poincare friedrichs type constants for operators involving grad curl and div theory and numerical experiments abstract we give some theoretical as well as computational results on laplace and maxwell constants i e on the smallest constants c n 0 arising in estimates of the form u l 2 c 0 grad u l 2 e l 2 c 1 curl e l 2 h l 2 c 2 div h l 2 besides the classical de rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of hilbert complexes we consider mixed boundary conditions and bounded lipschitz domains of arbitrary topology our numerical aspects are presented by examples for the de rham complex in 2d and 3d which not only confirm our theoretical findings but also indicate some interesting conjectures

the morley element for fourth order elliptic equations in any dimensions in this paper the well known nonconforming morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion the general n dimensional morley element consists of all quadratic polynomials defined on each n simplex with degrees of freedom given by the integral average of the normal derivative on each n 1 subsimplex and the integral average of the function value on each n 2 subsimplex explicit expressions of nodal basis functions are also obtained for this element on general n simplicial grids convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation

renormalisation on sierpinski type fractals presents a family of deterministic fractals which generalise the d dimensional sierpinski gaskets and establishes their order of ramification and their fractal and spectral dimensions random walks on these fractals are renormalisable and lead to rational not necessarily polynomial mapping

nodal finite element de rham complexes we construct 2d and 3d finite element de rham sequences of arbitrary polynomial degrees with extra smoothness some of these elements have nodal degrees of freedom and can be considered as generalisations of scalar hermite and lagrange elements using the nodal values the number of global degrees of freedom is reduced compared with the classical nedelec and brezzi douglas marini finite elements and the basis functions are more canonical and easier to construct our finite elements for h mathrm div with regularity r 2 coincide with the nonstandard elements given by stenberg numer math 115 1 131 139 2010 we show how regularity decreases in the finite element complexes so that they branch into known complexes the standard de rham complexes of whitney forms and their higher order version can be regarded as the family with the lowest regularity the construction of the new families is motivated by finite element systems

approximation of the eigenvalue problem for the time harmonic maxwell system by continuous lagrange finite elements we propose and analyze an approximation technique for the maxwell eigenvalue problem using h1 conforming finite elements the key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional sobolev space h with 1 2 1 the method is shown to be convergent and spectrally correct

local bounded cochain projections we construct projections from h k the space of differential k forms on which belong to l2 and whose exterior derivative also belongs to l2 to finite dimensional subspaces of h k consisting of piecewise polynomial differential forms defined on a simplicial mesh of thus their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom moreover these projections have the properties that they commute with the exterior derivative and are bounded in the h k norm independent of the mesh size h unlike some other recent work in this direction the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements in the spirit of the clement interpolant a double complex structure is introduced as a key tool to carry out the construction

fractal sequences derived from the self similar extensions of the sierpinski gasket the well known sierpinski gasket has been generalised by the authors into the generalised pascal sierpinski gaskets gpsg of orders k l where both k and l are or 2 it has been shown here that families of self similar sequences can be derived from these extensions of the sierpinski gasket when k 2 and l is a prime number

a systematic construction of finite element commuting exact sequences we present a systematic construction of finite element exact sequences with a commuting diagram property for the de rham complex in one two and three space dimensions we apply the construction in two space dimensions to rediscover two families of exact sequences for triangles and three for squares and to uncover one new family of exact sequence for squares and two new families of exact sequences for general polygonal elements we apply the construction in three space dimensions to rediscover two families of exact sequences for tetrahedra three for cubes and one for prisms and to uncover four new families of exact sequences for pyramids three for prisms and one for cubes

discontinuous finite element methods for a bi wave equation modeling d wave superconductors this paper concerns discontinuous nite element approximations of a fourth order bi wave equation arising as a simplied ginzburg landau type model for d wave superconductors in the absence of an applied magnetic eld in the rst half of the paper we construct a variant of the morley nite element method which was originally developed for approximating the fourth order biharmonic equation for the bi wave equation it is proved that unlike the biharmonic equation it is necessary to impose a mesh constraint and to include certain penalty terms in the method to guarantee convergence nearly optimal order o by a factor j lnhj error estimates in the energy norm and in the h 1 norm are established for the proposed morley type nonconforming method in the second half of the paper we develop a symmetric interior penalty discontinuous galerkin method for the bi wave equation using general meshes and prove optimal order error estimates in the energy norm finally numerical experiments are provided to gauge the eciency of the proposed methods and to validate the theoretical error bounds

