superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

approximation of boundary conditions for mimetic finite difference methods abstract the numerical solution of partial differential equations solved with finite difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results furthermore those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation unless special care is taken mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient curl and divergence operators at the computational boundary in this paper we describe how to incorporate boundary conditions into finite difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus the approach is valid for a wide class of partial differential equations of mathematical physics and will be described for poisson s equation with dirichlet neumann and robin boundary conditions we prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions

parallel multigrid preconditioning of the conjugate gradient method for systems of subsurface hydrology parallel preconditioners are considered for improving the convergence rate of the conjugate gradient method for solving sparse symmetric positive definite systems generated by finite element models of subsurface flow the difficulties of adapting effective sequential preconditioners to the parallel environment are illustrated by our treatment of incomplete cholesky preconditioning these difficulties are avoided with multigrid preconditioning which can be extended naturally to many processors so that the preconditioner remains global and effective the coarse grid correction which defines the multigrid preconditioner is outlined and its parallel implementation with the distributed finite element data structure is presented along with some examples of its use as a parallel preconditioner

underlying paths in interior point methods for the monotone semidefinite linear complementarity problem an interior point method defines a search direction at each interior point of the feasible region the search directions at all interior points together form a direction field which gives rise to a system of ordinary differential equations odes given an initial point in the interior of the feasible region the unique solution of the ode system is a curve passing through the point with tangents parallel to the search directions along the curve we call such curves off central paths we study off central paths for the monotone semidefinite linear complementarity problem sdlcp we show that each off central path is a well defined analytic curve with parameter ranging over 0 and any accumulation point of the off central path is a solution to sdlcp through a simple example we show that the off central paths are not analytic as a function of sqrt mu and have first derivatives which are unbounded as a function of at 0 in general on the other hand for the same example we can find a subset of off central paths which are analytic at 0 these nice paths are characterized by some algebraic equations

multilevel upscaling through variational coarsening 1 a new efficient multilevel upscaling procedure for single phase saturated flow in porous media is presented while traditional approaches to this problem have focused on the computation of an upscaled hydraulic conductivity here the coarse scale model is created explicitly from the fine scale model through the application of operator induced variational coarsening this technique which originated with robust multigrid solvers has been shown to accurately capture the influence of fine scale heterogeneous structure over the complete hierarchy of coarse scale models that it generates moreover implicit in this hierarchy is the construction of interpolation operators that provide a natural and complete multiscale basis for the fine scale problem thus this new multilevel upscaling methodology is similar to the multiscale finite element method and indeed it attains similar accuracy in computations of the fine scale hydraulic head and coarse scale normal flux on a variety of problems yet it is an order of magnitude faster on the examples considered here

support operator finite difference algorithms for general elliptic problems abstract an algorithm is developed for discretizing boundary value problems given by a general linear elliptic second order partial differential equation with general mixed or robin boundary conditions in general logically rectangular grids the continuum problem can be written as an operator equation where the operator is self adjoint and positive definite the discrete approximations have the same property consequently the matrices for the discrete problem are symmetric and positive definite also the scheme has a nearest neighbor stencil consequently the most powerful linear solvers can be applied in smooth grids the algorithm produces second order accurate solutions it is the generality of the problem general matrix coefficients general boundary conditions general logically rectangular grids that makes finding such an algorithm difficult the algorithm which is a combination of the method of support operators and the mapping method overcomes certain difficulties of the individual methods producing a high quality algorithm for solving general elliptic problems

the curvature integral and the complexity of linear complementarity problems in this paper we propose a predictor corrector type algorithm for solving the linear complementarity problem lcp and prove that the actual number of iterations needed by the algorithm is bounded from above and from below by a curvature integral along the central trajectory of the problem this curvature integral is not greater than and possibly smaller than the best upper bound obtained in the literature to date

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

high order mimetic finite difference method for diffusion problems on polygonal meshes the mimetic finite difference mfd methods mimic important properties of physical and mathematical models as a result conservation laws solution symmetries and the fundamental identities of the vector and tensor calculus are held for discrete models the mfd methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non convex and degenerate elements the existing mfd methods for solving diffusion type problems are second order accurate for the conservative variable temperature pressure energy etc and only first order accurate for its flux we developed new high order mfd methods which are second order accurate for both scalar and vector variables the second order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes

a probabilistic analysis of a measure of combinatorial complexity for the central curve we investigate certain combinatorial properties of the central curve associated with interior point methods for linear optimization we define a measure of complexity for the curve in terms of the number of turns or changes of direction that it makes in a geometric sense and then perform an average case analysis of this measure for p matrix linear complementarity problems we show that the expected number of nondegenerate turns taken by the central curve is bounded by n 2 n where the expectation is taken with respect to a sign invariant probability distribution on the problem data as an alternative measure of complexity we also consider the number of times the central curve intersects with a wide class of algebraic hypersurfaces including such objects as spheres and boxes as an example of the results obtained we show that the primal and dual variables in each coordinate of the central curve cross each other at most once on average as a further example we show that the central curve intersects any sphere centered at the origin at most twice on average

multiscale linear solvers for very large systems derived from pdes we present a novel linear solver that works well for large systems obtained from discretizing pdes it is robust and for the examples we studied the computational effort scales linearly with the number of equations the algorithm is based on a wavelength decomposition that combines conjugate gradient multiscaling and iterative splitting methods into a single approach on the surface the algorithm is a simple preconditioned conjugate gradient with all the sophistication of the algorithm in the choice of the preconditioning matrix the preconditioner is a very good approximate inverse of the linear operator it is constructed from the inverse of the coarse grained linear operator and from smoothing operators that are based on an operator splitting on the fine grid the coarse graining captures the long wavelength behavior of the inverse operator while the smoothing operator captures the short wavelength behavior the conjugate gradient iteration accounts for the coupling between long and short wavelengths the coarse grained operator corresponds to a lower resolution approximation to the pdes while the coarse grained inverse is not known explicitly the algorithm requires only that the preconditioner can be a applied to a vector the coarse inverse applied to a vector can be obtained as the solution of another preconditioned conjugate gradient solver that applies the same algorithm to the smaller problem thus the method is naturally recursive the recursion ends when the matrix is sufficiently small for a solution to be obtained efficiently with a standard solver the local feedback provided by the conjugate gradient step at every level makes the algorithm very robust in spite of the effort required for the coarse inverse the algorithm is efficient because the increased quality of the approximate inverse greatly reduces the number of times the preconditioner needs to be evaluated a feature of the algorithm is that the transition between coarse grids is determined dynamically by the accuracy requirement of the conjugate gradient solver at each level typically later iterations on the finer scales need fewer iterations on the coarser scales and the computational effort is proportional to n rather than n log n where n is the number of equations we have tested our solver on the porous flow equation on a workstation we have solved problems on grids ranging in dimension over three orders of magnitude from 103 to 106 and found that the linear scaling holds the algorithm works well even when the permeability tensor has spatial variations exceeding a factor of 109

existence of oscillatory solutions along the path of longitudinal flight equilibriums of an unmanned aircraft when the automatic flight control system fails abstract the motion around the center of mass of a rigid unmanned aircraft whose flight control system fails in an aero data model in a research environment is described by a set of nine nonlinear ordinary differential equations the longitudinal flight with constant forward velocity is described by a subset of three nonlinear differential equations obtained from the general system in this paper the existence of oscillatory solutions of this system of three differential equations is proved by means of coincidence degree theory and mawhin s continuation theorem

parallel preconditioned conjugate gradient algorithm on gpu we propose a parallel implementation of the preconditioned conjugate gradient algorithm on a gpu platform the preconditioning matrix is an approximate inverse derived from the ssor preconditioner used through sparse matrix vector multiplication the proposed preconditioner is well suited for the massively parallel gpu architecture as compared to cpu implementation of the conjugate gradient algorithm our gpu preconditioned conjugate gradient implementation is up to 10 times faster 8 times faster at worst

oscillation susceptibility analysis along the path of longitudinal flight equilibriums abstract in this paper the oscillation susceptibility of an aircraft in a longitudinal flight with constant forward velocity is analyzed in different flight models conditions which ensure such a flight and equations governing the flight are presented the stability of the equilibriums appearing is analyzed and the existence of hopf bifurcations and saddle node bifurcations is researched for two aircrafts in a simplified model it is shown that saddle node bifurcations are present and there are no hopf bifurcations it is shown that for the elevator deflection there are two turning points e e having the property that if e e e then the angle of attack and the pitch rate q oscillate with the same period while the pitch angle increases decreases tending to the behavior of the aircraft is simulated in the simplified model when the elevator deflection e varies in the range e e and when e leaves this range for one of the aircrafts the analysis is performed also in the not simplified model showing the differences between the results obtained in different models

a polynomial method of approximate centers for linear programming we present a path following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour this is done both for the problem in standard form and for its dual problem we also discuss some implementation strategies

existence of heteroclinic orbits in simplified longitudinal flight dynamics the existence of an infinite chain of heteroclinic orbits connecting saddle nodes is proved in the case of a simplified system of differential equations governing the longitudinal flight of an aircraft using fixed point theory for nonlinear volterra operators with convolution kernels numerical simulations are given and the importance of the obtained results is discussed from the point of view of the appearance of oscillatory behavior in longitudinal flight

on the complexity of following the central path of linear programs by linear extrapolation ii a class of algorithms is proposed for solving linear programming problems withm inequality constraints by following the central path using linear extrapolation with a special adaptive choice of steplengths the latter is based on explicit results concerning the convergence behaviour of newton s method to compute points on the central pathx r r 0 and this allows to estimate the complexity i e the total numbern n r of steps needed to go from an initial pointx r to a final pointx r 0 by an integral of the local weighted curvature of the primal dual path here the central curve is parametrized with the logarithmic penalty parameterr 0 it is shown that for large classes of problems the complexity integral i e the number of stepsn is not greater than constm log r where 1 2 e g 1 4 or 3 8 note that 1 2 gives the complexity of zero order methods we also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for 1 3 as a byproduct many analytical and structural properties of the primal dual central path are obtained there are for instance close relations between the weighted curvature and the logarithmic derivatives of the slack variables the dependence of these quantities on the parameterr is described also related results hold for a family of weighted trajectories into which the central path can be embedded

a preconditioned multilevel mimetic finite difference method for solving the pressure velocity system abstract a mimetic finite difference mfd implementation is proposed to solve the pressure velocity system iteratively the recursive procedure only solves for the velocity variable while the pressure is calculated by direct substitution simplifying the problem the symmetric positive definite system obtained is solved with an innovative implementation of a pre conditioned multilevel conjugate gradient algorithm where coarse algebraic objects are easily computed from the corresponding fine level ones based on fundamental geometric properties of the mfd technology a succinct eigen analysis shows second order convergence on the velocity variable for cartesian grids which is corroborated numerically also results consider different topologies and different medium properties showing the applicability of the method and optimal convergence rates for the velocity on cartesian and h 2 uniform curved grids

primal dual affine scaling interior point methods for linear complementarity problems a first order affine scaling method and two m th order affine scaling methods for solving monotone linear complementarity problems lcps are presented all three methods produce iterates in a wide neighborhood of the central path the first order method has o nl 2 lognl 2 log lognl 2 iteration complexity if the lcp admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with q order two if m omega log sqrt n l then both higher order methods have o sqrt n l iteration complexity the q order of convergence of one of the methods is m 1 for problems that admit a strict complementarity solution while the q order of convergence of the other method is m 1 2 for general monotone lcps

interior point algorithms for linear complementarity problems based on large neighborhoods of the central path in this paper we study a first order and a high order algorithm for solving linear complementarity problems these algorithms are implicitly associated with a large neighborhood whose size may depend on the dimension of the problems the complexity of these algorithms depends on the size of the neighborhood for the first order algorithm we achieve the complexity bound which the typical large step algorithms possess it is well known that the complexity of large step algorithms is greater than that of short step ones by using high order power series hence the name high order algorithm the iteration complexity can be reduced we show that the complexity upper bound for our high order algorithms is equal to that for short step algorithms

spectral properties and conservation laws in mimetic finite difference methods for pdes the mimetic finite difference mfd methods for pdes mimic crucial properties of mathematical systems duality and self adjointness of differential operators conservation laws and properties of the solution on general polytopal meshes in this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed in addition the numerical approximation of parabolic equations is discussed where the mfd approach is used in the space discretization while implicit method and explicit runge kutta chebyshev schemes are used in time discretization moreover we will show how the numerical solution preserves certain conservation laws of the theoretical solution

guaranteed properties of gain scheduled control for linear parameter varying plants abstract gain scheduling has proven to be a successful design methodology in many engineering applications however in the absence of a sound theoretical analysis these designs come with no guarantees on the robustness performance or even nominal stability of the overall gain scheduled design this paper presents such an analysis for one type of gain scheduled system namely a linear parameter varying plant scheduling on its exogenous parameters conditions are given which guarantee that the stability robustness and performance properties of the fixed operating point designs carry over to the global gain scheduled design these conditions confirm and formalize popular notions regarding gain scheduled design such as the scheduling variable should vary slowly