postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

polynomial approximation of certain biharmonic mappings abstract in this paper we first determine conditions on the analytic polynomials p and q such that the family of univalent sense preserving biharmonic polynomials of the form p z 2 p q is dense in the family of univalent sense preserving biharmonic mappings f defined on the unit disk d of the form f z 2 g next we consider the univalency of harmonic function g whenever f z 2 g is univalent in d finally we give a partial answer to the problem raised by muhanna about the radius of univalence of a family of sense preserving biharmonic mappings with the form f z 2 g h where g is univalent in d

finite element method for elliptic problems with edge singularities we consider tangentially regular solution of the dirichlet problem for an homogeneous strongly elliptic operator with constant coecients on an innite vertical polyhedral cylinder based on a bounded polygonal domain in the horizontal xy plane the usual complex blocks of singularities in the non tensor product singular decomposition of the solution are made more explicit by a suitable choice of the regularizing kernel this permits to design a well posed semi discrete singular function method sfm which diers from the usual sfm in that the dimension of the space of trial and test functions is innite partial fourier transform in the z direction of edges enables us to overcome the diculty of an innite dimension and to obtain optimal orders of convergence in various norms for the semi discrete solution asymptotic error estimates are also proved for the coecients of singularities for practical computations an optimally convergent full discretization approach which consists in coupling truncated fourier series in the z direction with the sfm in the xy plane is implemented other good though not optimal schemes which are based on a tensor product form of singularities are investigated as an illustration of the results we consider the laplace operator c 1999 elsevier science b v all rights reserved msc 65n30 65n15 65n99 35b65 35b99 35j99

on invariant subspaces of matrices a new proof of a theorem of halmos abstract lmos proved that if a is a matrix and if e is an a invariant subspace then there exist matrices b and c such that ba ab ca ac e is the kernel of b and e is the range of c we present an elementary proof of this result and show that there exist b and c that additionally satisfy bc cb o

high order convergence for collocation of second kind boundary integral equations on polygons we propose collocation methods with smoothest splines to solve the integral equation of the second kind on a plane polygon they are based on the bijectivity of the double layer potential between spaces of sobolev type with arbitrary high regularity and involving the singular functions generated by the corners if splines of order 2m 1 are used we get quasi optimal estimates in h m norm and optimal order convergence for the h k norm if 0 le k le m numerical experiments are presented

performance of the h p version of the finite element method with various elements the paper addresses the performance of square elements of type q p and q p q p and q p are elements of degree p analogous to the well known 9 and 8 noded elements for p 2 the performance is analysed theoretically for the class of analytic functions numerical experiments confirm the conclusions drawn from the theory the computational complexity of a solution algorithm is studied using timings of the computation on an alliant fx 8 computer the data show that high order elements are very effective for the considered class of solutions

characteristic invariant subspaces generated by a single vector abstract if f is an endomorphism of a finite dimensional vector space v over a field k then an invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f the characteristic hull of a subset w of v is defined to be the smallest characteristic subspace in v that contains w it is known that characteristic subspaces that are not hyperinvariant can only exist when k 2 in this paper we study subspaces x which are the characteristic hull of a single element in the case where k 2 we derive a necessary and sufficient condition such that x is hyperinvariant

a saddle point least squares approach for primal mixed formulations of second order pdes abstract we present a saddle point least squares spls method for discretizing second order elliptic problems written as primal mixed variational formulations a stability lbb condition and a data compatibility condition at the continuous level are automatically satisfied the proposed discretization method follows a general spls approach and has the advantage that a discrete inf sup condition is automatically satisfied for standard choices of the test and trial spaces for the proposed iterative processes a nodal basis for the trial space is not required efficient preconditioning techniques that involve inversion only on the test space can be considered stability and approximation properties for two choices of discrete spaces are investigated applications of the new approach include discretization of second order problems with highly oscillatory coefficient interface problems and higher order approximation of the flux for elliptic problems with smooth coefficients

an adaptive newton algorithm based on numerical inversion regularization as postconditioner we examine a class of approximate inversion processes satisfying estimates similar to those defined by finite element or truncated spectral approximations these are to be used as approximate right inverses for newton iteration methods when viewed at the operator level these approximations introduce a defect or loss of derivatives of order one or more regularization is introduced as a form of defect correction a superlinearly convergent approximate newton iteration is thereby obtained by using the numerical inversion adaptively i e with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect this adaptive choice makes possible ascribing an order to the convergent process and this is identified as essentially optimal for elliptic problems relative to complexity the design of the algorithm involves multi parameter selection thereby opening up interesting avenues for elliptic problems relative to complexity this applies also to the regularization which may be carried out in the fourier transform space and is band limited in the language of whittaker shannon sampling theory the norms employed in the analysis are of holder space type the iteration is an adaptation of nash moser interation and the complexity studies use vituskin s theory of information processing computational experience is described in the final section

characteristic subspaces and hyperinvariant frames abstract let f be an endomorphism of a finite dimensional vector space v over a field k an f invariant subspace is called hyperinvariant respectively characteristic if it is invariant under all endomorphisms respectively automorphisms that commute with f we assume k 2 since all characteristic subspaces are hyperinvariant if k 2 the hyperinvariant hull w h of a subspace w of v is defined to be the smallest hyperinvariant subspace of v that contains w the hyperinvariant kernel w h of w is the largest hyperinvariant subspace of v that is contained in w and the pair w h w h is the hyperinvariant frame of w in this paper we study hyperinvariant frames of characteristic non hyperinvariant subspaces w we show that all invariant subspaces in the interval w h w h are characteristic we use this result for the construction of characteristic non hyperinvariant subspaces

axisymmetric stokes equations in polygonal domains regularity and finite element approximations we study the regularity and finite element approximation of the axisymmetric stokes problem on a polygonal domain w in particular taking into account the singular coefficients in the equation and non smoothness of the domain we establish the well posedness and full regularity of the solution in new weighted sobolev spaces k m 1 m w using our a priori results we give a specific construction of graded meshes on which the taylor hood mixed method approximates singular solutions at the optimal convergence rate numerical tests are presented to confirm the theoretical results in the paper

graded mesh approximation in weighted sobolev spaces and elliptic equations in 2d we study the approximation properties of some general finite element spaces constructed using improved graded meshes in our results either the approximating function or the function to be approximated or both are in a weighted sobolev space we consider also the l p version of these spaces the finite element spaces that we define are obtained from em conformally invariant families of finite elements no affine invariance is used stressing the use of elements that lead to higher regularity finite element spaces we prove that for a suitable grading of the meshes one obtains the usual optimal approximation results we provide a construction of these spaces that does not lead to long skinny triangles our results are then used to obtain l 2 error estimates and h m quasi optimal rates of convergence for the fem approximation of solutions of strongly elliptic interface boundary value problems

on certain geometric properties of polyharmonic mappings abstract the main purpose of this paper is twofold first the three quarter theorem is established for a large class of polyharmonic mappings second a class of polyharmonic mappings with positive jacobian and being univalent in the unit disk d are constructed

analysis of periodic schrodinger operators regularity and approximation of eigenfunctions let v be a real valued potential that is smooth everywhere on r3 except at a periodic discrete set s of points where it has singularities of the coulomb type z r we assume that the potential v is periodic with period lattice l we study the spectrum of the schrodinger operator h v acting on the space of bloch waves with arbitrary but fixed wavevector k let t r3 l let u be an eigenfunction of h with eigenvalue and let 0 be arbitrarily small we show that the classical regularity of the eigenfunction u is u h5 2 t in the usual sobolev spaces and u k3 2 m t s in the weighted sobolev spaces the regularity index m can be as large as desired which is crucial for numerical methods for any choice of the bloch wavevector k we also show that h has compact resolvent and hence a complete eigenfunction expansion the case of the hydrogen atom suggests that our regularity results are optimal we present two applications to the numerical approximation of eigenvalues using wave functions and usi

weighted norm first order system least squares fosls for problems with corner singularities a weighted norm least squares method is considered for the numerical approximation of solutions that have singularities at the boundary while many methods suffer from a global loss of accuracy due to boundary singularities the least squares method can be particularly sensitive to a loss of regularity the method we describe here requires only a rough lower bound on the power of the singularity and can be applied to a wide range of elliptic equations optimal order discretization accuracy is achieved in weighted h 1 and functional norms and l 2 accuracy are retained for boundary value problems with a dominant div curl operator our analysis including interpolation bounds and several poincare type inequalities are carried out in appropriately weighted sobolev spaces numerical results confirm the error bounds predicted in the analysis

on the univalence of the log biharmonic mappings in this paper we establish the univalence and starlikeness connection between log biharmonic mappings and logharmonic mappings

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

finite element approximations in a non lipschitz domain part ii fil acosta gabriel universidad de buenos aires facultad de ciencias exactas y naturales departamento de matematica argentina

fully discrete convergence estimates for vortex methods in bounded domains in this paper the authors study vortex method for 2 dimensionsal euler equations of incompressible flow in bounded domains to approximate the initial vortex field by a sum of vortex blobs with arbitrary high accuracy this field is extended smoothly to a small neighborhood of the boundary and the computation is carried out in the extended domain to construct the velocity field from vorticity a second order isoparametric finite element method is applied and to solve the characteristic equations the explicit euler s scheme is considered optimal error bounds for this fully discrete scheme are obtained

dirichlet problems in polyhedral domains ii approximation by fem and bem the convergence of the classical finite element method fem and boundary element method bem is poor due to the edge and vertex singularities of the solution of the involved dirichlet problem relative to an elliptic operator in a polyhedron using the global regularity results of lubuma and nicaise 1994 we analyse refined fem and bem with optimal rates of convergence

biorthogonal series solution of stokes flow problems in sectorial regions in this paper we develop an eigenfunction expansion method for solving stokes flow problems in sectorial cavities that arise in fluid dynamics the method leads to the development of a set of eigenfunctions adjoint eigenfunctions biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion the resulting infinite system of linear equations is then solved by truncation these biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions which are used to compute the coefficients of the eigenfunction expansion solution the biorthogonality conditions are derived for a class of fourth order boundary value problems with variable coefficients that arise from separating variables of the governing stokes equation the method is applied to the slow steady flow in a two dimensional sectorial cavity the fluid is set into motion by the uniform translation of a covering plate or belt

on some properties of solutions of the biharmonic equation in the present paper the properties of the linear complex operator l f zfz z fz which is defined on the class of complex valued c1 functions in the plane are investigated it is shown that harmonicity and biharmonicity are invariant under the linear operatorl results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered the operator l can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way

quasi optimal rates of convergence for the generalized finite element method in polygonal domains we consider a mixed boundary value interface problem for the elliptic operator p i j i a i j j u f on a polygonal domain r 2 with straight sides we endowed the boundary of partially with dirichlet boundary conditions u 0 on d and partially with neumann boundary conditions i j i a i j j u 0 on n the coefficients a i j are piecewise smooth with jump discontinuities across the interface which is allowed to have singularities and cross the boundary of in particular we consider triple junctions and even multiple junctions our main result is to construct a sequence of generalized finite element spaces s n that yield h m quasi optimal rates of convergence m 1 for the galerkin approximations u n s n of the solution u more precisely we prove that u u n c dim s n m 2 f h m 1 where c depends on the data for the problem but not on f u or n and dim s n our construction is quite general and depends on a choice of a good sequence of approximation spaces s n on a certain subdomain w that is at some distance to the vertices in case the spaces s n are generalized finite element spaces then the resulting spaces s n are also generalized finite element spaces

symmetric interior penalty discontinuous galerkin methods for elliptic problems in polygons we analyze symmetric interior penalty discontinuous galerkin dg finite element methods for linear second order elliptic boundary value problems in polygons omega with straight edges where solutions exhibit singular behavior near corners and at boundary points where boundary conditions change to resolve corner singularities we admit both graded meshes and bisection refinement meshes we prove that judiciously chosen refinement parameters in these mesh families imply optimal asymptotic rates of convergence with respect to the total number of degrees of freedom n both for the dg energy norm error and the l 2 norm error the sharpness of our asymptotic convergence rate estimates is confirmed in a series of numerical experiments

combined effect of explicit time stepping and quadrature for curvature driven flows the flow of a closed surface of codimension 1 in bbb r n driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter varepsilon 0 conforming piecewise linear finite elements with mass lumping over a quasi uniform and weakly acute mesh of size h are further used for space discretization and combined with forward differences for time discretization with uniform time step tau the resulting explicit schemes are the basis for an efficient algorithm the so called dynamic mesh algorithm and exhibit finite speed of propagation and discrete ondegeneracy no iteration is required not even to handle the obstacle constraints the zero level set of the fully discrete solution is shown to converge past singularities to the true interface provided tau h 2 approx o varepsilon 4 and no fattening occurs if the more stringent relations tau h 2 approx o varepsilon 6 are enforced then an interface rate of convergence o varepsilon is derived in the vicinity of regular points along with a companion o varepsilon 1 2 for type i singularities for smooth flows an interface rate of convergence of o varepsilonq is proven provided tau h 2 approx o varepsilon 5 and exact integration is used for the potential term the analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature which bears some intrinsic interest lipschitz properties of viscosity solutions of the level set approach and discrete nondegeneracy these basic ingredients are also discussed

bloch constant and landau s theorem for planar p harmonic mappings abstract in this paper our main aim is to introduce the concept of planar p harmonic mappings and investigate the properties of these mappings first we discuss the p harmonic bloch mappings two estimates on the bloch constant are obtained which are generalizations of the main results in colonna 1989 9 as a consequence of these investigations we establish a bloch and landau s theorem for p harmonic mappings

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

cosine methods for nonlinear second order hyperbolic equations construction et analyse des methodes efficaces avec une precision d ordre eleve pour l approximation des solutions regulieres d une classe d equations hyperboliques non lineaires du second ordre application des discretisations de type galerkin et des schemas a deux pas estimations d erreurs d ordre optimal

landau s theorem for log p harmonic mappings abstract the main aim of this paper is to prove the existence of landau bloch constant for log p harmonic mappings

stokes flow in curved channels in this paper the problem of stokes flow in curved channels is solved by applying the matched eigenfunction expansion method the flow region is partitioned into a union of sectorial subregions then within each subregion the stream function is represented by an expansion of papkovich fadle eigenfunctions the coefficients in these expansions are obtained by requiring weak c 3 continuity of the stream function across common interfaces biorthogonality conditions in cylindrical coordinates are utilized

an adaptive finite element method for two phase stefan problems in two space dimensions i stability and error estimates a simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two phase stefan problems in 2d a typical triangula tion is coarse away from the discrete interface where discretization parameters satisfy a parabolic relation whereas it is locally refined in the vicinity of the dis crete interface so that the relation becomes hyperbolic several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations mesh selection is based upon equidistributing pointwise interpolation errors be tween consecutive meshes and imposing that discrete interfaces belong to the so called refined region consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated a general theory for interpolation between noncompatible meshes is set up in lp based norms the resulting scheme is stable in various sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods 3 2 2 on quasi uniform meshes namely 0 r as opposed to 0 x to achieve the same global asymptotic accuracy here r 0 is the uniform time step 112 a rate of convergence of essentially 0 x is derived in the natural energy spaces provided the total number of mesh changes is restricted to 0 x which in turn is compatible with the mesh selection procedure an auxiliary quasi optimal pointwise error estimate for the laplace operator is proved as well numerical results illustrate the scheme s efficiency in approximating both solutions and interfaces

an adaptively refined least squares finite element method for generalized newtonian fluid flows using the carreau model we implemented an adaptively refined least squares finite element approach for the navier stokes equations that govern generalized newtonian fluid flows using the carreau model to capture the flow region we developed an adaptive mesh refinement approach based on the least squares method the generated refined grids agree well with the physical attributes of the flows we also proved that the least squares approximation converges to the linearized versions solutions of the carreau model at the best possible rate model problems considered in the study are the flow past a planar channel and 4 to 1 contraction problems we presented the numerical results of the model problems revealing the efficiency of the proposed scheme and investigated the physical parameter effects

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

orbits of invariant subspaces of algebraic linear operators abstract let e be a vector space over a field k and a an algebraic k linear operator on e we describe the orbits under the action of the commutant of a and of the group of the invertible elements of the commutant of a on the lattice of a invariant subspaces

regularity estimates and optimal finite element methods in wp1 on polygonal domains abstract consider the poisson equation with the dirichlet boundary condition on a bounded convex polygonal domain r 2 we investigate the finite element approximation of singular solutions that are due to the non smoothness of the domain in the w p 1 norm 1 p in particular with analysis in weighted sobolev spaces and weighted holder spaces we provide regularity requirements on the given data and specific parameter selection criteria for graded meshes such that the resulting numerical approximation achieves the optimal convergence rate in w p 1 sample results from various numerical tests are provided to confirm the theory

the shape of stress free surfaces on a sheared block we obtain solutions for the shape of the free surface on the upper and lower boundaries of an initially rectangular incompressible linearly viscoelastic block when the block is sheared at the vertical sidewall to solve the problem when the vertical displacement of the sidewall is prescribed we use biorthogonal series of the fadle papkovich type the series has a point of novelty in that zero is an algebraically double but geometrically simple eigenvalue to expand arbitrary vector fields with two components it is necessary to include in the series the proper and generalized eigenvectors belonging to zero

regularity and multigrid analysis for laplace type axisymmetric equations consider axisymmetric equations associated with laplace type operators we establish full regularity estimates in high order kondrat vetype spaces for possible singular solutions due to the non smoothness of the domain and to the singular coefficients in the operator then we show suitable graded meshes can be used in high order finite element methods to achieve the optimal convergence rate even when the solution is singular using these results we further propose multigrid v cycle algorithms solving the system from linear finite element discretizations on optimal graded meshes we prove the multigrid algorithm is a contraction with the contraction number independent of the mesh level numerical tests are provided to verify the theorem

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

the fourier finite element method for the corner singularity expansion of the heat equation near the non convex vertex the solution of the heat equation is of the form u c e r sin w w l 2 r h 2 where c is the stress intensity function of the time variable t the convolution e x t r e r 2 4 t 2 t 3 a cutoff function and the opening angle of the vertex in this paper we use the fourier finite element method for approximating the stress intensity function c and the regular part w and derive the error estimates depending on the regularities of c and w we give some numerical examples confirming the derived convergence rates

on semi infinite spectral elements for poisson problems with re entrant boundary singularities abstract a spectral element method is described which enables poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi infinite rectangular regions two choices of trial functions are considered namely the eigenfunctions of the differential operator and chebyshev polynomials the coefficients in the series expansions are obtained by imposing weak c1 matching conditions across element interfaces singularities at re entrant corners are treated by a post processing technique which makes use of the known asymptotic behaviour of the solution at the singular point accurate approximations are obtained with few degrees of freedom

sous espaces invariants d un operateur nilpotent sur un espace de banach resume soit a un operateur lineaire continu nilpotent sur un espace de banach reel ou complexe nous montrons que pour tout sous espace invariant separable m de a il existe un commutant de a dont l image est dense dans m

description of characteristic non hyperinvariant subspaces over the field gf 2 given a square matrix a an a invariant subspace is called hyperinvariant respectively characteristic if and only if it is also invariant for all matrices t respectively nonsingular matrices t that commute with a shoda s theorem gives a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces for a nilpotent matrix in gf 2 here we present an explicit construction for all subspaces of this type

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

on the solution of a partial differential equation arising in stokes flow the partial differential equation e 4 j 0 that arises in the theory of fluids associated with axisymmetric stokes flow in a torus will be investigated in toroidal coordinates the solution j which represents the stream function will be shown to depend on legendre trigonometric and hyperbolic functions

finite element analysis for the axisymmetric laplace operator on polygonal domains abstract let l r 2 r r 2 z 2 we consider the equation l u f on a bounded polygonal domain with suitable boundary conditions derived from the three dimensional axisymmetric poisson s equation we establish the well posedness regularity and fredholm results in weighted sobolev spaces for possible singular solutions caused by the singular coefficient of the operator l as r 0 and by non smooth points on the boundary of the domain in particular our estimates show that there is no loss of regularity of the solution in these weighted sobolev spaces besides by analyzing the convergence property of the finite element solution we provide a construction of improved graded meshes such that the quasi optimal convergence rate can be recovered on piecewise linear functions for singular solutions the introduction of a new projection operator from the weighted space to the finite element subspace certain scaling arguments and a calculation of the index of the fredholm operator together with our regularity results are the ingredients of the finite element estimates

landau s theorem for certain biharmonic mappings abstract in this paper we show the existence of landau and bloch constants for biharmonic mappings of the form l f here l represents the linear complex operator l z z z z defined on the class of complex valued c 1 functions in the plane and f belongs to the class of biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

biorthogonality condition for creeping flow in wedge shaped trenches in the present paper we derive the biorthogonality condition for stokes flow in wedge shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge this condition is a property satisfied by the eigenfunctions and adjoint eigenfunctions needed to compute the coefficients of the eigenfunction expansion solution of the corresponding stokes flow problem

reflexivite d une extension d un operateur normal par un operateur nilpotent let t be an extension of a one to one normal operator a by a nilpotent operator n in this paper we calculate the defect of reflexivity of t we give a necessary and sufficient condition to insure the reflexivity of such extensions in particular it is shown that t is reflexive when n is reflexive

coefficients of the singularities for elliptic boundary value problems on domains with conical points iii finite element methods on polygonal domains in the two first parts of this work rairo model math anal numer 24 1990 pp 27e52 rairo model math anal numer 24 1990 pp 343 367 formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated now for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains the solution of such problems by the finite element method is considered in order to approximate the solution or the coefficients different methods are used based on the expressions of the coefficients that were obtained in the first two parts the dual singular function method is also generalized

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

optimal rates of convergence for degenerate parabolic problems in two dimensions fully discrete implicit schemes for the solution of degenerate parabolic equations in two dimensions are shown to satisfy error estimates of the form o tau ln 1 h alpha h where tau is the time step size and h is the mesh size these results apply to both the stefan problem and the porous medium equation

sur l orbite d un espace de banach sous l action du commutant d un operateur nilpotent resume soit a un operateur nilpotent sur un espace de banach e nous demontrons que l orbite de e par l action du commutant de a contient les f lat a qui sont de dimension finie

extensions of jordan bases for invariant subspaces of a matrix a characterization is obtained for the matrices a with the property that every some jordan basis of every a invariant subspace can be extended to a jordan basis of a these results are based on a criterion for a jordan basis of an invariant subspace to be extendable to a jordan basis of the whole space the criterion involves two concepts the constancy property and the depth property

the optimal convergence rate of a c1 finite element method for non smooth domains we establish optimal up to arbitrary e 0 convergence rates for a finite element formulation of a model second order elliptic boundary value problem in a weightedh 2 sobolev space with 5th degree argyris elements this formulation arises while generalizing to the case of non smooth domains an unconditionally stable scheme developed by liu et al j g liu j liu r l pego stability and convergence of efficient navier stokes solvers via a commutator estimate comm pure appl math 60 2007 pp 1443 1487 for the navier stokes equations we prove the optimality for both quasiuniform and graded mesh refinements and provide numerical results that agree with our theoretical predictions

finite elements with mesh refinement for wave equations in polygons error estimates for the space semi discrete approximation of solutions of the wave equation in polygons g r 2 are presented based on corner asymptotics of the solution it is shown that for continuous simplicial lagrangian finite elements of polynomial degree p 1 with either suitably graded mesh refinement or with bisection tree mesh refinement towards the corners of g the maximal rate of convergence o n p 2 which is afforded by the lagrangian finite element approximations on quasiuniform meshes for smooth solutions is restored dirichlet neumann and mixed boundary conditions are considered numerical experiments which confirm the theoretical results are presented generalizations to nonhomogeneous coefficients and elasticity and electromagnetics are indicated

cascadic multilevel algorithms for symmetric saddle point systems abstract in this paper we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems the algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems on each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a schur complement is implemented the level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error we prove new estimates that relate the iteration error and the residual for the constraint equation the new estimates are the key ingredients in imposing an efficient level change criterion the first iteration on each new level uses information about the best approximation of the discrete solution from the previous level the theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non increasing number of iterations on each level even though numerical results supporting the efficiency of the algorithms are presented for the stokes system the algorithms can be applied to a large class of boundary value problems including first order systems that can be reformulated at the continuous level as symmetric saddle point problems such as the maxwell equations

pseudodifferential operators on manifolds with a lie structure at infinity we define and study an algebra 0 v m0 of pseudodifferential operators canonically associated to a noncompact riemannian manifold m0 whose geometry at infinity is described by a lie algebra of vector fields v on a compactification m of m0 to a compact manifold with corners we show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to 0 v m0 we also consider the algebra diff m0 of differential operators on m0 generated by v and c m and show that 0 v m0 is a microlocalization of diff m0 our construction solves a problem posed by melrose in 1990 finally we introduce and study semi classical and suspended versions of the algebra 0 v m0

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

interior energy error estimates for the weak galerkin finite element method consider the poisson equation in a polytopal domain omega subset mathbb r d d 2 3 as the model problem we study interior energy error estimates for the weak galerkin finite element approximation to elliptic boundary value problems in particular we show that the interior error in the energy norm is bounded by three components the best local approximation error the error in negative norms and the trace error on the element boundaries this implies that the interior convergence rate can be polluted by solution singularities on the domain boundary even when the solution is smooth in the interior region numerical results are reported to support the theoretical findings to the best of our knowledge this is the first local energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes

on besov regularity of solutions to nonlinear elliptic partial differential equations abstract in this paper we study the regularity of the solutions of some nonlinear elliptic equations in kondratiev spaces on certain domains of polyhedral type general embedding theorems between kondratiev spaces and besov spaces will allow to avoid drawbacks to the standard sobolev regularity theory for those nonsmooth domains this will give us the opportunity to derive optimal rates for certain nonlinear approximation schemes

landau s theorem for polyharmonic mappings abstract in this paper we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk d then we obtain two versions of landau s theorem for polyharmonic mappings f and for the mappings of the type l f where l is the differential operator of abdulhadi abu muhanna and khuri examples and numerical estimates are given

a nonconforming finite element method for a two dimensional curl curl and grad div problem a numerical method for a two dimensional curl curl and grad div problem is studied in this paper it is based on a discretization using weakly continuous p 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries optimal convergence rates up to an arbitrary positive epsilon in both the energy norm and the l 2 norm are established on graded meshes the theoretical results are confirmed by numerical experiments

biorthogonality condition for axisymmetric stokes flow in a toroidal region in the present paper the biorthogonality condition for axisymmetric stokes flow in a toroidal geometry is derived this condition is needed to compute the coefficients of the eigenfunction expansion solution of the corresponding stokes flow problem

on the orbit of invariant subspaces of linear operators in finite dimensional spaces new proof of a halmos s result abstract p r halmos proved that for a linear operator a over a finite dimensional complex vector space e every a invariant subspace of e is the range of a commutant of a his proof was based on a generalization of the concept of eigenvector in this note we give an invariant proof of this halmos s theorem

a posteriori error estimation of hierarchical type for the schrodinger operator with inverse square potential we develop an a posteriori error estimate for mixed boundary value problems of the form delta fancyscript v u f v u f where the potential fancyscript v v may possess inverse square singularities at finitely many points in the domain we prove that our error estimate can be efficiently computed and is asymptotically identical to the actual error in the energy norm on a family of geometrically graded meshes appropriate for singular solutions of such problems therefore our estimate can be used for a practical stopping criterion a variety of numerical experiments support our theoretical results we also offer a direct convergence and effectivity comparison between the geometrically graded meshes which are based on a priori knowledge of possible singularities in the solution and adaptively refined meshes driven by local error indicators associated with our a posteriori error estimate

regularity estimates for elliptic boundary value problems in besov spaces we consider the dirichlet problem for poisson s equation on a nonconvex plane polygonal domain new regularity estimates for its solution in terms of besov and sobolev norms of fractional order are proved the analysis is based on new interpolation results and multilevel representations of norms on sobolev and besov spaces the results can be extended to a large class of elliptic boundary value problems some new sharp finite element error estimates are deduced

picturing the lattice of invariant subspaces of a nilpotent complex matrix abstract let q be a nilpotent transformation acting on a finite dimensional complex vector space a method is given by which a diagrammatic representation of lat q the lattice of invariant subspaces of q can be obtained basically the method consists of adding to the hasse diagram of hyperlat q the finite lattice of hyperinvariant subspaces of q in a specified way some applications are given

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings abstract in this paper we discuss the properties of the schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings first we introduce the schwarzian derivative s f for biharmonic mappings f z 2g h and obtain several necessary and sufficient conditions for s f to be analytic second we introduce the subordination of biharmonic mappings and obtain inequalities for integral means of subordinate biharmonic mappings finally we introduce the affine and linear invariant families of biharmonic mappings and prove several estimates related to the jacobian of functions in these invariant families

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh canonical edge problems for the biharmonic equation can be solved by separating variables the eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of r c smith 1952 we study the reduced system which is governed by a vector ordinary differential equation a solution of the biharmonic problem governed by a partial differential equation can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem the theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f y with values in 2 can be represented in terms of eigenvectors of the reduced problem this paper adds new theorems and conjectures to the theory we extend smith s generalization to fourth order problems of the methods introduced by titchmarsh 1946 to study eigenfunction expansions associated with second order problems we use this method to prove that if f y f1 y f2y 1 y 1 f y e c1 1 1 f e l2 1 1 then the series expressing f y converges uniformly to f y in the open interval 1 1 uniformly in 1 1 if f1 1 0 and in any case to 0 f2 1 f1 1 at y 1 this is unlike fourier series which converge to the mean value of the periodic extension of a function the series exhibits a gibbs phenomenon near the end points of discontinuity when f1 1 0

characteristic and hyperinvariant subspaces over the field gf 2 let f be an endomorphism of a vector space v over a field k an f invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f if k 2 then all characteristic subspaces are hyperinvariant if k 2 then there are endomorphisms f with invariant subspaces that are characteristic but not hyperinvariant in this paper we give a new proof of a theorem of shoda which provides a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces

a note on the eigenfunction expansion for the elastic strip the papkovitch fadle expansion of the biharmonic stress function for a semi infinite strip x leqq 1 y geqq 0 with free edges x pm 1 is obtained as the residue sum of a fourier integral for the case of end data corresponding to prescribed normal stress and transverse displacement which implies knowledge of phi xx x 0 equiv f 2 x and delta phi x 0 equiv f 4 x it is shown that if f f 2 f 4 is twice differentiable with bf f in l 2 0 1 and the data are compatible with the edge conditions at the corners x pm 1 y 0 the derived expansions for phi xx x y and delta phi x y converge to f 2 x f 4 x as y downarrow 0 for x on the closed interval 0 1 even when the compatibility conditions are not met the expansions converge to the data on the open interval 0 1

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis