improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

coupling stokes darcy flow with transport a mathematical and numerical model describing chemical transport in a stokes darcy flow system is discussed the flow equations are solved through domain decomposition using classical finite element methods in the stokes region and mixed finite element methods in the darcy region the local discontinuous galerkin ldg method is used to solve the transport equation models dealing with coupling between stokes and darcy equations have been extensively discussed in the literature this paper focuses on the approximation of the transport equation stability of the ldg scheme is analyzed and an a priori error estimate is proved several numerical examples verifying the theory and illustrating the capabilities of the method are presented

discontinuous finite volume methods for the stationary stokes darcy problem summary in this paper we analyze discontinuous finite volume methods for the stationary stokes darcy problem that models coupled fluid flow and porous media flow the discontinuous finite volume methods are combinations of finite volume method and discontinuous galerkin method with three interior penalty types incomplete symmetric nonsymmetric and symmetric briefly using discontinuous functions as trial functions in the finite volume method optimal error estimates in broken h1 norm are obtained for the three discontinuous finite volume methods optimal error estimates in the standard l2 norm are derived for the symmetric interior penalty discontinuous finite volume method numerical experiments are presented to confirm the theoretical results with non matching meshes across the common interface of stokes region and darcy region copyright 2015 john wiley sons ltd

the nonconforming virtual element method for the darcy stokes problem abstract we present the nonconforming virtual element method for the darcy stokes problem by using a gradient projection operator and a special polynomial subspace orthogonal to the gradient of some polynomial space we construct a h 1 nonconforming but h div conforming virtual element that allows us to compute the l 2 projection the optimal convergence is proved under the assumption of sufficient regularity and the uniform convergence is also obtained for the lowest order case besides we establish a discrete exact sequence of de rham complex related to the nonconforming virtual element finally we carry out some numerical tests to make a comparison of the convergence between different nonconforming virtual elements and confirm the optimal and uniform convergence of the nonconforming virtual element

global strong solution for the korteweg system in dimension n 2 this work is devoted to prove the existence of global strong solution in dimension n 2 for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see 16 which can be used as a phase transition model in a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span t of existence of the solution this one depends on the norm of the initial data 0 v 0 the second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient 2 and on the viscosity tensor which corresponds to the shallow water case 2 div du to do this we derivate different energy estimate on the density and the effective velocity v which ensures that the strong solution can be extended beyond t the main difficulty consists in controlling the vacuum or in other words to estimate the l norm of 1 the proof relies mostly on a method introduced by de giorgi 14 see also ladyzhenskaya et al in 30 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

on new exponential quadratically convergent iterative formulae a class of exponential iterative formulae having quadratic convergence is discussed in this paper the theoretical analysis and numerical experiments show that new exponential iterative formulae are effective and comparable to well known method of newton

global solutions of the super critical 2d quasi geostrophic equation in besov spaces abstract in this paper we study the super critical 2d dissipative quasi geostrophic equation we obtain some regularization effects allowing us to prove a global well posedness result for small initial data lying in critical besov spaces constructed over lebesgue spaces l p with p 1 local results for arbitrary initial data are also given

coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

strong solution for korteweg system in bmo 1 rn with initial density in l in this paper we investigate the question of the existence of strong solution in finite time for the korteweg system for small initial data provided that the initial momentum 0 u 0 belongs to bmo 1 t r n for t 0 and the initial density 0 is in l r n with n 1 and far away from the vacuum this result extends the so called koch tataru theorem for the korteweg system it is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes lipschitz for any t with t 0 we also prove the existence of global strong solution for initial data 0 1 0 u 0 b n 2 1 2 r n b n 2 2 r n l r n b n 2 1 2 r n n this result allows in particular to extend the notion of oseen solution corresponding to particular solution of the incompressible navier stokes system in dimension n 2 to the korteweg system provided that the vorticity of the momentum 0 u 0 is a dirac mass 0 with sufficiently small ihowever unlike the navier stokes equations the property of self similarity is not conserved for the korteweg system since there is no invariance by scaling because the term of pressure

convergence analysis of the mimetic finite difference method for elliptic problems we propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes the a priori error analysis relies on the connection between the mimetic formulation and the lowest order raviart thomas mixed finite element method the theoretical results are confirmed by numerical experiments

