on existence of positive solutions of sturm liouville boundary value problems for a nonlinear singular differential system abstract by employing the fixed point theorem of cone expansion and compression of norm type we investigate the existence of positive solutions of sturm liouville boundary value problems for a nonlinear singular differential system some well known results in the literature are generalized and improved an example is presented to illustrate the application of our main result

sharp error bounds for the derivatives of lidstone spline interpolation ii abstract in this paper we shall derive explicit error estimates in l norm between a given function f x pc n a b 4 n 6 and its quintic lidstone spline interpolate the results obtained are then used to establish precise error bounds for the approximated and biquintic lidstone spline interpolates we also include applications to integral equations and boundary value problems as well as sufficient numerical examples which dwell upon the sharpness of the obtained results

monotone convergence of iterative methods for right focal point boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for the right focal point boundary value problems converges monotonically an example which dwells upon the usefulness of this technique is also included

existence of positive solutions for a system of generalized lidstone problems in this paper we study the existence of positive solutions for the system of generalized lidstone problems for ordinary differential equations 1 mu 2 m f 1 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 1 nv 2 n f 2 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 a 0u 2 i 0 b 0u 2 i 1 0 a 1u 2 i 1 b 1u 2 i 1 1 0 i 0 1 m 1 a 0v 2 j 0 b 0v 2 j 1 0 a 1v 2 j 1 b 1v 2 j 1 1 0 j 0 1 n 1 here f 1 f 2 c 0 1 xr m n r r 0 our hypotheses imposed on the nonlinearities f 1 and f 2 are formulated in terms of nonnegative matrices we use fixed point index theory to establish our main results based on a priori estimates of positive solutions

a singular boundary value problem for odd order differential equations abstract the odd order differential equation 1 nx 2n 1 f t x x 2n together with the lidstone boundary conditions x 2j 0 x 2j t 0 0 j n 1 and the next condition x 2n 0 0 is discussed here f satisfying the local caratheodory conditions can have singularities at the value zero of all its phase variables existence result for the above problem is proved by the general existence principle for singular boundary value problems

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

a deconvolution enhancement of the navier stokes αβ model a deconvolution enhancement of the email protected b model for turbulent flow is introduced the energy and energy dissipation rate for the enhanced model are derived it is also shown that the consistency error relative to the navier stokes equations and the microscale of the enhanced model are less than those of the email protected b model the proposed model is used to simulate the taylor green vortex problem and results show a qualitatively improved representation of the mean square vorticity when compared to the email protected b model numerical studies of the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor for three dimensional turbulent flows with re 200 are used to explore the utility of the model a benchmark problem of a two dimensional channel flow over a step for re 600 also indicates that this model can be applied to more general flows than those involving periodic boundary conditions

higher order lidstone boundary value problems for elliptic partial differential equations the aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth order elliptic lidstone boundary value problems where the nonlinear functions depend on the higher order derivatives sufficient conditions are given for the existence and uniqueness of a solution it is also shown that there exist two sequences which converge monotonically from above and below respectively to the unique solution the approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions all the results are directly applicable to 2nth order two point lidstone boundary value problems

piecewise complementary lidstone interpolation and error inequalities the purpose of this paper is to develop piecewise complementary lidstone interpolation in one and two variables and establish explicit error bounds for the derivatives in l and l 2 norms

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

two point higher order bvps with singularities in phase variables the existence of solutions for singular higher order differential equations with the lidstone or the n p boundary conditions is proved the right hand sides of differential equations can have singularities in the zero value of their phase variables and so higher derivatives of solutions changing their signs can pass through these singularities proofs are based on the method of a priori estimates the degree theory arguments and on vitali s convergence theorem

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

katoʼs residue homomorphisms and reciprocity laws on arithmetic surfaces abstract we explicitly study kato s residue homomorphisms in milnor k theory which are closely related to contou carrere symbols as applications we establish several reciprocity laws for certain locally defined maps on k groups that are associated to arithmetic surfaces

explicit error bounds for the derivatives of piecewise lidstone interpolation abstract the purpose of this paper is to provide explicit error bounds for the derivatives of piecewise lidstone interpolation in l and l2 norms the results obtained are best possible sharp and supplement those established in earlier papers

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

polynomial approximation of cm functions by means of boundary values and applications a survey we collect classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only the problem is considered from the point of view of the existence of explicit formulas interpolation to boundary data bounds for the remainder and convergence of the polynomial series applications to some problems of numerical analysis are pointed out such as nonlinear equations numerical differentiation and integration formulas special associated differential boundary value problems some polynomial expansions for smooth enough functions defined in rectangles or in triangles of r 2 as well as in cuboids or in tetrahedrons in r 3 and their applications are also discussed

a sensitivity study of the navier stokes α model abstract we present a sensitivity study of the navier stokes model with respect to perturbations of the differential filter length the parameter sensitivity is evaluated using the sensitivity equations method once formulated the sensitivity equations are discretized and computed alongside the ns model using the same finite elements in space and crank nicolson in time we provide a complete stability analysis of the scheme along with the results of several benchmark problems in both 2d and 3d we further demonstrate a practical technique to utilize sensitivity calculations to determine the reliability of the ns model in problem specific settings lastly we investigate the sensitivity and reliability of important functionals of the velocity and pressure solutions

truncation of scales by time relaxation abstract we study a time relaxation regularization of flow problems proposed and tested extensively by stolz and adams the aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term our aim herein is to understand how this term by itself acts to truncate solution scales and to use this understanding to give insight into parameter selection

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

upper and lower solution methods for fully nonlinear boundary value problems abstract sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions the conditions assume the existence of a strong upper solution lower solution pair a concept that is defined in the paper the differential equation has nonlinear dependence on all lower order derivatives of the unknown in particular appropriate nagumo conditions are obtained and employed

on the accuracy of the rotation form in simulations of the navier stokes equations the rotation form of the navier stokes equations nonlinearity is commonly used in high reynolds number flow simulations it was pointed out by a few authors and not widely known apparently that it can also lead to a less accurate approximate solution than the usual email protected u form we give a different explanation of this effect related to i resolution of the bernoulli pressure and ii the scaling of the coupling between velocity and pressure error with respect to the reynolds number we show analytically that i the difference between the two nonlinearities is governed by the difference in the resolution of the bernoulli and kinematic pressures and ii a simple linear grad div stabilization ameliorates much of the bad scaling of the velocity error with respect to re the rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners thus the rotational form with grad div stabilization is a promising method we also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form

impact of the inherent separation of scales in the navier stokes αβ equations we study the effect of the length scales alpha and beta in the navier stokes alphabeta equations on the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor in three dimensional homogeneous and isotropic turbulent flows in a periodic cubic domain including the limiting cases of the navier stokes alpha and navier stokes equations a significant increase in the accuracy of the energy spectrum at large wave numbers arises for beta alpha the vorticity structures predicted by the navier stokes alphabeta equations also improve as beta decreases away from alpha however optimal choices for alpha and beta depend not only on the problem of interest but also on the grid resolution

existence of nodal solutions for lidstone eigenvalue problems abstract we are concerned with determining values of r for which there exist nodal solutions of the lidstone eigenvalue problem 1 m u 2 m x r a x f u x 0 x 1 u 2 i 0 u 2 i 1 0 i 0 1 2 m 1 the proof of our main result is based upon eigenvalue theory and bifurcation techniques

eigenvalues and the one dimensional p laplacian we consider the boundary value problem p u f t u 0 with p 1 t 0 1 u 0 u 1 0 and with 0 the value of is chosen so that the boundary value problem has a positive solution in addition we derive an explicit interval for such that for any in this interval the existence of a positive solution to the boundary value problem is guaranteed in addition the existence of two positive solutions for in an appropriate interval is also discussed

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

monotone iterative technique and positive solutions of lidstone boundary value problems the purpose of this paper is to investigate the existence and iteration of n symmetric positive solutions for lidstone boundary value problem 1 nw 2 n t f t w t 0

the camassa holm equations and turbulence abstract in this paper we will survey our results on the camassa holm equations and their relation to turbulence as discussed in s chen c foias d d holm e olson e s titi s wynne the camassa holm equations as a closure model for turbulent channel and pipe flow phys rev lett 81 1998 5338 s chen c foias d d holm e olson e s titi s wynne a connection between the camassa holm equations and turbulent flows in channels and pipes phys fluids in press in particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large reynolds numbers there are many facts connecting the camassa holm equations to turbulent fluid flows the dimension of the attractor agrees with the heuristic argument based on the kolmogorov statistical theory of turbulence the statistical properties of the energy spectrum agree in numerical simulation with the kolmogorov power law furthermore comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the camassa holm equations

an interpretation of the navier stokes alpha model as a frame indifferent leray regularization abstract we show in this note that the leray regularization of the navier stokes equations is almost frame indifferent but for one second order term neglecting this term we readily recover the navier stokes alpha equations introduced by holm et al adv math 137 1998 1 phys fluids 11 8 1999 2343 hence as an alternative to the exposition in adv math 137 1998 1 phys fluids 11 8 1999 2343 we propose to interpret the navier stokes alpha model as a frame indifferent perturbation of the leray regularization

tractions balances and boundary conditions for nonsimple materials with application to liquid flow at small length scales using a nonstandard version of the principle of virtual power we develop general balance equations and boundary conditions for second grade materials our results apply to both solids and fluids as they are independent of constitutive equations as an application of our results we discuss flows of incompressible fluids at small length scales in addition to giving a generalization of the navier stokes equations involving higher order spatial derivatives our theory provides conditions on free and fixed boundaries the free boundary conditions involve the curvature of the free surface among the conditions for a fixed boundary are generalized adherence and slip conditions each of which involves a material length scale we reconsider the classical problem of plane poiseuille flow for generalized adherence and slip conditions

sensitivities in large eddy simulation and improved estimates of turbulent flow functionals we consider a prototypical problem simulate velocity and pressure in a turbulent flow using large eddy simulation les and then use the results to estimate the force that the underlying turbulent flow exerts on an immersed body for eddy viscosity type les models we develop the appropriate continuous sensitivity equation and show how it can improve the functional estimate by using the sensitivity with respect to the user selected length scale to incorporate the effects of the underlying unresolved small scale fluctuations on the functionals sought

a new representation for the error function for the hermite interpolation and sharper pointwise and uniform error bounds for the derivatives let a a a r 2 2 be the given points and x e c a a n 2 2 be a given function satisfying the conditions a a j l j i 0 1 k j 1 2 r 1 1 where j kj r n let p r be the hermite interpolation polynomial of degree n 1 satisfying 1 1 for the error function e r x f p t everyone knows the integral representation in terms of peano s kernel green s function however the explicit representation of the green s function is unwieldy l 21 and hence except in some simple cases it cannot be used to obtain the bounds for the derivatives of e t the situation further worsens because the derivatives of the green s function change sign in the interval at a in fact even for the case n 6 r 2 k k 2 the computation is quite lengthy 3 the purpose of this paper is to give a new representation of the error function e t and use it to obtain pointwise as well as uniform error bounds for je k t 0 5 k 5 n 1 in terms of max ix t in

a numerical study of the navier stokes αβ model abstract we present a numerical study of the ns model which is a recently proposed multiscale variation of the ns model that attempts to recapture scales lost through over regularization by separately modeling dissipation range scales we develop a similarity theory for the new model which shows that it is better equipped than the ns model to capture smaller scale behavior next we propose and study an unconditionally stable optimally accurate and efficient finite element implementation for the ns model rigorous proofs for stability and convergence are provided finally we present results from two numerical experiments that demonstrate the advantages of the ns model over the ns model

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

numerical methods for unilateral problems abstract the theory of variational inequalities besides being elegant and synthetic also provides a unified general framework for applying the numerical methods for solving many unrelated free boundary value problems in this paper we describe numerical experience on the use of variational inequalities and pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle in the case of a known obstacle these problems can be alternatively formulated as nonlinear boundary value problems without constraints for which the technique of pade approximants can be successfully employed the variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution

eigenvalues of a general class of boundary value problem with derivative dependent nonlinearity we consider a general class of boundary value problem bvp comprising the differential equation y m t f t y t y t y t y q t t 0 1 where 1 q m 1 and 0 together with multi point boundary conditions y 0 y 0 y 0 y q 1 0 0 a i y q t j y q 1 t j y m 1 t j 0 j r 0 1 i m q where 0 t 0 t 1 t r 1 particular cases of this general bvp include the well known abel gontscharoff focal m p sturm liouville and complementary lidstone bvps it is noted that bvps with derivative dependent nonlinear terms are less investigated in the literature due to technical difficulty in this paper a new technique is developed to characterize the eigenvalues so that the bvp has a positive solution explicit eigenvalue intervals are also established we include several well known examples in the literature to illustrate the results obtained

discrete models of fluids spatial averaging closure and model reduction we consider semidiscrete ode models of single phase fluids and two fluid mixtures in the presence of multiple fine scale heterogeneities the size of these ode systems can be very large spatial averaging is then a useful tool for reducing computational complexity of the problem the averages satisfy exact balance equations of mass momentum and energy these equations do not form a satisfactory continuum model because evaluation of stress and heat flux requires solving the underlying odes to produce continuum equations that can be simulated without resolving microscale dynamics we recently proposed a closure method based on the use of regularized deconvolution here we continue the investigation of deconvolution closure with the long term objective of developing consistent computational upscaling for multiphase particle methods the structure of the fine scale particle solvers is reminiscent of molecular dynamics for this reason we use nonlinear averaging introduced for atomistic systems by noll ha

singular lidstone boundary value problem with given maximal values for solutions abstract the singular problem 1 nx 2n f t x x 2n 2 x 2j 0 x 2j t 0 0 j n 1 max x t 0 t t a depending on the parameter is considered here the positive caratheodory function f may be singular at the zero value of all its phase variables the paper presents conditions which guarantee that for any a 0 there exists a 0 such that the above problem with a has a solution x ac2n 1 0 t which is positive on 0 t the proofs are based on the regularization and sequential techniques and use the leray schauder degree and vitali s convergence theorem

analysis of numerical errors in large eddy simulation we consider the question of numerical errors in large eddy simulation it is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the reynolds number and dependence only on the resolution delta of the eddies sought this report considers this question analytically is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on delta but not re we consider the most common simplest and most mathematically tractable model and the most mathematically clear discretization in two cases we prove such an error estimate and try to explain why our technique of proof fails in the most general case our analysis aims to assume as little time regularity on the true solution as possible

eigenvalues of lidstone boundary value problems we consider the following boundary value problem 1 ny 2 n lf t y n 1 t 0 1 y 2 i 0 y 2 i 1 0 0 0 the values of l are characterized so that the boundary value problem has a positive solution in addition we derive explicit intervals of l such that for any l in the interval existence of a positive solution of the boundary value problem is guaranteed several examples are also included to dwell upon the importance of the results obtained

monotone convergence of iterative methods for n p and p n boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for n p and p n boundary value problems converges monotonically an example which illustrates the usefulness of this technique is also included

a family of new high order ns α models arising from helicity correction in leray turbulence models abstract this report gives a reinterpretation of the ns model which leads to a family of high order ns deconvolution models with ns as the zeroth order case first we show that the navier stokes model arises by adding helicity correction to leray model higher order leray models have recently been proposed in w layton r lewandowski a high accuracy leray deconvolution model of turbulence and its limiting behavior anal appl 6 1 2008 w layton c manica m neda l rebholz numerical analysis and computational testing of a high accuracy leray deconvolution model of turbulence numer methods partial differential equations in press the so called leray deconvolution models that employ van cittert approximate deconvolution to decrease consistency error we use an analogous helicity correction idea to develop a family of higher order accurate ns type models the ns deconvolution models we prove several mathematical and physical properties for this new family of models and discuss the design of efficient algorithms for them

polynomial approximation to a non local boundary value problem in this paper we construct a special class of polynomials which converge uniformly to the solution of a non local boundary value problem nbvp the use of this special class is justified by the physics of the model which is described by this nbvp this nbvp has been studied by palamides et al 2009 in 2 where the existence of solutions is established

numerical analysis and computational comparisons of the ns alpha and ns omega regularizations we study stability accuracy and efficiency of algorithms for a new regularization of the nse the ns model in which the vorticity term u is averaged given by u t u q q v u f u 0 this is similar to the ns model in which the nonlinear term is x v x u but the small difference opens attractive algorithmic possibilities we give tests both confirming the predicted rates of convergence and exhibiting some shared limitations of both models the experiments also show the discrete ns simulation has greater accuracy tables 3 and 4 at less cost table 2 and requires significantly fewer degrees of freedom section 1 1 than a comparable ns simulation the experiments suggest consideration of adding grad div stabilization and higher accuracy ns deconvolution models as a next logical step in fact this combination produced accurate results see figs 10 and 11 on the coarsest mesh mesh 1 upon which all other methods models and variants tested failed

a posteriori analysis of a b spline based finite element method for the stationary quasi geostrophic equations of the ocean abstract we provide a posteriori error analysis of a b spline based finite element method for the streamfunction formulation of the stationary quasi geostrophic equations to model the large scale wind driven ocean circulation the analysis gives rise to a posteriori error estimator for more efficient and accurate modeling of a strong western boundary layer via adaptive local refinement of the computational geometries to verify the capability of the error estimator we use hierarchical b splines in conjunction with dorfler marking strategy numerical studies are performed for several benchmark examples on rectangular and l shape ocean geometries

on 2nth order lidstone boundary value problems this paper is concerned with the solutions of a class of 2nth order lidstone boundary value problems sufficient conditions for the existence and uniqueness of a solution are given a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution the approach to the problem is by the method of upper and lower solutions with a new maximum principle

the iterative solutions of 2nth order nonlinear multi point boundary value problems abstract the aim of this paper is to investigate the existence of iterative solutions for a class of 2 n th order nonlinear multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique a monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution two examples are presented to illustrate the results

a fourth order compact finite difference method for higher order lidstone boundary value problems a compact finite difference method is proposed for a general class of 2nth order lidstone boundary value problems the existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term a monotone iteration process is provided for solving the resulting discrete system efficiently and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations the convergence of the finite difference solution and the fourth order accuracy of the proposed method are proved numerical results demonstrate the high efficiency and advantages of this new approach

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

existence and uniqueness of positive solutions for a higher order boundary value problem the goal of this paper is to study the existence and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 n 1u 2 n 2 a 0u 2 i 0 email protected 0u 2 i 1 0 0 i 0 1 n 1 a 1u 2 i 1 email protected 1u 2 i 1 1 0 i 0 1 n 1 the krein rutman theorem and the krasnoselskii zabreiko fixed point theorem are the main tools that have been used to develop our work

error estimates of b spline based finite element methods for the stationary quasi geostrophic equations of the ocean abstract this paper presents theoretical error estimates of b spline based finite element methods for the streamfunction formulation of the stationary quasi geostrophic equations which describe the large scale wind driven ocean circulation we introduce variational formulations of the streamfunction formulation inspired by the interior penalty discontinuous galerkin method dirichlet boundary conditions are weakly enforced in the formulations and stabilizations are achieved via nitsche s method existence and uniqueness of the approximation are proved and optimal error estimates in the energy norm are demonstrated under a small data assumption numerical experiments are performed to verify the theoretical error estimates on rectangular and l shape geometries

erratum turbulent kinetic energy and a possible hierarchy of length scales in a generalization of the navier stokes a theory phys rev e 75 056306 2007

upper and lower estimates for positive solutions of the higher order lidstone boundary value problem we consider the higher order lidstone boundary value problem new upper and lower estimates for positive solutions of the problem are obtained a discussion of the sharpness of the estimates is included

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

positive solutions of a 2nth order boundary value problem involving all derivatives via the order reduction method this paper is mainly concerned with the existence multiplicity and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 i 2 u i 1 n 1u 2 n 1 u 2 i 0 u 2 i 1 1 0 i 0 1 n 1 where n 2 and f c 0 1 xr 2 n r r 0 we first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first order integro differential equation and then use the fixed point index theory to prove the existence multiplicity and uniqueness of positive solutions for the resulting problem based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators finally as a byproduct our main results are applied for establishing the existence multiplicity and uniqueness of symmetric positive solutions for the lidstone problem involving all derivatives

navier stokes equations in rotation form a robust multigrid solver for the velocity problem the topic of this paper is motivated by the navier stokes equations in rotation form linearization and application of an implicit time stepping scheme results in a linear stationary problem of oseen type in well known solution techniques for this problem such as the uzawa or schur complement method a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion reaction type must be solved to update the velocity vector field in this paper we analyze a standard finite element method for the discretization of this coupled system and we introduce and analyze a multigrid solver for the discrete problem both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed we prove stability results and discretization error bounds for the galerkin finite element method we present a convergence analysis of the multigrid method which shows the robustness of the solver results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver