a newton kantorovich convergence theorem for the inverse free jarratt method in banach space abstract under the weaken conditions we establish the newton kantorovich convergence theorem of inverse free jarratt method in banach space which is used to solve the nonlinear operator equation finally some examples are provided to show the application of our theorem

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

a framework of verified eigenvalue bounds for self adjoint differential operators for eigenvalue problems of self adjoint differential operators a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues in the case of the laplacian operator by applying crouzeix raviart finite elements an efficient algorithm is developed to bound the eigenvalues for the laplacian defined in 1d 2d and 3d spaces moreover for nonconvex domains for which case there may exist singularities of eigenfunctions around re entrant corners the proposed algorithm can easily provide eigenvalue bounds by further adopting the interval arithmetic the explicit eigenvalue bounds from numerical computations can be mathematically correct

minimizing neumann fundamental tones of triangles an optimal poincare inequality abstract the first nonzero eigenvalue of the neumann laplacian is shown to be minimal for the degenerate acute isosceles triangle among all triangles of given diameter hence an optimal poincare inequality for triangles is derived the proof relies on symmetry of the neumann fundamental mode for isosceles triangles with aperture less than 3 antisymmetry is proved for apertures greater than 3

decay rate for the incompressible flows in half spaces we show that the time decay rate of l 2 norm of weak solution for the stokes equations and for the navier stokes equations on the half spaces are t frac n 2 frac 1 r frac 1 2 frac 1 2 if the initial data u 0 in l 2 cap l r and int mathbb r n y n u 0 y r dy infty for 1 r 2 we also show that the decay rate is determined by the linear part of the weak solution we use the heat kernel and ukai s solution formula for the stokes equations it has been known up to now that the decay rate on the half space was t frac n 2 frac 1 r frac 1 2 which was obtained by borchers and miyakawa 1 and ukai 9

remarks on a posteriori error estimation for finite element solutions we utilize the classical hypercircle method and the lowest order raviart thomas h div element to obtain a posteriori error estimates of the p1 finite element solutions for 2d poisson s equation a few other estimation methods are also discussed for comparison we give some theoretical and numerical results to see the effectiveness of the methods

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

weighted lq l1 estimate of the stokes flow in the half space abstract in this paper we derive the weighted l q l 1 estimate of the stokes solution u in the half space when 1 q we obtained that x r u l q c t 1 2 n 2 1 1 q x r x n a l 1 c t r 1 2 n 2 1 1 q x n a l 1 for 0 r n 1 1 1 q and x n r u l q c t 1 2 n 2 1 1 q x n r 1 a l 1 c t r 1 2 n 2 1 1 q x n a l 1 for 0 r n 1 1 q

an alternative approach to norm bound computation for inverses of linear operators in hilbert spaces abstract in the present paper we propose a computer assisted procedure to prove the invertibility of a linear operator in a hilbert space and to compute a verified norm bound of its inverse a number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates the approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications several verification examples that confirm the actual effectiveness of the proposed procedure are reported

bounds to eigenvalues of the laplacian on l shaped domain by variational methods in this study the bounds for eigenvalues of the laplacian operator on an l shaped domain are determined by adopting some special functions in goerisch method for lower bounds and in traditional rayleigh ritz method for upper bounds very accurate bounds to eigenvalues for the problem are obtained numerical results show that these functions can also be successfully used to solve the problem on the region with other reentrant angle

computable bounds for eigenvalues and eigenfunctions of elliptic differential operators we are concerned with bounds for the error between given approximations and the exact eigenvalues and eigenfunctions of self adjoint operators in hilbert spaces the case is included where the approximations of the eigenfunctions don t belong to the domain of definition of the operator for the eigenvalue problem with symmetric elliptic differential operators these bounds cover the case where the trial functions don t satisfy the boundary conditions of the problem the error bounds suggest a certain defectminization method for solving the eigenvalue problems the method is applied to the membrane problem

asymptotic behavior for the navier stokes equations in 2d exterior domains we show that the lp spatial temporal decay rates of solutions of incompressible flow in an 2d exterior domain when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in he xin c he z xin weighted estimates for nonstationary navier stokes equations in exterior domain methods appl anal 7 3 2000 443 458 and our previous results h o bae b j jin asymptotic behavior of stokes solutions in 2d exterior domains j math fluid mech in press h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains submitted for publication for the spatial decay rate estimate we first extend temporal decay rate result of the navier stokes solutions for general lp space when the initial velocity is in l 2 lr 1 r q 1 r q