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

a domain decomposition method for the steady state navier stokes darcy model with beavers joseph interface condition this paper proposes and analyzes a robin type multiphysics domain decomposition method ddm for the steady state navier stokes darcy model with three interface conditions in addition to the two regular interface conditions for the mass conservation and the force balance the beavers joseph condition is used as the interface condition in the tangential direction the major mathematical difficulty in adopting the beavers joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system y cao et al comm math sci 8 2010 pp 1 25 y cao et al siam j numer anal 47 2010 pp 4239 4256 in this paper the well posedness of the navier stokes darcy model with beavers joseph condition is analyzed by using a branch of nonsingular solutions by following the idea in y cao et al numer math 117 2011 pp 601 629 the three physical interface conditions are utilized together to construct the robin type boundary conditions on th

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

the nonrelativistic limit in radiation hydrodynamics i weak entropy solutions for a model problem abstract this paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions the system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integral type source term to a family of radiation transport equations we first show existence of entropy solutions to cauchy problems of the model system in the framework of functions of bounded variation this is done by using difference schemes and discrete ordinates then we establish strong convergence of the entropy solutions indexed with c as c goes to infinity the limit function satisfies a scalar integro differential equation

a hybrid discontinuous galerkin method for elliptic problems a new family of hybrid discontinuous galerkin methods is studied for second order elliptic equations our proposed method is a generalization of the cell boundary element cbe method y jeon and e j park appl numer math 58 2008 pp 800 814 which allows high order polynomial approximations our method can be viewed as a hybridizable discontinuous galerkin method b cockburn j gopalakrishnan and r lazarov siam j numer anal 47 2009 pp 1319 1365 using a bauman oden type local solver the method conserves the mass in each element and the average flux is continuous across the interelement boundary for even degree polynomial approximations optimal order error estimates measured in the energy norm are proved numerical examples are presented to show the performance of the method

a parallel robin robin domain decomposition method for the stokes darcy system we propose a new parallel robin robin domain decomposition method for the coupled stokes darcy system with beavers joseph saffman jones interface boundary condition in particular we prove that with an appropriate choice of parameters the scheme converges geometrically independent of the mesh size

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

modifications of higher order convergence for solving nonlinear equations abstract in liang fang guoping he some modifications of newton s method with higher order convergence for solving nonlinear equations j comput appl math 228 2009 296 303 the authors pointed out that the iteration constructed in y m ham c b chun and s g lee some higher order modifications of newton s method for solving nonlinear equations j comput appl math 222 2008 477 486 failed when p 2 they gave some counterexamples and obtained a modified result however they did not show the essential reason which leads to the incorrect result in this paper we shall show that reason and present more general results than the above mentioned papers

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

global solutions of a high dimensional system for korteweg materials the global existence of solutions to a high dimensional system of korteweg materials is established when the initial data are small the complete model was proposed by dunn and serrin and the local existence of the solution was obtained in an earlier paper by the authors

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

global existence in critical spaces for flows of compressible viscous and heat conductive gases we are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat conductive gases the initial data are supposed to be close to a stable equilibrium with constant density and temperature using uniform estimates for the linearized system with a convection term we get global well posedness in a functional setting invariant with respect to the scaling of the associated equations in space dimension n 3 we also show a smoothing effect on the velocity and the temperature and a decay on the difference between the density and the constant reference state these results extend a previous paper devoted to the barotropic case see 5

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

improving the order of convergence of iteration functions abstract improving methods for the order of convergence of iteration functions are give using these methods new third or fourth order root finding methods for a single equation with a multiple root are derived numerical examples are given

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

relative entropy in multi phase models of 1d elastodynamics convergence of a non local to a local model abstract in this paper we study a local and a non local regularization of the system of nonlinear elastodynamics with a non convex energy we show that solutions of the non local model converge to those of the local model in a certain regime the arguments are based on the relative entropy framework and provide an example of how local and non local regularizations may compensate for non convexity of the energy and enable the use of the relative entropy stability theory even if the energy is not quasi or poly convex

some third order families of iterative methods for solving nonlinear equations abstract in this paper we present a convenient third order iteration formula improving newton s method for solving nonlinear equations with any two iteration functions of order two the formula provides one parameter third order family of methods each family member is free from second derivatives and per iteration it requires three evaluations of the given function and its derivative several numerical examples are given to illustrate the efficiency and performance of some of the presented methods

robin robin domain decomposition methods for the stokes darcy coupling in this paper we consider a coupled system made of the stokes and darcy equations and we propose some iteration by subdomain methods based on robin conditions on the interface we prove the convergence of these algorithms and for suitable finite element approximations we show that the rate of convergence is independent of the mesh size h special attention is paid to the optimization of the performance of the methods when both the kinematic viscosity nu of the fluid and the hydraulic conductivity tensor k of the porous medium are very small

the derivation of hybridizable discontinuous galerkin methods for stokes flow in this paper we introduce a new class of discontinuous galerkin methods for the stokes equations the main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries we present four ways of hybridizing the methods which differ by the choice of the globally coupled unknowns classical methods for the stokes equations can be thought of as limiting cases of these new methods

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

a least squares virtual element method for second order elliptic problems abstract in this paper a least squares virtual element method is presented for approximating the vector and scalar variables of second order elliptic problems the h div conforming and scalar conforming virtual elements are used to approximate the vector and scalar variables respectively the method allows the use of very general polygonal meshes and leads to a symmetric positive definite system the optimal a priori error estimates are established for the vector variable in h div norm and the scalar variable in h 1 norm a simple a posteriori error estimator is also presented and proved to be reliable and efficient the virtual element method handles the hanging nodes naturally thus the local mesh post processing to remove hanging nodes is not required numerical experiments are conducted to verify the accuracy of the method and show the effectiveness and flexibility of the adaptive strategy driven by the proposed estimator and suitable mesh refinement strategy

construction of newton like iteration methods for solving nonlinear equations in this paper we present a simple and yet powerful and easily applicable scheme in constructing the newton like iteration formulae for the computation of the solutions of nonlinear equations the new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations it provides a tool to develop new newton like iteration methods or to improve the existing iteration methods which contains the well known newton iteration formula in logic those all improve the newton method the orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of maple some numerical tests are given to support the theory developed in this paper

on a lagrangian method for the convergence from a non local to a local korteweg capillary fluid model in the present article we are interested in further investigations for the barotropic compressible navier stokes system endowed with a non local capillarity we studied in 7 thanks to an accurate study of the associated linear system using a lagrangian change of coordinates we provide more precise energy estimates in terms of hybrid besov spaces naturally depending on a threshold frequency which is determined in function of the physical parameter distinguishing the low and the high regimes it allows us in particular to prove the convergence of the solutions from the non local to the local korteweg system another mathematical interest of this article is the study of the effect of the lagrangian change on the non local capillary term

h text div h div and h mathbf curl h curl conforming virtual element methods in the present paper we construct virtual element spaces that are h mathrm div h div conforming and h mathbf curl h curl conforming on general polygonal and polyhedral elements these spaces can be interpreted as a generalization of well known finite elements we moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations finally we discuss the construction of exact sequences of vem spaces

on the solution of the coupled navier stokes and darcy equations abstract this paper introduces and analyzes two mathematical models for coupling the incompressible navier stokes equations with the porous media flow equations a numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium is proposed existence and uniqueness results under small data condition of the numerical solution are proved optimal a priori error estimates are derived numerical examples comparing the two models under varying physical parameters are provided

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

travelling waves for the gross pitaevskii equation ii the purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the gross pitaevskii equation in dimensions two and three our arguments based on minimization under constraints yield a full branch of solutions and extend earlier results see 3 4 8 where only a part of the branch was built in dimension three we also show that there are no travelling wave solutions of small energy

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a virtual element method for the acoustic vibration problem we analyze in this paper a virtual element approximation for the acoustic vibration problem we consider a variational formulation relying only on the fluid displacement and propose a discretization by means of mathrm h mathrm div h div virtual elements with vanishing rotor under standard assumptions on the meshes we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates with this end we prove approximation properties of the proposed virtual elements we also report some numerical tests supporting our theoretical results

a divergence free weak virtual element method for the stokes darcy problem on general meshes abstract this paper presents a weak virtual element method on general meshes for the stokes darcy problem with the beavers joseph saffman interface condition the velocity is discretized by the h div virtual element the pressure is approximated by discontinuous piecewise polynomials besides a polynomial space on the element faces is introduced to approximate the tangential trace of the velocity in the stokes equations the velocity on the discrete level is exactly divergence free and thus the exact mass conservation is preserved in the discretization the well posedness of the discrete problem is proved and an a priori error estimate is derived that implies the error for the velocity in a suitable norm does not depend on the pressure a series of numerical experiments are reported to illustrate the performance of the method

fourth order variants of cauchy s method for solving non linear equations abstract in this paper we present some new variants of cauchy s method in which the second derivative is replaced by a finite difference between the first derivatives analysis of convergence shows that the new methods have fourth order convergence per iteration the new methods require one evaluations of the function and two of its first derivative numerical results show that the new methods are efficient

local in time results for local and non local capillary navier stokes systems with large data abstract in this article we study three capillary compressible models the classical local navier stokes korteweg system and two non local models for large initial data bounded away from zero and with a reference pressure state which is not necessarily stable p can be non positive we prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non local models towards the local korteweg model the results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients

existence of global strong solution for korteweg system with large infinite energy initial data this work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by j e dunn and j serrin 1985 see cite fds which can be used as a phase transition model we will prove the existence of local and global under a condition of smallness on the initial data strong solutions with discontinuous initial density where ln rho 0 belongs to the besov space b n 2 infty r n in passing our result allow to consider initial data with discontinuous interfaces up our knowledge in the literature the results of existence of strong solutions require generally continuous initial density it implies also that we can work with initial data space of infinite energy our result relies on the fact that the density can be written as the sum of the solution rho l associated to linear system and a remainder density bar rho which is more regular than rho l by taking into account the regularizing effects induced on the bilinear convection term the main difficulty concerns the proof of new estimate of maximum principle type for the linear system associated to the korteweg system the proof is based on a characterization of the besov space in terms of the semi group associated to this linear system let also mention that we prove the existence of global strong solution with a smallness hypothesis which is subcritical in terms of the scaling of the equations it allows us to exhibit a family of large energy initial data for the scaling of the equations providing global strong solution in particular for the first time up our knowledge we show the existence of global strong solution for some large energy initial data when n 2 we finish this paper by introducing the notion of quasi solutions for the korteweg s system a tool which has been developed in the framework of the compressible navier stokes equations cite arxiv arxiv1 hal cras1 cras2 which enables us to improve the previous result and by obtaining the existence of global strong solution with large initial velocity in b n 1 2 infty in particular when n 2 it implies the existence of global strong solution with large energy data as a corollary we get global existence and uniqueness for highly compressiblekorteweg system when n geq2 it means that for any large initial data under an irrotational condition on the initial velocity we have the existence of global strong solution provided that the pressure is sufficiently highly compressible

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

a new continuation newton like method and its deformation this paper presents a family of new continuation newton like methods and its deformations for computing approximate solutions of the non linear algebraic equation f x 0 from a practical point of view the new methods are vast superior

analysis of fully mixed finite element methods for the stokes darcy coupled problem in this paper we analyze fully mixed finite element methods for the coupling of fluid flow with porous media flow flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the fully mixed concept employed here refers to the fact that we consider dual mixed formulations in both the stokes domain and the darcy region which means that the main unknowns are given by the pseudostress and the velocity in the fluid together with the velocity and the pressure in the porous medium in addition the transmission conditions become essential which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated lagrange multipliers we apply the fredholm and babuska brezzi theories to derive sufficient conditions for the unique solvability of the resulting continuous formulation since the equations and unknowns can be ordered in several different ways we choose the one yielding a doubly mixed structure for which the inf sup conditions of the off diagonal bilinear forms follow straightforwardly next adapting to the discrete case the arguments of the continuous analysis we are able to establish suitable hypotheses on the finite element subspaces ensuring that the associated galerkin scheme becomes well posed in addition we show that the existence of uniformly bounded discrete liftings of the normal traces simplifies the derivation of the corresponding stability estimates a feasible choice of subspaces is given by raviart thomas elements of lowest order and piecewise constants for the velocities and pressures respectively in both domains together with continuous piecewise linear elements for the lagrange multipliers this example confirms that with the present approach the stokes and darcy flows can be approximated with the same family of finite element subspaces without adding any stabilization term finally several numerical results illustrating the good performance of the method with these discrete spaces and confirming the theoretical rate of convergence are provided

robin robin domain decomposition methods for the steady state stokes darcy system with the beavers joseph interface condition domain decomposition methods for solving the coupled stokes darcy system with the beavers joseph interface condition are proposed and analyzed robin boundary conditions are used to decouple the stokes and darcy parts of the system then parallel and serial domain decomposition methods are constructed based on the two decoupled sub problems convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence

a simply constructed third order modifications of newton s method in this paper we present a simple and easily applicable approach to construct some third order modifications of newton s method for solving nonlinear equations it is shown by way of illustration that existing third order methods can be employed to construct new third order iterative methods the proposed approach is applied to the classical chebyshev halley methods to derive their second derivative free variants numerical examples are given to support that the methods thus obtained can compete with known third order methods

a strongly conservative finite element method for the coupling of stokes and darcy flow we consider a model of coupled free and porous media flow governed by stokes and darcy equations with the beavers joseph saffman interface condition this model is discretized using divergence conforming finite elements for the velocities in the whole domain discontinuous galerkin techniques and mixed methods are used in the stokes and darcy subdomains respectively this discretization is strongly conservative in h d i v w and we show convergence numerical results validate our findings and indicate optimal convergence orders

global strong solution for the korteweg system with quantum pressure in dimension n ge 2 this work is devoted to prove the existence of global strong solution in dimension n ge 2 for a isothermal model of capillary fluids derived by dunn and serrin see arch ration mech anal 88 2 95 133 1985 which can be used as a phase transition model we will restrict us to the case of the so called compressible navier stokes system with quantum pressure which corresponds to the korteweg system with capillary coefficient such that kappa rho frac kappa 1 rho with kappa 1 0 and rho the density in a first part we prove the existence of strong solution in finite time for large initial data in critical besov spaces with precise estimate on the life span t the second part consists in proving the existence of global strong solution with large initial data for a specific choice on the capillary coefficient kappa 1 mu 2 with mu the viscosity coefficient to do this we derive different energy estimates on the density and the effective velocity v which allow us to extend the strong solution beyond t the main difficulty consists in estimating the l infty norm of frac 1 rho the proof relies mostly on a method introduced by de giorgi mem accad sci torino cl sci fis mat nat 3 3 25 43 1957 see also ladyzhenskaya et al linear and quasilinear equations of parabolic type ams translations providence 1968 for the parabolic case to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

discontinuous galerkin and mimetic finite difference methods for coupled stokes darcy flows on polygonal and polyhedral grids we study locally mass conservative approximations of coupled darcy and stokes flows on polygonal and polyhedral meshes the discontinuous galerkin dg finite element method is used in the stokes region and the mimetic finite difference method is used in the darcy region dg finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces optimal convergence estimates for the numerical scheme are derived results from computational experiments supporting the theory are presented

on a class of quadratically convergent iteration formulae in this paper a new class of iterative formulae having quadratic convergence is presented furthermore these algorithms are comparable to the well known method of newton and the computed results support this theory

sixteenth order method for nonlinear equations modification of newton s method with higher order convergence is presented the modification of newton s method is based on king s fourth order method the new method requires three step per iteration analysis of convergence demonstrates that the order of convergence is 16 some numerical examples illustrate that the algorithm is more efficient and performs better than classical newton s method and other methods

on the global well posedness of the euler boussinesq system with fractional dissipation abstract we study the global well posedness of the euler boussinesq system with the term dissipation d on the temperature equation we prove that for 1 the coupled system has a global unique solution for initial data with critical regularities