally the numerical illustrations agree completely with the theoretical expectations

quasi nonexpansive iterations on the affine hull of orbits from mann s mean value algorithm to inertial methods fixed point iterations play a central role in the design and the analysis of a large number of optimization algorithms we study a new iterative scheme in which the update is obtained by applying a composition of quasi nonexpansive operators to a point in the affine hull of the orbit generated up to the current iterate this investigation unifies several algorithmic constructs including mann s mean value method inertial methods and multilayer memoryless methods it also provides a framework for the development of new algorithms such as those we propose for solving monotone inclusion and minimization problems

two grid mfeas for the incompressible stokes type variational inequality with damping abstract in this paper we study two grid mixed finite element algorithms mfeas for the stationary incompressible stokes equations with damping and nonlinear slip boundary conditions we set up the weak formulation and obtain the well posedness then employing the mixed finite element method some optimal error estimates between the exact solution and the approximation solution are derived two kinds of two grid mfeas based on the stokes correction and newton type correction are established some error estimates for two grid mfeas are obtained several numerical results are presented to confirm our theoretical findings

convergence of a relaxed inertial proximal algorithm for maximally monotone operators in a hilbert space mathcal h given a mathcal h rightarrow 2 mathcal h a maximally monotone operator we study the convergence properties of a general class of relaxed inertial proximal algorithms this study aims to extend to the case of the general monotone inclusion ax ni 0 the acceleration techniques initially introduced by nesterov in the case of convex minimization the relaxed form of the proximal algorithms plays a central role it comes naturally with the regularization of the operator a by its yosida approximation with a variable parameter a technique recently introduced by attouch peypouquet math program ser b 2018 https doi org 10 1007 s10107 018 1252 x for a particular class of inertial proximal algorithms our study provides an algorithmic version of the convergence results obtained by attouch cabot j differ equ 264 7138 7182 2018 in the case of continuous dynamical systems

a coherent analysis of stokes flows under boundary conditions of friction type this paper is concerned with the stationary and nonstationary flow of viscous incompressible fluid under boundary conditions of friction type which are certain nonlinear boundary conditions similar to the so called signorini boundary condition in elasticity we assume that the flow is governed by the linear stokes equation while the boundary condition is nonlinear from the methodological viewpoint the analysis is carried out in a coherent way starting from study of the related boundary value problems for the stationary flow by means of the theory of variational inequalities and getting to wellposedness of the initial boundary value problems for the nonstationary flow by means of the nonlinear semigroup theory from the viewpoint of applications we mention original motivations and include some new generalizations like the cases of anisotropic friction and inhomogeneous boundary value

on a strong solution of the non stationary navier stokes equations under slip or leak boundary conditions of friction type abstract strong solutions of the non stationary navier stokes equations under non linearized slip or leak boundary conditions are investigated we show that the problems are formulated by a variational inequality of parabolic type to which uniqueness is established using galerkin s method and deriving a priori estimates we prove global and local existence for 2d and 3d slip problems respectively for leak problems under no leak assumption at t 0 we prove local existence in 2d and 3d cases compatibility conditions for initial states play a significant role in the estimates

semi discrete stabilized finite element methods for navier stokes equations with nonlinear slip boundary conditions based on regularization procedure based on the pressure projection stabilized methods the semi discrete finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions is considered in this paper because this class of boundary condition includes the subdifferential property then the variational formulation is the navier stokes type variational inequality problem using the regularization procedure we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter varepsilon which means that the solution of the regularized problem converges to the solution of the navier stokes type variational inequality problem as the parameter varepsilon longrightarrow 0 moreover some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data the pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived

on multilevel iterative methods for optimization problems this paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces the construction of auxiliary problems as well as applications to elasto plastic model and linear programming are described the auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates equivalents of this condition are presented in special cases

worst case analysis of set union algorithms this paper analyzes the asymptotic worst case running time of a number of variants of the well known method of path compression for maintaining a collection of disjoint sets under union we show that two one pass methods proposed by van leeuwen and van der weide are asymptotically optimal whereas several other methods including one proposed by rein and advocated by dijkstra are slower than the best methods

remarks on the generalized newton method we give some convergence results for the generalized newton method for the computation of zeros of nondifferentiable functions which we proposed in an earlier work our results show that the generalized method can converge quadratically when used to compute the zeros of the sum of a differentiable function and the multivalued subgradient of a lower semicontinuous proper convex function the method is therefore effective for variational inequalities and can be used to find the minimum of a function which is the sum of a twice differentiable convex function and a lower semicontinuous proper convex function a numerical example is given

bregman monotone optimization algorithms a broad class of optimization algorithms based on bregman distances in banach spaces is unified around the notion of bregman monotonicity a systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block iterative surrogate bregman projection schemes another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of bregman monotonicity and to include the operators commonly found in bregman optimization methods special emphasis is placed on the viability of the algorithms and the importance of legendre functions in this regard various applications are discussed

convergence analysis of a conforming adaptive finite element method for an obstacle problem the adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual based a posteriori error estimator each cycle of the adaptive loop consists of the steps solve estimate mark and refine the techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of galerkin orthogonality we establish r linear convergence of the part of the energy above its minimal value if there is appropriate control of the data oscillations surprisingly the adaptive mesh refinement algorithm is the same as in the unconstrained case of a linear pde in fact there is no modification near the discrete free boundary necessary for r linear convergence the arguments are presented for a model obstacle problem with an affine obstacle and homogeneous dirichlet boundary conditions the proof of the discrete local efficiency is more involved than in the unconstrained case numerical results are given to illustrate the performance of the error estimator

a primal dual splitting algorithm for finding zeros of sums of maximal monotone operators we consider the primal problem of finding the zeros of the sum of a maximal monotone operator and the composition of another maximal monotone operator with a linear continuous operator by formulat

numerical analysis of a characteristic stabilized finite element method for the time dependent navier stokes equations with nonlinear slip boundary conditions abstract based on a characteristic method this work is concerned with a finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions since this slip boundary condition of friction type contains a subdifferential property its continuous variational problem is formulated as an inequality which can turn into an equality problem by using a powerful regularized method then a fully discrete characteristic scheme under the stabilized lower order finite element pairs is proposed for the equality problem optimal error estimates for velocity and pressure are derived under the corresponding l 2 h 1 norms finally a smooth problem test is reported to demonstrate the theoretically predicted convergence order and the expected slip phenomena and the simulation of a bifurcated blood flow model is displayed to illustrate the efficiency of the proposed method

regularity of solutions to a non homogeneous boundary value problem for general stokes systems in r n we give a simple and very complete proof of the existence of a strong h2 solution to the non homogeneous problem 1 1 under the non homogeneous boundary conditions 1 6 here we consider the half space case rn n 3 see theorem 1 2 this regularity result was previously obtained by solonnikov and scadilov in reference 33 for the classical stokes system 0 g x 0 in the 3 d homogenous case a 0 b 0 and a suitable open subset of r3

random assignment problems abstract analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem s solutions feasible region and optimal values especially in large scale cases a class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems and many important problems in operations research and computer science can be formulated as assignment problems this paper presents an overview of the recent results and developments in the area of probabilistic assignment problems including the linear and multidimensional assignment problems quadratic assignment problem etc

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

a multi grid continuation strategy for parameter dependent variational inequalities for parameter dependent nonlinear elliptic obstacle problems a path following multi grid continuation strategy is developed combining a nested iteration type scheme as predictor with a subsequent multi grid method as corrector the performance of the algorithm is illustrated by some numerical results for the bratu problem

composition duality and maximal monotonicity this paper shows how the attouch thera duality principle for operator inclusions can be extended to compositions of multifunctions so that the primal and dual inclusions may involve operators between different pairs of spaces we first present the extension and give an example from computational economics to demonstrate that it has practical utility then we consider a particular case often found in applications involving real hilbert spaces and maximal monotone operators we show that in this case our duality framework includes those previously developed by rockafellar mosco and gabay finally we demonstrate that in this case under very simple hypotheses the duality transformation preserves the maximal monotonicity of the operators involved this proof uses an apparently new criterion for maximal monotonicity of operators of the form l tl where t is maximal monotone and l is linear and continuous with adjoint l

weak galerkin finite element methods for a second order elliptic variational inequality abstract a weak galerkin finite element method is used to determine approximate solutions of an elliptic variational inequality three sets of basis functions are employed the first has constant values inside each element and on each edge the second has constant values inside each element but is a linear polynomial on each edge and the third has linear polynomials inside each element and on each edge error estimates including convergence rates are derived for all three cases numerical results were provided to illustrate the theoretical results to show that the choice of how the obstacle function is approximated affects those rates and to show the super convergence for piecewise constant basis functions

existence and algorithm of solutions for generalized mixed implicit quasi variational inequalities1this project was supported by the national science foundation of people s republic china 1 in this paper by applying the auxiliary variational principle technique some existence theorems of solutions for a class of generalized mixed implicit quasi variational inequalities are proved in hilbert spaces a novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed the convergence criteria are also given as special cases of these results the open problem put forward by noor is answered positively and some results of existence and algorithm of solutions for generalized implicit quasi complementarity problems are also obtained our results are new and generalize a number of known results of mixed variational inequalities mixed quasi variational inequalities and quasi complementary problems involving single valued and set valued mappings in recent literatures