a continuous framework for feti dp with a mesh independent condition number for the dual problem abstract we present an abstract setting for a variation of the feti dp method applied to second order elliptic problems in this setting the matrix iteration for the dual problem has a mesh independent condition number this work improves the ideas recently introduced by bernardi et al c bernardi t chacon rebollo e chacon vera a feti method with a mesh independent condition number for the iteration matrix comput methods appl mech engrg 197 13 16 2008 1410 1429 the continuity restrictions for the displacement field across interfaces are weakly imposed using the natural trace norm instead of pointwise continuity although a slight coupling between subdomains is introduced still a parallel computation is possible some numerical tests are also performed to illustrate the method

locally discontinuous but globally continuous galerkin methods for elliptic problems abstract we propose and analyze a stabilized hybrid finite element method for elliptic problems consisting of locally discontinuous galerkin problems in the primal variable coupled to a globally continuous problem in the multiplier numerical analysis shows that the proposed formulation preserves the main properties of the associate dg method such as consistency stability boundedness and optimal rates of convergence in the energy norm and in the l 2 norm for adjoint consistent formulations for using an element based data structure it has basically the same complexity and computational cost of classical conforming finite element methods convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here with accuracy equivalent or even better than the corresponding dg approximations

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

a high order elliptic pde based level set reinitialisation method using a discontinuous galerkin discretisation abstract in this paper an efficient high order accurate level set reinitialisation method is proposed based on the elliptic reinitialisation method basting and kuzmin 2013 1 which is discretised spatially using the discontinuous galerkin dg symmetric interior penalty method sipg in order to achieve this a number of improvements have been made to the elliptic reinitialisation method including reformulation of the underlying minimisation problem driving the solution adoption of a lagrange multiplier approach for enforcing a dirichlet boundary condition on the implicit level set interface and adoption of a narrow band approach numerical examples confirm the high order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the sipg method that is h p 1 and h p in the l 2 and dg norms respectively furthermore the degree to which the level set function satisfies the eikonal equation improves proportionally to h p and the often ignored homogeneous dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h 2 for all polynomial orders

a meshless galerkin method for dirichlet problems using radial basis functions in this paper a numerical method is given for partial differential equations which combines the use of lagrange multipliers with radial basis functions it is a new method to deal with difficulties that arise in the galerkin radial basis function approximation applied to dirichlet also mixed boundary value problems convergence analysis results are given several examples show the efficiency of the method using tps or sobolev splines

trudinger moser inequality with remainder terms abstract the paper gives the following improvement of the trudinger moser inequality 0 1 sup u 2 d x u 1 u c 0 e 4 u 2 d x r 2 related to the hardy sobolev mazya inequality in higher dimensions we show 0 1 with u v x u 2 d x for a class of v 0 that includes v r 1 4 r 2 log 1 r 2 max log 1 r 1 which refines two previously known cases of 0 1 proved by adimurthi and druet 2 and by wang and ye 23 in addition we verify 0 1 for u u p 2 as well as give an analogous improvement for the onofri beckner inequality for the unit disk beckner 6

improved moser trudinger type inequalities in the hyperbolic space hn we establish an improved version of the moser trudinger inequality in the hyperbolic space hnhn n 2n 2 namely we prove the following result for any 0 n 1nn then we have supu c0 hn hn gu gndvolg hn u ndvolg 1 hn n n u nn 1 dvolg where n n n 11n 1 n 1 n 1 denotes the surface area of the unit sphere in rnrn and n t et j 0n 2tjj this improves the moser trudinger inequality in hyperbolic spaces obtained recently by mancini and sandeep 2010 by mancini et al 2013 and by adimurthi and tintarev 2010 for 0 0 in the limiting case n 1n n we prove a moser trudinger inequality with exact growth in hnhn supu c0 hn hn gu gndvolg n 1n n hn u ndvolg 11 hn u ndvolg hn n n u nn 1 1 u nn 1dvolg this improves the moser trudinger inequality with exact growth in hnhn established by lu and tang 2016 these inequalities are achieved from the comparison of the symmetric non increasing rearrangement of a function both in the hyperbolic and the euclidean spaces and the same inequalities in the euclidean space this approach seems to be new comparing with the previous ones

the hardy rellich inequality for polyharmonic operators the hardy rellich inequality given here generalizes a hardy inequality of davies from the case of the dirichlet laplacian of a region n to that of the higher order polyharmonic operators with dirichlet boundary conditions the inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents

boundary lagrange multipliers in finite element methods error analysis in natural norms a finite element procedure for circumventing the babuska brezzi condition in mixed formulations with lagrange multipliers defined on the boundary is presented residual terms constructed from the euler lagrange equations are added to the classical galerkin formulation in order to attain coercivity in a mesh dependent norm convergence is proven for the primal variable and the multiplier in the natural mesh independent norm of the problem generalizing results of a previous paper

on some strong poincare inequalities on riemannian models and their improvements abstract we prove second and fourth order improved poincare type inequalities on the hyperbolic space involving hardy type remainder terms since theirs l h s only involve the radial part of the gradient or of the laplacian they can be seen as stronger versions of the classical poincare inequality we show that such inequalities hold true on model manifolds as well under suitable curvature assumptions and sharpness of some constants is also discussed

a note on the meshless method using radial basis functions in this paper we prove some inverse inequalities for the finite dimensional subspace interpolated by radial basis functions the properties of these inequalities are worse than those in finite element spaces these bad properties result in a huge condition number for both the collocation method and galerkin method using radial basis functions as applications theoretical estimates of the condition numbers of a meshless method using radial basis functions are given

fourier analysis and optimal hardy adams inequalities on hyperbolic spaces of any even dimension abstract recently the second and third authors established the sharp hardy adams inequalities of second order derivatives with the best constants on the hyperbolic space b 4 of dimension four 40 adv math 2017 a key method used in dimension four in 40 is that the following hardy operator when n 4 and k 2 b n k u 2 d x i 1 k 2 i 1 2 b n u 2 1 x 2 2 k d x can be decomposed exactly as the product of fractional laplacians however such a decomposition in dimension four is no longer possible in the case n 4 and 2 k n 2 and the situation here is thus considerably more difficult and complicated therefore it remains open as whether a sharp hardy adams inequality still holds on higher dimensional hyperbolic space b n for n 4 the main purpose of this paper is to use a substantially new method of estimating the hardy operator to establish the sharp hardy adams inequalities on hyperbolic spaces b n for all even dimension n and n 4 as applications of such inequalities we will improve substantially the known adams inequalities on hyperbolic space b n in the literature and also strengthen the classical adams inequality and the hardy inequality on euclidean balls in any even dimension the later inequality can be viewed as the borderline case of the sharp hardy sobolev maz ya inequalities for higher order derivatives in high dimensions obtained recently by the second and third authors 41

wavelet stabilization of the lagrange multiplier method we propose here a stabilization strategy for the lagrange multiplier formulation of dirichlet problems the stabilization is based on the use of equivalent scalar products for the spaces h 1 2 partial omega and h 1 2 partial omega which are realized by means of wavelet functions the resulting stabilized bilinear form is coercive with respect to the natural norm associated to the problem a uniformly coercive approximation of the stabilized bilinear form is constructed for a wide class of approximation spaces for which an optimal error estimate is provided finally a formulation is presented which is obtained by eliminating the multiplier by static condensation this formulation is closely related to the nitsche s method for solving dirichlet boundary value problems

a space time formulation for multiscale phenomena abstract we develop subgrid scale models for a class of nonsymmetric linear evolution operators by applying the variational multiscale method in space time the results generalize those of hughes 14 which were confined to the steady case the subgrid scale models are shown to be a paradigm for bubble function finite element methods and provide a theoretical and practical framework for the development of so called stabilized methods

generalized schwinger duality in bound state theory it is shown that an exactly solvable bound state problem is the generator of a nonterminable sequence of partially exactly solvable problems the reversible passage from exact to partial solvability is realized through a class of admissible nonlinear coordinate transformations of which the parabolic schwinger transformation that relates the coulomb and oscillator problems is a particular case interesting spectral features of a novel set of partially solvable problems that emerge through the present considerations are also pointed out

a feti method with a mesh independent condition number for the iteration matrix abstract we introduce a framework for feti methods using ideas from the decomposition via lagrange multipliers of h 0 1 derived by raviart thomas p a raviart j m thomas primal hybrid finite element methods for second order elliptic equations math comput 31 138 1977 391 413 and complemented with the detailed work on polygonal domains developed by grisvard p grisvard singularities in boundary value problems rech math appl 22 1992 we compute the action of the lagrange multipliers using the natural h 00 1 2 scalar product therefore no consistency error appears our analysis allows to deal with cross points and floating subdomains in a natural manner as a byproduct we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning this result improves the standard asymptotic bound for this condition number given by 1 log h h 2 where h and h are the characteristic subdomain size and element size respectively shown by mandel tezaur in j mandel r tezaur convergence of a substructuring method with lagrange multipliers numer math 73 1996 473 487 numerical results that confirm our theoretical analysis are presented

a new fictitious domain approach inspired by the extended finite element method the purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by moes dolbow and belytschko in internat j numer methods engrg 46 1999 pp 131 150 an optimal method is obtained thanks to an additional stabilization technique some a priori estimates are established and numerical experiments illustrate different aspects of the method the presentation is made on a simple poisson problem with mixed neumann and dirichlet boundary conditions the extension to other problems or boundary conditions is quite straightforward

imposing dirichlet boundary conditions in hierarchical cartesian meshes by means of stabilized lagrange multipliers this is the pre peer reviewed version of the following article tur m albelda j nadal e and rodenas j j 2014 imposing dirichlet boundary conditions in hierarchical cartesian meshes by means of stabilized lagrange multipliers int j numer meth engng 98 399 417 which has been published in final form at http dx doi org 10 1002 nme 4629 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

fem bem coupling for the large body limit in micromagnetics we present and analyze a coupled finite element boundary element method for a model in stationary micromagnetics the finite element part is based on mixed conforming elements for two and three dimensional settings we show well posedness of the discrete problem and present an a priori error analysis for the case of lowest order elements

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

a lagrange multiplier method for the finite element solution of elliptic interface problems using non matching meshes in this paper we propose a lagrange multiplier method for the finite element solution of multi domain elliptic partial differential equations using non matching meshes the interface lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes e g we will consider global polynomials as multiplier the ideas are illustrated using poisson s equation as a model and the proposed method is shown to be stable and optimally convergent numerical experiments demonstrating the theoretical results are also presented

adams inequality on the hyperbolic space in this article we establish the following adams type inequality in the hyperbolic space hn supu cc hn hn pku udvg 1 hn e u2 1 dvg iff 0 n k where 2k n pk is the critical gjms operator in hn and 0 n k is as defined in 1 3 as an application we prove the asymptotic behaviour of the best constants in sobolev inequalities when 2k n and also prove some existence results for the qk curvature type equation in hn

xfem based fictitious domain method for linear elasticity model with crack reduction of computational cost of solutions is a key issue for crack identification or crack propagation problems one of the solutions is to avoid remeshing the domain when the crack position changes or when the crack extends to avoid remeshing we propose a new finite element approach for the numerical simulation of discontinuities of displacements generated by cracks inside elastic media the approach is based on a fictitious domain method originally developed for dirichlet conditions for the poisson problem and for the stokes problem which is adapted to the neumann boundary conditions of crack problems the crack is represented by level set functions numerical tests are made with a mixed formulation to emphasize the accuracy of the method as well as its robustness with respect to the geometry enforced by a stabilization technique in particular an inf sup condition is theoretically proven for the latter a realistic simulation with a uniformly pressurized fracture inside a volcano is given for i

stabilized method of imposing dirichlet boundary conditions using a recovered stress field abstract this paper proposes a new formulation to impose dirichlet boundary conditions on immersed boundary cartesian finite element meshes the method uses a recovered stress field calculated by superconvergent patch recovery to stabilize the lagrange multiplier formulation of the problem the optimal convergence of the method and the convergence of the proposed iterative procedure are demonstrated the proposed method is also suitable for problems with non linear material behavior some numerical examples are included to confirm the theoretical results

a fictitious domain approach for a mixed finite element method solving the two phase stokes problem with surface tension forces abstract in this article we study a mixed finite element formulation for solving the stokes problem with general surface forces that induce a jump of the normal trace of the stress tensor on an interface that splits the domain into two subdomains equality of velocities is assumed at the interface the interface conditions are taken into account with multipliers a suitable lagrangian functional is introduced for deriving a weak formulation of the problem a specificity of this work is the consideration of the interface with a fictitious domain approach the latter is inspired by the xfem approach in the sense that cut off functions are used but it is simpler to implement since no enrichment by singular functions is provided in that context getting convergence for the dual variables defined on the interface is non trivial for that purpose an augmented lagrangian technique stabilizes the convergence of the multipliers which is important because their value would determine the dynamics of the interface in an unsteady framework theoretical analysis is provided where we show that a discrete inf sup condition independent of the mesh size is satisfied for the stabilized formulation this guarantees optimal convergence rates that we observe with numerical tests the capacity of the method is demonstrated with robustness tests and with an unsteady model tested for deformations of the interface that correspond to ellipsoidal shapes in dimension 2

weak imposition of the slip boundary condition on curved boundaries for stokes flow we study the finite element approximation of two methods to weakly impose a slip boundary condition for incompressible fluid flows the lagrange multiplier method and nitsche s method for each method we can distinguish several formulations depending on the values of some real parameters in the case of a spatial domain with a polygonal or polyhedral boundary we prove convergence results of their finite element approximations extending previous results of verfurth 33 and we show numerical results confirming them in the case of a spatial domain with a smooth curved boundary numerical results show that approximations computed on polygonal domains approximating the original domain may not converge to the exact solution depending on the values of the aforementioned parameters and on the finite element discretization these negative results seem to highlight babuska s like paradox due to the approximation of the boundary by polygonal ones in particular they seem to contradict some of verfurth s theoretical convergence results

trudinger moser inequalities on harmonic an groups under lorentz norms abstract the main purpose of this paper is to establish sharp trudinger moser type inequalities on harmonic a n groups s for functions whose gradient is in the lorentz space l n q our results show that even if s is not with strictly negative sectional curvature one can also replace the norm s s u n d v s u n d v by s s u n d v in the trudinger moser type inequalities

a stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics in this work we consider a stabilized lagrange or kuhn tucker multiplier method in order to approximate the unilateral contact model in linear elastostatics the particularity of the method is that no discrete inf sup condition is needed in the convergence analysis we propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method

on the connection between the stabilized lagrange multiplier and nitsche s methods we derive nitsche s method for the domain decomposition through the stabilized lagrange multiplier method taking material parameters carefully into account this derivation naturally introduces parameter weighted average flux and stabilizing terms to nitsche s method we show stability and a priori analyses in the mesh dependent norms for both the stabilized method and nitsche s method and discuss connections between the proposed methods

min max level estimate for a singular quasilinear polyharmonic equation in r2m abstract using the framework first presented by ruf and sani we give a proof of an adams type inequality which can be applied to the functional j u 1 2 r 2 m m u 2 0 m 1 a x u 2 d x r 2 m f x u x d x r 2 m h u d x under two kinds of assumptions on the nonlinearity we estimate the min max level of the functional as an application a multiplicity result for the related singular quasilinear elliptic equation is proved

parameter free weak imposition of dirichlet boundary conditions and coupling of trimmed and non conforming patches we present a parameter free domain sewing approach for low order as well as high order finite elements its final form contains only primal unknowns that is the approach does not introduce additional unknowns at the interface additionally it does not involve problem dependent parameters which require an estimation the presented approach is symmetry preserving that is the resulting discrete form of an elliptic equation will remain symmetric and positive definite it preserves the order of the underlying discretization and we demonstrate high order accuracy for problems of non matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p we also demonstrate how the method may be used to model material interfaces which may be curved and for which the interface does not coincide with the underlying mesh this novel approach is presented in the context of the p version and b spline version of the finite cell method an embedded domain method of high order and compared with more classical methods such as the penalty method or nitsche s method

sharp hardy adams inequalities for bi laplacian on hyperbolic space of dimension four abstract we establish sharp hardy adams inequalities on hyperbolic space b 4 of dimension four namely we will show that for any 0 there exists a constant c 0 such that b 4 e 32 2 u 2 1 32 2 u 2 d v 16 b 4 e 32 2 u 2 1 32 2 u 2 1 x 2 4 d x c for any u c 0 b 4 with b 4 h 9 4 h u u d v 1 as applications we obtain a sharpened adams inequality on hyperbolic space b 4 and an inequality which improves the classical adams inequality and the hardy inequality simultaneously the later inequality is in the spirit of the hardy trudinger moser inequality on a disk in dimension two given by wang and ye in 46 and on any convex planar domain by lu and yang in 33 the fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work

a stabilized lagrange multiplier method for the enriched finite element approximation of tresca contact problems of cracked elastic bodies abstract in this paper we propose a local projection stabilized lagrange multiplier method in order to approximate the two dimensional linear elastostatics unilateral contact problem with tresca friction in the framework of the extended finite element method x fem this last method allows to perform finite element computations on cracked domains by using meshes of the non cracked domain the advantage of the used stabilization technique is to affect only the equation on multipliers and thus to be equation independent we study the existence uniqueness and a priori error estimate of three hybrid discrete formulations