error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions we study here the convergence of a finite volume scheme for a diffusion convection equation on an open bounded set of r dim dim 2 or 3 for which we consider dirichlet neumann or robin boundary conditions we consider unstructured meshes which include voronoi or triangular meshes we use for the diffusion term an s points where s is the number of sides of each cell finite volume scheme and for the convection term an upstream finite volume scheme assuming the exact solution at least in h2 we prove error estimates in a discrete h 1 0 norm of order the size of the mesh discrete poincare inequalities then allow one to prove error estimates in the l2 norm

homotopy approach for solving constrained optimization problems a homotopy approach for solving constrained parameter optimization problems is examined the first order necessary conditions with the complementarity conditions represented using a technique due to mangasarian 1967 are solved the equations are augmented to avoid singularities which occur when the active constraint changes the chow yorke 1978 algorithm is used to track the homotopy path leading to the solution to the desired problem at the terminal point a simple example which illustrates the technique and an application to a fuel optimal orbital transfer problem are presented

a semismooth equation approach to the solution of nonlinear complementarity problems in this paper we present a new algorithm for the solution of nonlinear complementarity problems the algorithm is based on a semismooth equation reformulation of the complementarity problem we exploit the recent extension of newton s method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration we include numerical tests which show the viability of the approach

a new cement to glue nonconforming grids with robin interface conditions the finite element case robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in the convergence non conforming grids make the grid generation much easier and faster since it is then a parallel task the goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of robin interface conditions with non matching grids we consider both a symmetric definite positive operator and the convection diffusion equation discretized by finite volume schemes numerical results are shown

effects of random motility on microbial growth and competition in a flow reactor the authors investigate the effects of random motility on the ability of a microbial population to survive in pure culture and to be a good competitor for scarce nutrient in mixed culture in a flow reactor model consisting of a nonlinear parabolic system of partial differential equations for pure culture a single population a sharp condition is derived which distinguishes between the two outcomes 1 washout of the population from the reactor or 2 persistence of the population and the existence of a unique single population steady state the simulations suggest that this steady state is globally attracting for the case of two populations competing for scarce nutrient they obtain sufficient conditions for the uniform persistence of the two populations for the existence of a coexistence steady state and for the ability of one population to competitively exclude a rival extensive simulations are reported which suggest that 1 all solutions approach some steady state solution 2 all possible outcomes exhibited by the classical competitive lotka volterra ode model can occur in the model and 3 the outcome of competition between two bacterial strains can depend rather subtly on their respective random motility coefficients

steady states and dynamics of an autocatalytic chemical reaction model with decay abstract the dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates upper lower solution method and bifurcation theory the effects of decay order decay rate and diffusion rates to the dynamical behavior are discussed

a globally convergent probability one homotopy for linear programs with linear complementarity constraints a solution of the standard formulation of a linear program with linear complementarity constraints lpcc does not satisfy a constraint qualification a family of relaxations of an lpcc associated with a probability one homotopy map proposed here is shown to have several desirable properties the homotopy map is nonlinear replacing all the constraints with nonlinear relaxations of ncp functions under mild existence and rank assumptions 1 the lpcc relaxations rlpcc lambda have a solution for 0 le lambda le1 2 rlpcc 1 is equivalent to lpcc 3 the kuhn tucker constraint qualification is satisfied at every local or global solution of rlpcc lambda for almost all 0 le lambda 1 4 a point is a local solution of rlpcc 1 and lpcc if and only if it is a kuhn tucker point for rlpcc 1 and 5 a homotopy algorithm can find a kuhn tucker point for rlpcc 1 since the homotopy map is a globally convergent probability one homotopy robust and efficient numerical algorithms exist to f

maintaining lu factors of a general sparse matrix abstract the authors describe a set of procedures for computing and updating an lu factorization of a sparse matrix a where a may be square possibly singular or rectangular the procedures include a markowitz factorization and a bartels golub update similar to those of reid 1976 1982 the updates provided are addition deletion or replacement of a row or column of a and rank one modification previously column replacement has been the only update available various design features of the implementation lusol are described and computational comparisons are made with the la05 and ma28 packages of reid 1976 and duff 1977 keywords sparse matrix lu factors matrix factorization matrix updates rank one modification fortran software lower and upper triangular