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

infinitesimal cr automorphisms of real hypersurfaces let m be an analytic real hypersurface through the origin in c n 1 and let hol m denote the space of vector fields x re z m tangent to m in a neighborhood of the origin where z is a holomorphic vector field defined in a neighborhood of the origin the hypersurface m is holomorphically nondegenerate at the origin if there is no holomorphic vector field tangent to m in a neighborhood of the origin the main result of the paper is that hol m is finite dimensional if and only if m is holomorphically nondegenerate at the origin

on the spatial fourier transforms of the pascal sierpinski gaskets it has been shown here that the fourier transform which can be obtained experimentally of an evolving fractal can be manipulated to yield the fractal dimension although only the pascal sierpinski gaskets have been considered here because of their generality it is expected that this technique can be utilized to identify other planar fractals as well

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

defining equations for real analytic real hypersurfaces in ⁿ a defining function for a real analytic real hypersurface can be utliquely written as 2 re h e where h is holomorphic and e contains no pure terms we study how h and e change when we perform a local biholomorphic change of coordinates or multiply by a unit one of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of e expanded in terms of h beyond eoo that serve as obstructions to writing a defining equation as 2 re h e where e is independent of h we also find necessary pluriharmonic obstructions to doing this which arise from the easier case of attempting to straighten the hypersurface

regular decomposition and a framework of order reduced methods for fourth order problems this paper is devoted to the construction of order reduced method of fourth order problems a constructive framework is presented such that a problem on the high regularity space can be deduced to an equivalent system on three low regularity spaces which are connected by a regular decomposition corresponding to a decomposition of the regularity of the high order space the generated numerical schemes based on the deduced problems can be of lower complicacy and the framework is fit for various fourth order problems three fourth order problems are then discussed under the framework including one in two dimension and two in three dimension they are each corresponding to a regular decomposition and thus are discretised based on the discretised analogues of the regular decomposition optimal error estimates are given

a posteriori error analysis for the morley plate element with general boundary conditions this paper introduces and analyses a local residual based a posteriori error indicator for the morley finite element method of the biharmonic kirchhoff plate bending problem in the theoretical part of the paper a recent approach presented by the authors for clamped boundaries is extended to general boundary conditions the error indicator is proven to be both reliable and efficient the numerical part of the paper presents a set of results on various benchmark computations with different kinds of domains and boundary conditions these tests verify the reliability and efficiency of the error estimator and illustrate the robustness of the method for adaptive mesh refinements copyright 2009 john wiley sons ltd

a quadrilateral morley element for biharmonic equations in this paper we propose a morley type finite element for quadrilateral meshes to solve biharmonic problems for each quadrilateral q the finite element space is defined by the span of p 2 q plus two functions in p 3 q each of the cubic polynomials is the product of a pair of equations of opposite edges and the equation of the bimedian between them the degrees of freedom consist of the values at vertices and integrals of normal derivatives over edges optimal orders of convergence are proved both in discrete h 2 and h 1 seminorms several numerical tests confirm the convergence analysis

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

on the necessity and sufficiency of the patch test for convergence of nonconforming finite elements this paper proposes a weak patch test that improves both the patch test and the generalized patch test under a weak superapproximation assumption which can be obtained from certain approximation and weak continuity property it is shown that the weak patch test is equivalent to the generalized patch test thus if a nonconforming finite element passes the patch test and satisfies some approximation and weak continuity property its convergence is then guaranteed furthermore the consistency term of such a nonconforming element can be proved to be of order o h it is shown that if a nonconforming element does not pass the patch test and a condition relative to subdividing a domain by the element is true then one can find a family of triangulations for which the element is divergent

nodal based finite element modeling of maxwell s equations weak forms are derived for maxwell s equations which are suitable for implementation on conventional c sup 0 elements with scalar bases the governing equations are expressed in terms of general vector and scalar potentials for the electric field intensity vector gauge theory is invoked to close the system and dictates the continuity requirements for the potentials at material interfaces as well as the blend of boundary conditions at exterior boundaries two specific gauges are presented both of which lead to helmholtz weak forms which are parasite free and enjoy simple physically meaningful boundary conditions a general and numerically efficient procedure for enforcing the jump discontinuities on the normal components of vector fields at dielectric interfaces and boundary conditions on curved surfaces is also given

a nonconforming morley finite element method for the fully nonlinear monge ampere equation in this paper we study finite element approximations of the viscosity solution of the fully nonlinear monge ampere equation det d 2 u f 0 using the well known nonconforming morley element our approach is based on the vanishing moment method which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and feng j sci comput 38 1 74 98 2009 the vanishing moment method approximates the monge ampere equation by the fourth order quasilinear equation epsilon delta 2u epsilon rm det d 2u epsilon f with appropriate boundary conditions we develop a finite element scheme using the n dimensional morley element introduced in wang and xu numer math 103 155 169 2006 to approximate the regularized fourth order problem in two and three dimensions and then derive optimal order error estimates

the lower bound property of the morley element eigenvalues in this paper we prove that the morley element eigenvalues approximate the exact ones from below on regular meshes including adaptive local refined meshes for the fourth order elliptic eigenvalue problems with the clamped boundary condition in any dimension and we implement the adaptive computation with the morley element to obtain lower bounds of eigenvalues for the vibration problem of clamped plate under tension and a fourth order elliptic eigenvalue problem with variable coefficients

local and parallel algorithms for fourth order problems discretized by the morley wang xu element method this paper systematically studies numerical solution of fourth order problems in any dimensions by use of the morley wang xu mwx element discretization combined with two grid methods xu and zhou math comp 69 881 909 1999 since the coarse and fine finite element spaces are nonnested two intergrid transfer operators are first constructed in any dimensions technically based on which two classes of local and parallel algorithms are then devised for solving such problems following some ideas in xu and zhou math comp 69 881 909 1999 the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems huang et al appl numer math 37 519 533 2001 huang et al sci china ser a 49 109 120 2006 and the error estimates for the intergrid transfer operators we prove that the discrete energy errors of the two classes of methods are of the sizes o h h 2 and o h h 2 h h d 1 2 respectively here h and h denote respectively the mesh sizes of the coarse and fine finite element triangulations and d indicates the space dimension of the solution region numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators

finite elements for symmetric tensors in three dimensions we construct finite element subspaces of the space of symme tric tensors with square integrable divergence on a three dimensional domain these spaces can be used to approximate the stress field in the classical hellinger reissner mixed formulation of the elasticty equations when stan dard discontinous finite element spaces are used to approximate the displace ment field these finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain and there is one for each positive value of the polynomial degree used for the displacements for each degree these provide a stable finite element discretization the construction of the spaces is closely tied to discretizations of the elasticity complex and can be viewed as the three dimensional analogue of the triangular element family for plane elasticity previously proposed by arnold and winther

a quadratic finite element method for solving biharmonic problems in ir n a family of simplicial finite element methods having the simplest possible structure is introduced to solve biharmonic problems in n n 3 using the primal variable the family is inspired in the morley triangle for the two dimensional case and in some sense this element can be viewed as its member corresponding to the valuen 2

stokes complexes and the construction of stable finite elements with pointwise mass conservation two families of conforming finite elements for the two dimensional stokes problem are developed guided by two discrete smoothed de rham complexes which we coin stokes complexes we show that the finite element pairs are inf sup stable and also provide pointwise mass conservation on very general triangular meshes

boundary invariants of pseudoconvex domains the purpose of this paper is to describe an invariant ax z which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in cn the motivation for defining the invariant ax z referred to as the multitype of z comes from the study of the boundary regularity properties of solutions of the d neumann problem on domains of finite type by a domain of finite type we mean one that satisfies the definition given by d angelo 5 namely a domain such that at any boundary point the maximum order of contact of one dimensional complex analytic varieties with the boundary is bounded in 4 the author showed how one can give a proof of global regularity of the a neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper in a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains before giving the definition of the multitype ax z we must introduce some

the local l 2 projected c 0 finite element method for maxwell problem an element local l 2 projected c 0 finite element method is presented to approximate the nonsmooth solution being not in h 1 of the maxwell problem on a nonconvex lipschitz polyhedron with reentrant corners and edges the key idea lies in that element local l 2 projectors are applied to both curl and div operators the c 0 linear finite element enriched with certain higher degree bubble functions is employed to approximate the nonsmooth solution the coercivity in l 2 norm is established uniform in the mesh size and the condition number cal o h 2 of the resulting linear system is proven for the solution and its curl in h r with r 1 we obtain an error bound cal o h r in an energy norm numerical experiments confirm the theoretical error bound

studies of the spectral dimension of the vicsek snowflake fractal by an exact real space renormalization group method the authors obtain the spectral dimension of the vicsek snowflake fractal the result seems to cast some doubt on the range of validity of the einstein relation on the deterministic fractal

a robust finite element method for darcy stokes flow finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed the system is approximately a linear stokes problem when the perturbation parameter is large while it degenerates to a mixed formulation of poisson s equation as the perturbation parameter tends to zero it is established basically by numerical experiments that most of the proposed finite element methods for stokes problem or the mixed poisson s system are not well behaved uniformly in the perturbation parameter this is used as the motivation for introducing a new robust finite element which exhibits this property

a nonstandard mixed finite element family we show that standard mixed finite element methods of second or higher degree for second order elliptic equations can be modified by imposing additional continuity conditions for the flux which reduces the dimension of the space this reduced space still gives a stable method with an optimal order of convergence we recall our postprocessing method and the a posteriori error estimator based on this

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

necessary conditions for subellipticity of the overline partial neumann problem let a be a a closed form of type p q with l2 coefficients on a smoothly bounded domain q in c the problem of determining both the existence and regularity properties of the solution u of au a where u is orthogonal to the null space of a on p q l forms is known as the a neumann problem one of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates let u be a neighborhood of a point z0 in the boundary of q a subelliptic estimate is said to hold in u if the estimate

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

mixed finite element methods for linear elasticity with weakly imposed symmetry in this paper we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima tions to both stresses and displacements the methods are based on a modified form of the hellinger reissner variational principle that only weakly imposes the symmetry condition on the stresses although this approach has been previously used by a number of authors a key new ingredient here is a constructive derivation of the elasticity complex starting from the de rham complex by mimicking this construction in the discrete case we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de rham complex these elements appear to be simpler than the ones previously derived for example we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

convergence of an adaptive mixed finite element method for kirchhoff plate bending problems some reliable and efficient a posteriori error estimators are produced for a mixed finite element method the hellan herrmann johnson h h j method for kirchhoff plate bending problems cf k hellan acta polytech scand civil engrg ser 46 1967 pp 1 28 l herrmann j eng mech div asce 93 1967 pp 13 26 c johnson numer math 21 1973 pp 43 62 based on these results with k 0 1 where k denotes the polynomial order of the discrete moment field space an adaptive mixed finite element method amfem is set up and its convergence and complexity are studied thoroughly the key points of the theoretical analysis include achieving a discrete helmholtz decomposition and a discrete inf sup condition which serve as the main tools in deducing the quasi orthogonality of the moment field and the discrete reliability of the estimator it is shown that the amfem is a contraction for the sum of the moment field error in an energy norm and the scaled error estimator between two consecutive adaptive loops moreover an estimate for the amfem s complexity via the number of elements is developed

geometric decompositions and local bases for spaces of finite element differential forms abstract we study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions these spaces are generalizations of the classical finite element spaces for vector fields frequently referred to as raviart thomas brezzi douglas marini and nedelec spaces in the present paper we derive geometric decompositions of these spaces which lead directly to explicit local bases for them generalizing the bernstein basis for ordinary lagrange finite elements the approach applies to both families of finite element spaces for arbitrary polynomial degree arbitrary order of the differential forms and an arbitrary simplicial triangulation in any number of space dimensions a prominent role in the construction is played by the notion of a consistent family of extension operators which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis

weighted regularization of maxwell equations in polyhedral domains we present a new method of regularizing time harmonic maxwell equations by a bf grad div term adapted to the geometry of the domain this method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions in the presence of reentrant corners or edges the usual regularization is known to produce wrong solutions due the non density of smooth fields in the variational space we get rid of this undesirable effect by the introduction of special weights inside the divergence integral standard finite elements can then be used for the approximation of the solution this method proves to be numerically efficient

a discrete helmholtz decomposition with morley finite element functions and the optimality of adaptive finite element schemes the discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it one approach for this is the careful design of an intermediate triangulation with one level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation this paper enfolds the second possibility of some novel discrete helmholtz decomposition for the nonconforming morley finite element method this guarantees the optimality of a standard adaptive mesh refining algorithm for the biharmonic equation numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre asymptotic range of the adaptive morley finite element method

critical spin wave dynamics for two dimensional sierpinski type fractal the dynamic critical exponent z for a system of nearest neighbour interacting heisenberg spins on a two dimensional sierpinski type fractal is obtained by applying a real space renormalisation group transformation of the equations of motion

a posteriori error estimates for the morley plate bending element a local a posteriori error indicator for the well known morley element for the kirchhoff plate bending problem is presented the error indicator is proven to be both reliable and efficient the technique applied is general and it is shown to have also other applications