the application of an inverse free jarratt type approximation to nonlinear integral equations of hammerstein type abstract we consider an inverse free jarratt type approximation whose order of convergence is four for solving nonlinear equations the convergence of this method is analysed under two different types of conditions we use a new technique based on constructing a system of real sequences finally this method is applied to the study of hammerstein s integral equations

numerical verification method for positive solutions of elliptic problems abstract the purpose of this paper is to propose methods for verifying the positivity of a weak solution u of an elliptic problem assuming h 0 1 error estimation u u h 0 1 given some numerical approximation u and an explicit error bound we provide a sufficient condition for the solution to be positive and analyze the range of application of our method for elliptic problems with polynomial nonlinearities we present numerical examples where our method is applied to some important problems

numerical verifications of solutions for elliptic equations in nonconvex polygonal domains in this paper methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied in order to verify solutions using computer it is necessary to determine some constants which appear in a priori error estimations we propose some methods for determination of these constants in numerical examples calculating these constants for anl shaped domain we verify the solution of a nonlinear elliptic equation

temporal decays in l 1 and l for the stokes flow we estimate the time decay rates in l1 in the hardy space and in l of solutions for the stokes equations on the half spaces for the estimates in the hardy space we adopt the ideas in math z 231 1999 383 temporal and spatial decays for the stokes flows submitted for publication and also use the heat kernel and the solution formula for the stokes equations

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

sharp numerical inclusion of the best constant for embedding h 0 1 ź ź l p ź on bounded convex domain in this paper we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding h 0 1 l p on a bounded convex domain in r 2 we estimate the best constant by computing the corresponding extremal function using a verified numerical computation verified numerical inclusions of the best constant on a square domain are presented

algebraic decay of weak solutions to 3d navier stokes equations in general unbounded domains abstract in this paper we study the decay properties of a weak solution to the nonstationary navier stokes equations by means of the spectral decomposition method of fractional powers of the stokes operator in an arbitrary unbounded domain the explicit l 2 decay rates are given as t which improves greatly the decay results in 8 9

weighted decay properties for the incompressible stokes flow and navier stokes equations in a half space abstract using the stokes solution formula and l q l r estimates of the stokes operator semigroup we establish the weighted decay properties for the stokes flow and navier stokes equations including their spatial derivatives in half spaces in addition the unboundedness of the projection operator p l r n l r n is overcome by employing a decomposition for the nonlinear term and l asymptotic behavior for the second derivatives of navier stokes flows in half spaces is given

optimal weighted estimates of the flows in exterior domains abstract in this paper we prove some decay properties of global solutions for the navier stokes equations in an exterior domain r n n 2 3 when a domain has a boundary the pressure term is troublesome since we do not have enough information on the pressure near the boundary to overcome this difficulty by multiplying a special form of test functions we obtain an integral equation he xin 2000 12 first introduced this method and then bae jin 2006 2007 1 13 modified their method to obtain better decay rates also bae roh 2009 11 improved bae jin s results unfortunately their results were not optimal because there exists an unpleasant positive small in their rates in this paper we obtain the following optimal rate without x u t l p c t n 2 1 r 1 p 2 for the initial condition u 0 with x u 0 l r and for any n

a new convergence theorem for the jarratt method in banach space abstract in this study we approximate a locally unique solution of a nonlinear equation in banach space using the jarratt method sufficient convergence conditions for this method have already been given by several authors when the equation is defined on the real line or complex plane 1 3 or in banach space 1 4 7 if a certain newton kantorovich type hypothesis is satisfied then the jarratt method converges to a solution of the equation with order four the verification of some of the earlier hypotheses is too difficult or too expensive here using lipschitz conditions on the second frechet derivative of the operator involved we provide a convergence theorem for the jarratt method which uses conditions that are very easy to check see the example and remark 4 finally a numerical example is provided to show that our results apply to solve a nonlinear equation where others fail

analyticity and asymptotics for the stokes solutions in a weighted space abstract we first show the analyticity of stokes operator in a weighted space l p r 3 we also show that the hodge decomposition holds on lp r 3 then we estimate the asymptotic behavior of the stokes solutions in space and time directions for 1 r 3 u x t q x q dx 1 q c t 3 2 1 r 1 q 1 2 2 where is a constant with 0

the jarratt method in banach space setting abstract in this note we extend the jarratt method of order four into banach spaces we also establish a kantorovich type convergence theorem see the references and give an explicit expression for the error bound of the method jarratt 1966 1969

recurrence relations for the super halley method in this paper we give sufficient conditions in order to assure the convergence of the super halley method in banach spaces we use a system of recurrence relations analogous to those given in the classical newton kantorovich theorem or those given for chebyshev and halley methods by different authors 1998 elsevier science ltd all rights reserved

guaranteed a posteriori bounds for eigenvalues and eigenvectors multiplicities and clusters this paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second order self adjoint elliptic linear operators with compact resolvent given a cluster of eigenvalues we estimate the error in the sum of the eigenvalues as well as the error in the eigenvectors represented through the density matrix i e the orthogonal projector on the associated eigenspace this allows us to deal with degenerate multiple eigenvalues within the framework all the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues we show how this assumption can be numerically checked our bounds are guaranteed and converge with the same speed as the exact errors they can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available which is presented in two particular cases the laplace eigenvalue problem discretized with conforming finite elements and a schrodinger operator with periodic boundary conditions of the form v discretized with planewaves for these two cases numerical illustrations are provided on a set of test problems

verified eigenvalue evaluation for the laplacian over polygonal domains of arbitrary shape the finite element method fem is applied to bound leading eigenvalues of the laplace operator over polygonal domains compared with classical numerical methods most of which can only give concrete eigenvalue bounds over special domains of symmetry our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape even when the eigenfunction has a singularity the problem of eigenvalue estimation is solved in two steps first we construct a computable a priori error estimation for the fem solution of poisson s problem which holds even for nonconvex domains with reentrant corners second new computable lower bounds are developed for the eigenvalues because the interval arithmetic is implemented throughout the computation the desired eigenvalue bounds are expected to be mathematically correct we illustrate several computation examples such as the cases of an l shaped domain and a crack domain to demonstrate the efficiency and flexibility of the proposed method

a boundary integral algorithm for the laplace dirichlet neumann mixed eigenvalue problem we present a novel integral equation algorithm for evaluation of zaremba eigenvalues and eigenfunctions that is eigenvalues and eigenfunctions of the laplace operator with mixed dirichlet neumann boundary conditions of course slight modifications of our algorithms are also applicable to the pure dirichlet and neumann eigenproblems expressing the eigenfunctions by means of an ansatz based on the single layer boundary operator the zaremba eigenproblem is transformed into a nonlinear equation for the eigenvalue for smooth domains the singular structure at dirichlet neumann junctions is incorporated as part of our corresponding numerical algorithm which otherwise relies on use of the cosine change of variables trigonometric polynomials and to avoid the gibbs phenomenon that would arise from the solution singularities the fourier continuation method fc the resulting numerical algorithm converges with high order accuracy without recourse to use of meshes finer than those resulting from the cosine transformation for non smooth lipschitz domains in turn an alternative algorithm is presented which achieves high order accuracy on the basis of graded meshes in either case smooth or lipschitz boundary eigenvalues are evaluated by searching for zero minimal singular values of a suitably stabilized discrete version of the single layer operator mentioned above the stabilization technique is used to enable robust non local zero searches the resulting methods which are fast and highly accurate for high and low frequencies alike can solve extremely challenging two dimensional dirichlet neumann and zaremba eigenproblems with high accuracies in short computing times enabling in particular evaluation of thousands of eigenvalues and corresponding eigenfunctions for a given smooth or non smooth geometry with nearly full double precision accuracy

long time behavior for the nonstationary navier stokes flows in l1 r n the large time decay for the navier stokes flows in l1 r n is a long standing unsolved question the main difficulties are that usual lq lr estimates for the stokes flow fail in this case and the projection operator p l1 r n l 1 r n becomes unbounded using the stokes solution formula we find a crucial and new estimate for the stokes flow in l1 r n which plays a fundamental role in studying the time l1 behavior for the navier stokes equations in addition we decompose the operator p into two parts and reduce its unboundedness to establish an l1 estimate for an elliptic problem with neumann boundary condition which is overcome by using the weighted estimates of the gaussian kernel s convolution the main results in this article are motivated by bae s works 2006 2008 1 2 and brandolese s work 2001 11

decay results of the nonstationary navier stokes flows in half spaces there is a long standing problem on the nonstationary navier stokes equations which pertains to how to characterize the l1 time asymptotic expansion of the navier stokes flows in the half space beyond a few partial results new progress has not yet to be made on this open question in this article we give a confirmed answer to this problem namely a thorough characterization on l1 summability is revealed in order to prove this result we need to avoid the unboundedness of the projection operator which is overcome by treating an elliptic neumann problem finally using the weighted estimates on the heat kernel s convolution we obtain the exact profile structure of the asymptotic expansion in l 1 mathbb r n in addition some crucial estimates on the fractional spatial derivatives of the non stationary stokes and navier stokes flows are established for the first time which allows for a better understanding of the l1 decay problem

flux reconstructions in the lehmann goerisch method for lower bounds on eigenvalues abstract the standard application of the lehmann goerisch method for lower bounds on eigenvalues of symmetric elliptic second order partial differential operators relies on determination of fluxes i that approximate co gradients of exact eigenfunctions scaled by corresponding eigenvalues fluxes i are usually computed by solving a global saddle point problem with mixed finite element methods in this paper we propose a simpler global problem that yields fluxes i of the same quality the simplified problem is smaller it is positive definite and any h div conforming finite elements such as raviart thomas elements can be used for its solution in addition these global problems can be split into a number of independent local problems on patches which allows for trivial parallelization the computational performance of these approaches is illustrated by numerical examples for laplace and steklov type eigenvalue problems these examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions

explicit eigenvalue bounds of differential operators defined by symmetric positive semi definite bilinear forms abstract recently the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues in this paper the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi definite bilinear forms as an application the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections

upper and lower bounds of temporal and spatial decays for the navier stokes equations we obtain the lower bounds of the temporal spatial decays for weak solutions of the navier stokes equations c0 1 t 5 2i4 1 2 i 2u t l2 c1 1 t 5 2i4 for i 1 2 the upper bound parts are estimated in several papers for example bae and jin submitted and he and xin proc roy soc edinburg section a 131 3 2001 597 by the interpolation arguments we also have c0 1 t 54 2 1 x 2 2u t l2 c1 1 t 54 2 for 0 2

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

space time decay of navier stokes flows invariant under rotations we show that the solutions to the non stationary navier stokes equations in open image in new window d 2 3 which are left invariant under the action of discrete subgroups of the orthogonal group o d decay much faster as open image in new window than in generic case and we compute for each subgroup the precise decay rates in space time of the velocity field

an optimal adaptive fem for eigenvalue clusters the analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods a first analysis for multiple eigenvalues of the recent work dai et al arxiv preprint 1210 1846v2 2013 introduces an adaptive method whose marking strategy is based on the element wise sum of local error estimator contributions for multiple eigenvalues this paper proves the optimality of a practical adaptive algorithm based on a lowest order conforming finite element method for eigenvalue clusters for the eigenvalues of the laplace operator in terms of nonlinear approximation classes all estimates are explicit in the initial mesh size the eigenvalues and the cluster width to clarify the dependence of the involved constants

large time behavior for the nonstationary navier stokes flows in the half space abstract asymptotic behavior of higher order spatial derivatives in l r is established for incompressible navier stokes flows in the half space which is a long time unsolved problem the main tools employed for solving this problem are the nonstationary stokes solution formula and the a priori estimates of the steady stokes system in the half space another main result is devoted to studying time l 1 decay and a partial answer to this open problem is given by means of a crucial and new estimate for the stokes flow two theorem 1 1 theorem 1 2 in this article can be regarded as a great improvement and powerful extension of the work in 6 38

lower bounds of rates of decay for solutions to the navier stokes equations where a p 3 2 p 1 and the constant c depends on the l2 and lnorms of the initial data u0 this paper deals with the more subtle problem of deriving lower bounds on the energy decay rates we show that for a certain class of initial data the solutions u x t to the 2d and 3d navier stokes equations admit an algebraic lower bound on the energy decay specifically there are two cases to consider in the first case the average of the initial data f u0 dx is nonzero this case was treated in the earlier paper 5 where it was established that

estimation of interpolation error constants for the p0 and p1 triangular finite elements abstract we give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles for the piecewise linear one we mainly analyze the conforming case but some results are also given for the non conforming case we obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles in particular we explicitly determine the babuska aziz constant which plays an essential role in the interpolation error estimation of the linear triangular finite element the equation for determination is the transcendental equation tan 0 so that the solution can be numerically obtained with desired accuracy and verification such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions

global ln solution and its decay property for the navier stokes equations in half space r n soit r n x 1 x 2 x n 1 x n x n 0 n 2 dans q r n 0 on considere les equations de navier stokes u t u u u 0 dans q divu 0 dans q u x u n 0 0 u 0 a on demontre l existence et l unicite d une l n solution globale forte pour des donnees initiales petites

solutions of the 2d quasi geostrophic equation in holder spaces abstract the 2d quasi geostrophic equation t u 0 u r is a two dimensional model of the 3d hydrodynamics equations when 1 2 the issue of existence and uniqueness concerning this equation becomes difficult it is shown here that this equation with either 0 or 0 and 0 1 2 has a unique local in time solution corresponding to any initial datum in the space c r l q for r 1 and q 1