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

the multiscale hybrid mixed method in general polygonal meshes this work extends the general form of the multiscale hybrid mixed mhm method for the second order laplace darcy equation to general non conforming polygonal meshes the main properties of the mhm method i e stability optimal convergence and local conservation are proven independently of the geometry of the elements used for the first level mesh more precisely it is proven that piecewise polynomials of degree k and k 1 k ge 0 for the lagrange multipliers flux along with continuous piecewise polynomial interpolations of degree k 1 posed on second level sub meshes are stable if the latter is fine enough with respect to the mesh for the lagrange multiplier we provide an explicit sufficient condition for this restriction also we prove that the error converges with order k 1 and k 2 in the broken h 1 and l 2 norms respectively under usual regularity assumptions and that such estimates also hold for non convex or even non simply connected elements numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides

non conforming curved finite element schemes for time dependent elastic acoustic coupled problems high order numerical methods for solving time dependent acoustic elastic coupled problems are introduced these methods based on finite element techniques allow for a flexible coupling between the fluid and the solid domain by using non conforming meshes and curved elements since characteristic waves travel at different speeds through different media specific levels of granularity for the mesh discretization are required on each domain making impractical a possible conforming coupling in between advantageously physical domains may be independently discretized in our framework due to the non conforming feature consequently an important increase in computational efficiency may be achieved compared to other implementations based on conforming techniques namely by reducing the total number of degrees of freedom differently from other non conforming approaches proposed so far our technique is relatively simpler and requires only a geometrical adjustment at the coupling interface at a preprocessing stage so that no extra computations are necessary during the time evolution of the simulation on the other hand as an advantage of using curvilinear elements the geometry of the coupling interface between the two media of interest is faithfully represented up to the order of the scheme used in other words higher order schemes are in consonance with higher order approximations of the geometry concerning the time discretization we analyze both explicit and implicit schemes these schemes are energy conserving and for the explicit case the stability is guaranteed by a cfl condition in order to illustrate the accuracy and convergence of these methods a set of representative numerical tests are presented

a simple model for sustained oscillations in isothermal branched chain or autocatalytic reactions in a well stirred open system i stationary states and local stabilities the behaviour of a simple autocatalytic or branched chain reaction a b rightarrow 2b text rate k 1 ab coupled with a surface controlled decay of the catalyst b rightarrow c text rate k 2 b 1 rb is considered under isothermal well stirred open conditions the c s t r this system has a unique physically acceptable stationary state at any given residence time if the saturation term r is zero this solution is always stable although it may exhibit damped oscillatory responses characteristic of a stable focus small non zero values of r do not introduce qualitative changes however for sufficiently large surface effects such that r k 1 k 2 the stable focus may become unstable at a point of hopf bifurcation this leads to the appearance of sustained oscillations in the concentrations a and b for relatively stable catalysts such that k 2 is non zero but very small compared with k 1 a 0 where a 0 is the concentration of reactant a in the feed to the reactor the governing kinetic equations can be reduced to especially simple asymptotic forms these in turn allow the analytical determination of the point of hopf bifurcation in terms of the residence time t res at which the change in stability occurs numerical solution of the full equations reveals that the asymptotic expressions provide a very good prediction for the dependence of t res on k 1 a 0 k 2 and r even when k 2 approx k 1 a 0 systems which have some autocatalyst b in the feed show an important qualitative change at long residence times there is a second point of hopf bifurcation at which the focal stationary state becomes stable again oscillations die out at this point and in the limit of infinitely long residence times the behaviour approaches that of the corresponding closed system

theory of globally convergent probability one homotopies for nonlinear programming for many years globally convergent probability one homotopy methods have been remarkably successful on difficult realistic engineering optimization problems most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective convergence theory has been derived for a few particular problems and considerable fixed point theory exists but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking this paper derives some probability one homotopy convergence theorems for unconstrained and inequality constrained optimization for linear and nonlinear inequality constraints and with and without convexity some insight is provided into why the homotopies used in engineering practice are so successful and why this success is more than dumb luck by presenting the theory as variations on a prototype probability one homotopy convergence theorem the essence of such convergence theory is elucidated

analysis of chemical kinetic systems over the entire parameter space ii isothermal oscillators two simple but similar isothermal kinetic schemes are studied by using recently developed qualitative techniques with the aim of finding all of the qualitatively distinct behaviour that the system can exhibit the first scheme which can be regarded as one of the simplest isothermal models of interest is based on the quadratic autocatalytic step a b rightarrow 2b text rate k q ab where a and b are the concentrations of a and b respectively combined with a nonlinear termination step of the form b rightarrow c text rate k t b 1 rb where r is some parameter the second scheme is a simplified by treating one parameter as a constant version of a model that has been proposed for the oxidation of carbon monoxide co this scheme has a termination step of the same form as above and like the quadratic model the highest nonlinearity apart from this term is quadratic for the quadratic model we find 10 qualitatively distinct phase portraits and with our choice of bifurcation parameter 22 distinct bifurcation diagrams the most complex phase portraits contain 3 steady states and 2 limit cycles or 2 steady states and 3 limit cycles for the co model we find 8 distinct phase portraits but leading to only 5 separate bifurcation diagrams in both models much of the behaviour occurs over only extremely small regions in parameter space

a cell functional minimization scheme for domain decomposition method on non orthogonal and non matching meshes we are interested in a robust and accurate domain decomposition algorithm with interface conditions of robin type on non matching multiblock grids using a cell functional minimization scheme which has a good performance on non orthogonal meshes in order to treat the non matching grids at the interface we introduce the l 2 l 2 projection operator to ensure weak continuity of the primary unknown and of the normal flux across the non matching interface furthermore we prove the wellposedness of local and global problems and obtain as well an error estimate of first order in a discrete h 1 h 1 norm only using the l 2 l 2 projection operator on the non matching interface as done in the matching case numerical results are presented in confirmation of the theoretical results

mathcal h matrices compressed multiplicative schwarz preconditioner for nonconformal fem bem ddm an mathcal h matrices compressed multiplicative schwarz ms preconditioner for the nonconformal finite element method fem boundary element method bem domain decomposition method ddm is proposed to analyze the electromagnetic radiation and scattering from periodic structures such as frequency selective surfaces fsss or antenna arrays to improve the overall convergence an ms preconditioner is proposed for the ddm matrix the hierarchical matrix mathcal h matrices technique is introduced into the preconditioned system to compress the dense matrices to further improve the efficiency moreover a finite element tearing and interconnecting like algorithm is adopted to eliminate the interior volumetric fem unknowns and the multilevel fast multipole algorithm is integrated to reduce the computational complexity of bem numerical examples are presented to validate the proposed method it is observed that this method exhibits superior performance and consumes far less memory and cpu time than the traditional implementation

an iterative procedure with interface relaxation for domain decomposition methods a domain decomposition method for second order elliptic problems is considered an iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed at each iteration a relaxation is accomplished at the subdomain interfaces in several circumstances a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found moreover when such a value is not available an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed this iterative method is then applied to the spectral collocation approximation of the differential problem the same kind of convergence results are proven many numerical experiments show the effectiveness of the method proposed here

review v i arnol d s m gusein zade and a n varchenko singularities of differentiable maps vol 1 the classification of critical sets caustics and wave fronts

optimized schwarz waveform relaxation and discontinuous galerkin time stepping for heterogeneous problems we design and analyze a schwarz waveform relaxation algorithm for domain decomposition of advection diffusion reaction problems with strong heterogeneities the interfaces are curved and we use optimized ventcell transmission conditions we analyze the semidiscretization in time with discontinuous galerkin as well we also show two dimensional numerical results using generalized mortar finite elements in space

optimal schwarz waveform relaxation for the one dimensional wave equation we introduce a nonoverlapping variant of the schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension we derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains independently of the length of the time interval these optimal transmission conditions are in general nonlocal but we show that the nonlocality depends on the time interval under consideration and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed we show that convergence in two iterations can be achieved independently of the number of subdomains in that case the algorithm thus scales optimally with the number of subdomains provided the time windows are chosen appropriately for continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates we then introduce a finite volume discretization which permits computations on nonmatching grids and we prove convergence of the fully discrete schwarz waveform relaxation algorithm we finally illustrate our analysis with numerical experiments

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

implementation of a continuation method for normal maps this paper presents an implementation of a nonsmooth continuation method of which the idea was originally put forward by alexander et al we show how the method can be computationally implemented and present numerical results for variational inequality problems in up to 96 variables

a finite volume ventcell schwarz algorithm for advection diffusion equations this paper provides a new fully discrete domain decomposition algorithm for the advection diffusion reaction equation it relies on the optimized ventcell schwarz algorithm with a finite volume discretization of the subdomain problems the scheme includes a wide range of advection fluxes with a special treatment on the boundary a complete analysis of the scheme is presented and the convergence of the algorithm to a discrete approximation of the equation using a modified convective flux is proved numerical illustrations of the efficiency of the discrete ventcell schwarz algorithm are given

newton s method for computing a normalized equilibrium in the generalized nash game through fixed point formulation we consider the generalized nash equilibrium problem gnep where not only the players cost functions but also their strategy spaces depend on the rivals decision variables existence results for gneps are typically shown by using a fixed point argument for a certain set valued function here we use a regularization of this set valued function in order to obtain a single valued function that is easier to deal with from a numerical point of view we show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria this fixed point formulation is then used to develop a nonsmooth newton method for computing a normalized equilibrium the method uses a so called computable generalized jacobian that is much easier to compute than clarke generalized jacobian or b subdifferential we establish local superlinear quadratic convergence of the method under the constant rank constraint qualification which is weaker than the frequently used linear independence constraint qualification and a suitable second order condition some numerical results are presented to illustrate the performance of the method

regular pseudo smooth ncp and bvip functions and globally and quadratically convergent generalized newton methods for complementarity and variational inequality problems the nonlinear complementarity problem ncp can be reformulated as a system of semismooth equations by some ncp functions a well known ncp function is the fischer burmeister function which is a strongly semismooth function it is smooth everywhere except at the origin the generalized newton direction of the system of semismooth equations formulated with the fischer burmeister function is always a descent direction at a nonsolution point the generalized jacobian of the system is nonsingular under mild conditions efficient algorithms have been developed based upon these nice properties in this paper we define a class of ncp functions called regular pseudo smooth ncp functions and show that they have these nice properties regular pseudo smooth ncp functions can be easily identified they include the fischer burmeister function the tseng luo ncp function family and the kanzow kleinmichel ncp function family we give two new regular pseudo smooth ncp function families the ratio generated ncp function family and the c curve generated ncp function family we then discuss the box constrained variational inequality problem bvip we define a class of bvip functions called regular pseudo smooth bvip functions and show that they have these nice properties too we present three different approaches to generate regular pseudo smooth bvip functions from regular pseudo smooth ncp functions globally and quadratically convergent generalized newton methods are established for solving the ncp and the bvip based upon regular pseudo smooth ncp and bvip functions

critical sets in parametric optimization we deal with one parameter families of optimization problems in finite dimensions the constraints are both of equality and inequality type the concept of a generalized critical point g c point is introduced in particular every local minimum kuhn tucker point and point of fritz john type is a g c point under fairly weak even generic conditions we study the set consisting of all g c points due to the parameter the set is pieced together from one dimensional manifolds the points of can be divided into five characteristic types the subset of nondegenerate critical points first type is open and dense in nondegenerate means strict complementarity nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints a nondegenerate critical point is completely characterized by means of four indices the change of these indices along is presented finally the kuhn tucker subset of is studied in more detail in particular in connection with the failure of the mangasarian fromowitz constraint qualification

a preconditioned dual primal finite element tearing and interconnecting method for solving three dimensional time harmonic maxwell s equations a new preconditioned dual primal nonoverlapping domain decomposition method is proposed for the finite element solution of three dimensional large scale electromagnetic problems with the aid of two lagrange multipliers the new method converts the original volumetric problem to a surface problem by using a higher order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation similar to the previous version a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration which further increases the rate of convergence in addition a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable perfectly matched layers pmls are considered for the boundary truncation when solving open region problems the influence of the pml truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the pml numerical examples including wave propagation and antenna radiation problems truncated with pmls are presented to demonstrate the validity and the capability of this method

coupling spectral and finite elements for second order elliptic three dimensional equations this paper deals with the approximation of elliptic second order problems with a nonconforming domain decomposition method that allows for coupling spectral and finite element discretizations in three dimensions we present two kinds of matching conditions at the interfaces between the subdomains the detailed numerical analysis of the error committed upon the exact solution turns out to be optimal we describe also the implementation and give some numerical results confirming the theoretical predictions

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

a fast domain decomposition method for the solution of electromagnetic scattering by large objects a nonoverlapping domain decomposition method is proposed for the finite element solution of the scattering problem by electrically large inhomogeneous infinite cylinders of arbitrary cross section to minimize the size of the total computational domain a second order absorbing boundary condition abc is applied upon an outer boundary of arbitrary shape which may be conformal to the surface of the scatterer this domain is then partitioned into concentric subdomains circumscribing the object a second order transmission condition derived from the abc is prescribed upon the interfaces between two adjacent subdomains this particular configuration is responsible for the fast convergence of the domain decomposition iterative algorithm which is parallelizable numerical results obtained with a nonparallelized computer code are presented which emphasize the superiority of this technique in terms of memory storage requirements and computing times over the standard finite element approach as well as over the rigorous hybrid finite element integral equation formulation

feasible descent algorithms for mixed complementarity problems in this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems the main features of this framework are a it is well defined for an arbitrary mixed complementarity problem b it generates only feasible iterates c it has a strong global convergence theory and d it is locally fast convergent under standard regularity assumptions this framework is applied to the path solver in order to show viability of the approach numerical results for an appropriate modification of the path solver indicate that this framework leads to substantial computational improvements

space time domain decomposition methods for diffusion problems in mixed formulations this paper is concerned with global in time nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem two approaches are considered one uses the time dependent steklov poincare operator and the other uses optimized schwarz waveform relaxation oswr based on robin transmission conditions for each method a mixed formulation of an interface problem on the space time interfaces between subdomains is derived and different time grids are employed to adapt to different time scales in the subdomains demonstrations of the well posedness of the robin subdomain problems involved in the oswr method and a convergence proof of the oswr algorithm are given for the mixed formulation numerical results for two dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods

a homotopy based algorithm for mixed complementarity problems this paper develops an algorithm for solving mixed complementarity problems that is based upon probability one homotopy methods after the complementarity problem is reformulated as a system of nonsmooth equations a homotopy method is used to solve a sequence of smooth approximations to this system of equations the global convergence properties of this approach are qualitatively different from those of other recent methods which rely upon decrease of a merit function this enables the algorithm to reliably solve certain classes of problems that prove troublesome for other methods to improve efficiency the homotopy algorithm is embedded in a generalized newton method

a globally convergent successive approximation method for severely nonsmooth equations this paper presents a globally convergent successive approximation method for solving f x 0 where f is a continuous function at each step of the method f is approximated by a smooth function f k with pa f k f pa rightarrow 0 as k rightarrow infty the direction f k x k 1 f x k is then used in a line search on a sum of squares objective the approximate function f k can be constructed for nonsmooth equations arising from variational inequalities maximal monotone operator problems nonlinear complementarity problems and nonsmooth partial differential equations numerical examples are given to illustrate the method

the evolution of travelling waves in fractional order autocatalysis with decay ii the initial boundary value problem in this paper we examine an initial boundary value problem for a singular reaction diffusion equation arising from a model of isothermal chemical autocatalysis with termination we consider the case where the reaction and termination orders m and n respectively are fractional that is 0 0 k being a parameter which measures the relative strength of the termination step to that of the autocatalysis we show that solutions approach the equilibrium state 1 k for k 1 we demonstrate that the problem exhibits solutions which decay to zero with contracting support in finite t finally for m n we show that traveling waves with semi infinite support occ

a schwarz alternating procedure for singular perturbation problems we show that the schwarz alternating procedure offers a good algorithm for the numerical solution of singular perturbation problems provided the domain decomposition is properly designed to resolve the boundary and transition layers we give sharp estimates for the optimal position of the domain boundaries and study the convergence rates of the algorithm for various linear second order singular perturbation problems we report on implementation results for a turning point problem and a combustion problem

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

a probability one homotopy algorithm for nonsmooth equations and mixed complementarity problems convergence theory for a new probability one homotopy algorithm for solving non smooth equations is given this algorithm is able to solve problems involving highly nonlinear equations where the norm of the residual has nonglobal local minima the algorithm is based on constructing homotopy mappings that are smooth in the interior of their domains the algorithm is specialized to solve mixed complementarity problems mcp through the use of mcp functions and associated smoothers this specialized algorithm includes an option to ensure that all iterates remain feasible easily satisfiable sufficient conditions are given to ensure that the homotopy zero curve remains feasible and global convergence properties for the mcp algorithm are proved computational results on the mcplib test library demonstrate the effectiveness of the algorithm

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

a non mortar mixed finite element method for elliptic problems on non matching multiblock grids abstract we consider the approximation of second order elliptic equations on domains that can be described as a union of sub domains or blocks we assume that a grid is defined on each block independently so that the resulting grid over the entire domain need not be conforming i e match across the block boundaries several techniques have been developed to approximate elliptic equations on multiblock grids that utilize a mortar finite element space defined on the block boundary interface itself we define a mixed finite element method that does not use such a mortar space the method has an advantage in the case where adaptive local refinement techniques will be used in that there is no mortar grid to refine as is typical of mixed methods our method is locally conservative element by element it is also globally conservative across the block boundaries theoretical results show that the approximate solution converges at the optimal rate to the true solution we present computational results to illustrate and confirm the theory

the evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay abstract in this paper we study a reaction diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step a b 2 b and a decay step b c where a b and c are the reactant the autocatalyst and the inner product respectively previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant a initially distributes uniformly in the whole space then a pair of waves will be generated and will propagate outwards from the initial reaction zone one crucial feature of this phenomenon is that for the strong decay case the formation of waves is independent of the amount of the autocatalyst b introduced into the system it is this phenomenon of kpp type which we would like to address in this paper to study the propagation of reactant and autocatalyst analytically we first use the tail behavior of waves to construct a pair of generalized super sub solutions for the approximate system of the autocatalytic reaction system note that the autocatalytic reaction system does not enjoy comparison principle together with a family of truncated problems we can establish the existence of a family of traveling waves with the minimal speed second we use this pair of generalized super sub solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of aronson weinberger formulation which in turn confirms the aforementioned numerical and experimental results

a local regularization operator for triangular and quadrilateral finite elements this paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes this operator is a variant of the regularization operator of clement however ours is constructed via a local projection in a reference domain we prove in this paper that it has the same optimal approximation properties as the standard interpolation operator and we present some applications

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

a cell functional minimization scheme for parabolic problem we are interested in a robust and accurate finite volume scheme for 2 d parabolic problems derived from the cell functional minimization approach the scheme has a local stencil is locally conservative treats discontinuity rigorously and leads to a symmetric positive definite linear system since the scheme has both cell centered unknowns and cell edge unknowns the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach the interface condition is of the dirichlet robin type and has a parameter l by choosing this parameter properly the convergence of the iteration process could be sped up numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes

a fem domain decomposition method for photonic and electromagnetic band gap structures this paper introduces an efficient domain decomposition algorithm for the solution of time harmonic electromagnetic fields arising in three dimensional finite size photonic band gap and electromagnetic band gap structures the method is based on the finite element approximation and a nonoverlapping domain decomposition method a set of cement unknowns on the inter domain interfaces has been explicitly introduced to enforce the appropriate field continuities the introduction of these extra unknowns allows for nonconforming nonmatching triangulations across domain eliminating the need for periodic mesh to ensure and improve the convergence of the outer iteration loop robin transmission condition is used to communicate information across interfaces the resulting system of equations is solved with a fast algorithm that loosely resembles the well known finite element tearing and interconnecting algorithm in this algorithm the method solves in the preprocessing step for the robin primal subdomain problems multiple times by exciting one dual unknown at a time this step generates an iteration matrix that is then used to update the dual unknowns in the outer loop iteration the present method becomes extremely efficient for problems with geometric repetitions such as photonic and electromagnetic band gap structures

homotopy method for solving ball constrained variational inequalities abstract in this paper we present a smoothing homotopy method for solving ball constrained variational inequalities by utilizing a similar chen harker kanzow smale function to smooth robinson s normal equation without any monotonicity condition on the defining map f for the starting point chosen almost everywhere in r n the existence and convergence of the homotopy pathway are proven numerical experiments illustrate that the method is feasible and effective

existence of traveling waves of auto catalytic systems with decay abstract this article establishes the existence of traveling waves of a class of reaction diffusion systems which model the pre mixed isothermal autocatalytic chemical reaction of order m m 1 between two chemical species a reactant and an auto catalyst and a linear decay moreover our result shows that the set of speed is contained in a bounded interval for any fixed initial value at x this is in strong contrast to either the reaction diffusion systems of autocatalytic chemical reaction of the order m without decay or to the systems which have the same order of decay which were shown by various authors e g 8 13 17 26 that the set of traveling wave speeds contains c for some c 0 the same systems also appear in a mathematical model of microbial growth and competition in a flow reactor see 2 24

review lecture instabilities and oscillations in chemical reactions in closed and open systems this lecture is chiefly concerned with isothermal oscillations in homogeneous systems and with finding and validating self consistent models emphasis is laid on satisfying basic physical principles and on preserving chemical reasonableness in as simple a manner as possible the goal is to devise a scheme that can be applied to a wide variety of circumstances to predict new behaviour as well as to interpret what is known and to form a foundation for later developments and elaborations this core of the scheme called the autocatalator combines an autocatalytic step a 2b rightarrow 3b rate k 1 ab 2 with catalyst decay b rightarrow c rate k 2 b these core reactions may display reversibility and they may be accompanied by uncatalysed conversion a rightarrow b text rate k u a particular stress is given to the problem of closed systems in closed systems oscillations cannot be indefinitely sustained in former days it was customary to ignore this inconvenience and to repeal the law of conservation of matter today a less unsatisfactory account is possible

a modified standard embedding with jumps in nonlinear optimization the paper deals with a combination of pathfollowing methods embedding approach and feasible descent direction methods so called jumps for solving a nonlinear optimization problem with equality and inequality constraints since the method that we propose here uses jumps from one connected component to another one more than one connected component of the solution set of the corresponding one parametric problem can be followed numerically it is assumed that the problem under consideration belongs to a generic subset which was introduced by jongen jonker and twilt there already exist methods of this type for which each starting point of a jump has to be an endpoint of a branch of local minimizers in this paper the authors propose a new method by allowing a larger set of starting points for the jumps which can be constructed at bifurcation and turning points of the solution set the topological properties of those cases where the method is not successful are analyzed and the role of constraint qualifications in this context is discussed furthermore this new method is applied to a so called modified standard embedding which is a particular construction without equality constraints finally an algorithmic version of this new method as well as computational results are presented

finite volume methods for domain decomposition on nonmatching grids with arbitrary interface conditions we are interested in a robust and accurate domain decomposition method with arbitrary interface conditions on nonmatching multiblock grids using a finite volume discretization to take into account the nonmatching grids at the interface we introduce transmission operators dirichlet neumann interface conditions and arbitrary equivalent interface conditions for example robin interface conditions under a compatibility assumption on the transmission operators we prove the equivalence between the different types of interface conditions and the well posedness of the local and global problems then two error estimates are proven in terms of the discrete h1 norm the first in o h 1 2 with transmission operators based on piecewise constant functions and the second in o h as in the matching case with transmission operators using a piecewise linear rebuilding in conclusion numerical results are presented in confirmation of the theory

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions