global weak solution to the inhomogeneous navier stokes vlasov equations abstract the inhomogeneous navier stokes vlasov equations for fluid particle flows are considered in the three dimensional space the coupling in the fluid particle system arises from the drag force in the fluid equations and the acceleration in the vlasov equation an initial boundary value problem is studied in a bounded domain with large initial data the existence of global weak solution is established through an approximation scheme a fixed point argument energy estimates and a weak convergence method

approximation order of bivariate spline interpolation for arbitrary smoothness by using the algorithm of nurnberger and riessinger 1995 we construct hermite interpolation sets for spaces of bivariate splines sqr 1 of arbitrary smoothness defined on the uniform type triangulations it is shown that our hermite interpolation method yields optimal approximation order for q 3 5r 1 in order to prove this we use the concept of weak interpolation and arguments of birkhoff interpolation

on smooth multivariate spline functions in this paper the dimensions of bivariate spline spaces with simple cross cut grid partitions are determined and expressions of their basis functions are given consequently the closures of these spaces over all partitions of the same type can be determined a somewhat more detailed study on bivariate splines with rectangular grid partitions is included the results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions

construction of local c 1 quartic spline elements for optimal order approximation this paper is concerned with a study of approximation order and construction of locally supported elements for the space s 4 1 of c 1 pp piecewise polynomial functions on an arbitrary triangulation of a connected polygonal domain in r 2 it is well known that even when a is a three directional mesh 1 the order of approximation of s 4 1 1 is only 4 not 5 the objective of this paper is two fold i a local clough tocher refinement procedure of an arbitrary triangulation a is introduced so as to yield the optimal fifth order of approximation where locality means that only a few isolated triangles need refinement and ii locally supported hermite elements are constructed to achieve the optimal order of approximation

scattered data interpolation using c 2 supersplines of degree six we show how c2 supersplines of degree six can be used to interpolate hermite data at the vertices of a quadrangulation we also present error bounds that show that our method has full approximation order seven and compare its efficiency with other c2interpolation methods in the literature

on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

a c r trivariate macro element based on the worsey farin split of a tetrahedron we construct trivariate c r macro elements for any r ge 1 over the worsey farin refinement of any tetrahedral partition this extends the construction of c 1 cubic worsey farin elements and c 2 elements of degree nine to the c r situation with r 2 we also show that the degree of polynomials used for our macro elements is the lowest

dynamic theory of suspensions with brownian effects we consider a suspension of particles in a fluid settling under the influence of gravity and dispersing by brownian motion a mathematical description is provided by the stokes equations and a fokker planck equation for the one particle phase space density this is a nonlinear system that depends on a number of parametric functions of the spatial concentration of the particles these functions are known only empirically or for dilute suspensions we analyze the system its stability its asymptotic behavior under different scalings and its validity from more microscopic description we summarize our conclusions at the end

hermite interpolation of nonsmooth functions preserving boundary conditions this article is devoted to the construction of a hermite type regularization operator transforming functions that are not necessarily c 1 into globally c 1 finite element functions that are piecewise polynomials this regularization operator is a projection it preserves appropriate first and second order polynomial traces and it has approximation properties of optimal order as an illustration it is used to discretize a nonhomogeneous navier stokes problem with tangential boundary condition

swapping edges of arbitrary triangulations to achieve the optimal order of approximation in the representation of scattered data by smooth pp piecewise polynomial functions perhaps the most important problem is to find an optimal triangulation of the given sample sites called vertices of course the notion of optimality depends on the desirable properties in the approximation or modeling problems in this paper we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions we will only consider c1 pp approximation with r 1 and k 4 the main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites such that these sample sites are the only vertices of the triangulation and that for any discrete data given at these sample sites there is a c1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation

c 1 hierarchical bases abstract in this note we derive estimates for the condition numbers of stiffness matrices relative to certain c1 conforming hierarchical bases for fourth order elliptic boundary value problems

approximation by cubic c 1 splines on arbitrary triangulations in this paper we present an efficient representation for bivariate piecewise cubicc 1 splines on arbitrary triangulations a numerical method is discussed for computing the dimension of the spaces 3 1 of these splines we consider subspaces ofs 3 1 satisfying certain boundary conditions some applications are given where piecewise cubicc 1 functions are used to solve interpolation problems and least squares approximation problems

optimal order approximation by mixed three directional spline elements this paper is concerned with a study of approximation order and construction of locally supported elements for the space s14 of c1 quartic pp piecewise polynomial functions on a triangulation of a connected polygonal domain in r2 it is well known that when is a three directional mesh 1 the order of approximation of s14 1 is only 4 not 5 though a local clough tocher refinement procedure of an arbitrary triangulation yields the optimal fifth order of approximation from the space s14 see 1 it needs more data points in addition to the vertex set of the triangulation in this paper we will introduce a particular mixed three directional mesh 3 and construct so called mixed three directional elements we prove that the space s14 3 achieves its optimal order of approximation by constructing an interpolation scheme using mixed three directional elements

d modules on rigid analytic spaces i we introduce a sheaf of infinite order differential operators d on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle we show that the sections of this sheaf over sufficiently small affinoid varieties are frechet stein algebras and use this to define co admissible sheaves of d modules we prove analogues of cartan s theorems a and b for co admissible d modules

hydrodynamic limit for the inhomogeneous incompressible navier stokes vlasov fokker planck equations abstract in this paper we study the hydrodynamic limit for the inhomogeneous incompressible navier stokes vlasov fokker planck equations in a two or three dimensional bounded domain when the initial density is bounded away from zero the proof relies on the relative entropy argument to obtain the strong convergence of macroscopic density of the particles n in l 0 t l 1 which extends the works of goudon jabin vasseur 15 and mellt vasseur 26 to inhomogeneous incompressible navier stokes vlasov fokker planck equations precisely the relative entropy estimates in 15 and 26 give the strong convergence of u and n and n respectively however we only obtain the strong convergence of n and u from the relative entropy estimate and we use another way to obtain the strong convergence of via the convergence of u furthermore when the initial density may vanish taking advantage of compactness result l m h 1 of orlicz spaces in 2d we obtain the convergence of n in l 0 t h 1 which is used to obtain the relative entropy estimate thus we also show the hydrodynamic limit for 2d inhomogeneous incompressible navier stokes vlasov fokker planck equations when there is initial vacuum

linear independence and stability of piecewise linear prewavelets on arbitrary triangulations in this paper we establish linear independence and stability of certain piecewise linear prewavelets over arbitrary bounded triangulations these prewavelets are natural generalizations of the locally supported element constructed by kotyczka and oswald for an infinite three directional mesh

construction of quintic powell sabin spline quasi interpolants based on blossoming abstract by blossoming marsden s identity we investigate local quasi interpolation schemes for c 2 continuous quintic powell sabin splines represented with a normalized b spline basis as applications various families of discrete and differential quasi interpolants reproducing quintic polynomials are presented

nonparametric fitting of aerodynamic data using smoothing thin plate splines this paper introduces a nonparametric fitting method for the interpolation of aerodynamic observations over a large range of multiple angles of attack the method is based on the employment of smoothing thin plate spline class functions a well renewed mathematical tool for multivariate data mining based on the generalization of the univariate natural cubic splines in which a roughness penalty criterion is used to produce very smooth predictive hypersurfaces compared with other methods such as parametric or even conventional nonparametric methods the use of a smoothing thin plate spline is more effective in that the predictive surface comes directly from the observed points thus minimizing any intervention of the analyst aimed at introducing model parameters this forms the basis for a very reliable fitting technique in which model construction can be relatively easy to implement an application of the method is carried out on a case study representative of some experimental data coming from a wind tunnel campaign on a typical three dimensional fuselage shaped body aimed at the acquisition of its aerodynamic coefficients over a rather extensive attitude range specifically the application is focused on the body lift coefficient as a function of both angle of attack and sideslip angle the data set is also interpolated using concurrent response surface methods namely a linear model a bivariate spline a radial basis function network a support vector regression technique a regression kriging and a moving least squares approach alternatively known as local polynomial regression results of data fitting are assessed using a cross validation approach and reveal a clear superiority of smoothing thin plate spline over the other methods leading to a more regular fitted surface and a more reliable prediction tool even when some observations are omitted this is important per se but acquires even more significance when an aerodynamic test campaign is to be planned with the minimum number of experimental observations

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

a domain decomposition method for computing bivariate spline fits of scattered data a domain decomposition method for solving large bivariate scattered data fitting problems with bivariate minimal energy discrete least squares and penalized least squares splines is described the method is based on splitting the domain into smaller domains solving the associated smaller fitting problems and combining the coefficients to get a global fit explicit error bounds are established for how well our locally constructed spline fits approximate the global fits some numerical examples are given to illustrate the effectiveness of the method

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

scattered data interpolation by bivariate splines with higher approximation order given a set of scattered data we usually use a minimal energy method to find a lagrange interpolation in a bivariate spline space over a triangulation of the scattered data locations it is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation to improve this order of approximation we propose several new schemes in this paper mainly we follow the ideas of clamped cubic interpolatory splines and not a knot interpolatory splines in the univariate setting and extend them to the bivariate setting in addition instead of the energy functional of the second order we propose to use higher order versions we shall present some theoretical analysis as well as many numerical results to demonstrate that our new interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory spline

on the optimal approximation rates for criss cross finite element spaces abstract let denote the triangulation of the plane obtained by multi integer translates of the four lines x 0 y 0 x y and x y by l k h we mean the space of all piecewise polynomials of degree k with respect to the scaled triangulation h having continuous partial derivatives of order on r 2 we show that the approximation properties of l k h are completely governed by those of the space spanned by the translates of all so called box splines contained in l k h combining this fact with fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where is the largest degree of smoothness for which these spaces are dense as h tends to zero furthermore we study the question of local linear dependence of the translates of the box splines for the above criss cross triangulations

smooth macro elements on powell sabin 12 splits macro elements of smoothness c r are constructed on powell sabin 12 splits of a triangle for all r 0 these new elements complement those recently constructed on powell sabin 6 splits and can be used to construct convenient superspline spaces with stable local bases and full approximation power that can be applied to the solution of boundary value problems and for interpolation of hermite data

asymptotic preserving schemes for kinetic fluid modeling of disperse two phase flows abstract we consider a system coupling the incompressible navier stokes equations to the vlasov fokker planck equation such a problem arises in the description of particulate flows we design a numerical scheme to simulate the behavior of the system this scheme is asymptotic preserving thus efficient in both the kinetic and hydrodynamic regimes it has a numerical stability condition controlled by the non stiff convection operator with an implicit treatment of the stiff drag term and the fokker planck operator yet consistent to a standard asymptotic preserving fokker planck solver or an incompressible navier stokes solver only the conjugate gradient method and fast poisson and helmholtz solvers are needed numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme with several interesting applications

composite finite elements of class ck let be a triangulation of d r2 with vertices a ai 1 i n and pkn d v ck d t v t pn where pn polynomials of total degree n for u cr d r k and v finite dimensional subspace of ck d we consider the hermite problem hk a v find v v d v ai d u ai 1 i n k this problem is solvable in v pkn d iff n 4k 1 enisek when v pkn d 3 where 3 is a subtriangulation of obtained by subdividing each t into 3 triangles the problem has a solution for n 4k 1 e g the c1 cubic hct triangle and a p27 triangle with 31 parameters when v pkn d 6 where 6 is a subtriangulation of obtained by subdividing each t into 6 triangles the problem has a solution for n 3k 1 e g the c1 quadratic ps triangle and a p25 triangle with 31 parameters in both cases the construction needs the partial derivatives d u ai for 2k 1 the domain d is assumed to be polygonal compact and connected

optimal lagrange interpolation by quartic c1 splines on triangulations we develop a local lagrange interpolation scheme for quartic c 1 splines on triangulations given an arbitrary triangulation d we decompose d into pairs of neighboring triangles and add diagonals to some of these pairs only in exceptional cases a few triangles are split based on this simple refinement of d we describe an algorithm for constructing lagrange interpolation points such that the interpolation method is local stable and has optimal approximation order the complexity for computing the interpolating splines is linear in the number of triangles for the local lagrange interpolation methods known in the literature about half of the triangles have to be split

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

approximation error estimates on a bounded domain in r n interpolating and smoothing discrete d m splines considerant un espace discretv h associe ah m une fonctionf h m 1 et lad m spline d interpolation discrete h d def dansv h cf 1 on etablit des estimations de l erreurf eh d en fonction de la distance de hausdorffd de et de l ensemblea d des points de donnees du type h d l o d m l en utilisant des resultats de duchon 5

stein domains in banach algebraic geometry abstract in this article we give a homological characterization of the topology of stein spaces over any valued base field in particular when working over the field of complex numbers we obtain a characterization of the usual euclidean transcendental topology of complex analytic spaces for non archimedean base fields the topology we characterize coincides with the topology of the berkovich analytic space associated to a non archimedean stein algebra because the characterization we used is borrowed from a definition in derived geometry this work should be read as a derived perspective on analytic geometry

on the stability of normalized powell sabin b splines in this paper we show that the normalized powell sabin b splines form a stable basis for the max norm the approximation constants depend only on the smallest angle in the underlying triangulation since the b splines refer to the size of the powell sabin triangles we find that small powell sabin triangles yield better approximation constants than big powell sabin triangles next in addition to the max norm we treat the lp norm here the approximation constants depend also on a fraction proper to the triangulation thus the b splines are not stable for the lp norm finally as a special case we consider the b spline bases obtained from powell sabin triangles with minimal area and pay extra attention to the approximation constants for the max norm

smooth macro elements based on clough tocher triangle splits summary macro elements of smoothness c r on clough tocher triangle splits are constructed for all r ge 0 these new elements are improvements on elements constructed in 11 in that disproving a conjecture made there certain unneeded degrees of freedom have been removed numerical experiments on hermite interpolation with the new elements are included

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

stability of optimal order approximation by bivariate splines over arbitrary triangulations let a be a triangulation of some polygonal domain in r2 and skr a the space of all bivariate cr piecewise polynomials of total degree 3r 2 that can be used to provide the highest order of approximation with the property that the approximation constant of this order is independent of the geometry of a with the exception of the smallest angle in the partition this result is obtained by means of a careful choice of locally supported basis functions which however require a very technical proof to justify their stability in optimal order approximation a new formulation of smoothness conditions for piecewise polynomials in terms of their b net representations is derived for this purpose

increasing the approximation order of spline quasi interpolants abstract in this paper we show how by a very simple modification of bivariate spline discrete quasi interpolants we can construct a new class of quasi interpolants which have remarkable properties such as high order of regularity and polynomial reproduction more precisely given a spline discrete quasi interpolation operator q d which is exact on the space p m of polynomials of total degree at most m we first propose a general method to determine a new differential quasi interpolation operator q r d which is exact on p m r q r d uses the values of the function to be approximated at the points involved in the linear functional defining q d as well as the partial derivatives up to the order r at the same points from this result we then construct and study a first order differential quasi interpolant based on the c 1 cubic b spline on the equilateral triangulation with a hexagonal support when the derivatives are not available or extremely expensive to compute we approximate them by appropriate finite differences to derive new discrete quasi interpolants q d we estimate with small constants the quasi interpolation errors f q r d f and f q d f in the infinity norm finally numerical examples are used to analyze the performance of the method

multivariate spline spaces in principle all of the properties and applications of multivariate spline functions can be derived from the basic theories in chapter 1 the expressions of the multivariate spline functions in 2 of chapter 1 are inconvenient to be applied directly because the expressions also rely on the solutions of the global conformality conditions so as to determine all the smooth cofactors

rational finite elements of class ck over a quadrangulation abstract in this paper we give a constructive method for rational c k finite element method in r 2 for quadrilateral for k 2 the usual polynomials splines require some supersmoothness at the vertices of the quadrilateral even when dealing with macro elements this is no more a requirement for our rational approach the process of the construction is illustrated with the help of two examples namely the c 1 case and the c 2 case

gaussian quadrature for c1 cubic clough tocher macro triangles abstract a numerical integration rule for multivariate cubic polynomials over n dimensional simplices was designed by hammer and stroud 1956 the quadrature rule requires n 2 quadrature points the barycentre of the simplex and n 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex in the planar case this particular rule belongs to a two parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles we prove that this rule is exact for a larger space namely the c 1 cubic clough tocher spline space over macro triangles if and only if the split point is the barycentre this results in a factor of three reduction in the number of quadrature points needed to integrate the clough tocher spline space exactly

upper and lower bounds on the dimension of multivariate spline spaces we give upper and lower bounds on the dimensions of multivariate spline spaces defined on triangu lations in mathbb r k the bounds are optimal in a certain sense

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

an explicit basis for c 1 quartic by various bivariate splines we establish the dimension of the space of c 1 bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain our approach is to construct a minimal determining set and an associated explicit basis for the space for general triangulations the minimal determining set must be defined globally

simulation of fluid and particles flows asymptotic preserving schemes for bubbling and flowing regimes in this work we propose asymptotic preserving numerical schemes for the bubbling and flowing regimes of particles immersed in a fluid treated by two phase flow models the description comprises compressible euler equations for the dense phase fluid and a kinetic fokker planck equation for the disperse phase particles coupled through friction terms we show numerical simulations in the relevant case of gravity in the one dimensional case demonstrating the overall behavior of the schemes

global existence and large time behavior of classical solutions to the euler maxwell vlasov fokker planck system abstract we consider global classical solutions to the cauchy problem for an euler maxwell vlasov fokker planck system which is a fluid particle model describe the evolutions of disperse two phase flows in these two phase flows the compressible euler maxwell equations modeling a dense phase fluid and the vlasov fokker planck equation for the dispersed phase particles which is usually considered from the statistical view of point they are coupled through the friction force global existence of classical solutions to the cauchy problem in r 3 is established when initial data is a small perturbation of some given equilibrium the algebraic rate of convergence of solution toward the equilibrium state is also obtained under additional conditions on initial data the proof is based on the classical energy estimates and fourier multiplier technique which are considerably complicated and some new ideas and techniques are thus required

on multivariate polynomials in bernstein bezier form and tensor algebra the bernstein bezier representation of polynomials is a very useful tool in computer aided geometric design in this paper we make use of multilinear tensors to describe and manipulate multivariate polynomials in their bernstein bezier form as an application we consider hermite interpolation with polynomials and splines

optimal bivariate c1 cubic quasi interpolation on a type 2 triangulation in a guessab o nouisser g schmeisser multivariate approximation by a combination of modified taylor polynomials j comput appl math 196 2006 162 179 a general method is proposed to increase the approximation order of approximation operators in this work by using these enhancement techniques we introduce and study new schemes based on a c 1 spline quasi interpolant on a type 2 triangulation they are designed for approximating real valued functions defined on r 2 the proposed method is based on the following idea from a discrete quasi interpolant defined by the quadratic box spline exact on p 2 and by judiciously choosing the first order taylor coefficients we derive a cubic differential quasi interpolant yielding optimal approximation order in addition when the derivatives are not available or are extremely expensive to compute we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi interpolants as an essential difference to some of the existing methods we only use the given data values and then we do not modify the original triangulation finally we present some numerical tests which confirm the efficiency of the newly quasi interpolant and demonstrate good visual quality in particular we compare it with a differential quasi interpolant done by lai m j lai approximation order from bivariate c 1 cubics on a four directional mesh is full comput aided geom design 11 2 1994 215 223 which is also exact on p 3 but uses third order partial derivatives

quadrilateral finite elements of fvs type and class c rho let cal p be some partition of a planar polygonal domain omega into quadrilaterals given a smooth function u we construct piecewise polynomial functions upsilon in c rho omega of degree n 3 rho for rho odd and n 3 rho 1 for rho even on a subtriangulation tau 4 of cal p the latter is obtained by drawing diagonals in each q in cal p and upsilon vert q is a composite quadrilateral finite element generalizing the classical c 1 cubic fraeijs de veubeke and sander or fvs quadrilateral the function upsilon interpolates the derivatives of u up to order rho rho 2 at the vertices of cal p polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements

on c 2 quintic spline functions over triangulations of powell sabin s type abstract given a triangulation of a polygonal domain we find a refinemet of of by choosing t in a neighborhood of the center of the inscribed circle of each triangle of t connecting t to the vertices of the triangle t and connecting t to t if t shares an interior edge with t or to the midpoint e of any boundary edge e of t the resulting triangulation is a triangulation of powell sabin s type we investigate a c 2 quintic spline space s 5 2 whose elements are c 3 only at t s we give a dimension formula for this spline space show how to construct a locally supported basis display an interpolation scheme and prove that this spline space has the full approximation order

construction of global surfaces by variational evolutionary pde splines this paper deals with the construction and characterization of discrete variational pde splines to formulate the problem we need an evolutionary pde equation certain boundary conditions and a set of points to approximate we show the existence and the uniqueness of the solution of such a problem and we establish a convergence result of a discrete variational evolutionary pde spline to a given function we present several numerical and graphical examples of construction and approximation of surfaces in order to prove the validity of the presented method

c1 macroelements in adaptive finite element methods some interesting properties of composite basis functions for c1 macroelements are investigated including their use for constructing conforming c1 function spaces on non conforming adaptively refined meshes of particular interest is the classical cubic hsieh clough tocher 3 split triangle because of its simplicity and convergence properties in fourth order problems as compared with the powell sabin heindl 6 split and 12 split triangles a posteriori error indicators for adaptive refinement are developed numerical experiments demonstrate convergence rates and adaptive refinement performance based on a simplified error indicator is tested extensibility to analogous three dimensional tetrahedral elements is briefly discussed copyright 2006 john wiley sons ltd

interpolation by c 1 splines of degree q 4 on triangulations let be an arbitrary regular triangulation of a simply connected compact polygonal domain ir and let s1q denote the space of bivariate polynomial splines of degree q and smoothness 1 with respect to we develop an algorithm for constructing point sets admissible for lagrange interpolation by s1q if q 4 in the case q

spherical splines for data interpolation and fitting we study minimal energy interpolation and discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

bivariate spline interpolation at grid points we describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible the splines are defined on rectangular partitions adding one or two diagonals to each rectangle the interpolation sets are selected in such a way that the grid points of the partition are contained in these sets and no large linear systems have to be solved our method is to generate a net of line segments and to choose point sets in these segments which satisfy the schoenberg whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied in order to include the grid points in the interpolation sets we give a sufficient schoenberg whitney type condition for interpolation by bivariate splines supported in certain cones this approach is completely different from the known interpolation methods for bivariate splines of degree at most three our method is illustrated by some numerical examples

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

a bivariate c1 cubic super spline space on powell sabin triangulation a bivariate c 1 cubic super spline is constructed on powell sabin type 1 split with the additional smoothness at vertices in the original triangulation being c 2 which permits the hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation the locally supported dual basis and computational details using derivatives around each vertex are given

macro elements and stable local bases for splines on clough tocher triangulations macro elements of arbitrary smoothness are constructed on clough tocher triangle splits these elements can be used for solving boundary value problems or for interpolation of hermite data and are shown to be optimal with respect to spline degree we conjecture they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over clough tocher refinements of arbitrary triangulations these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

on numerical quadrature for c1 quadratic powell sabin 6 split macro triangles abstract the quadrature rule of hammer and stroud 1956 for cubic polynomials has been shown to be exact for a larger space of functions namely the c 1 cubic clough tocher spline space over a macro triangle if and only if the split point is the barycentre of the macro triangle kosinka and barto 2018 we continue the study of quadrature rules for spline spaces over macro triangles now focusing on the case of c 1 quadratic powell sabin 6 split macro triangles we show that the 3 node gaussian quadrature s for quadratics can be generalised to the c 1 quadratic powell sabin 6 split spline space over a macro triangle for a two parameter family of inner split points not just the barycentre as in kosinka and barto 2018 the choice of the inner split point uniquely determines the positions of the edge split points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature consequently the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three for the inner split point at the barycentre we prove that the two 3 node quadratic polynomial quadratures of hammer and stroud exactly integrate also the c 1 quadratic powell sabin spline space if and only if the edge split points are at their respective edge midpoints for other positions of the inner and edge split points we provide numerical examples showing that three nodes suffice to integrate the space exactly but a full classification and a closed form solution in the generic case remain elusive

c 1 hierarchical riesz bases of lagrange type on powell sabin triangulations in this paper we construct c1 continuous piecewise quadratic hierarchical bases on powell sabin triangulations of arbitrary polygonal domains in r2 our bases are of lagrange type instead of the usual hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable riesz bases for the sobolev spaces hs with s 1 5 2 especially the case s 2 is of interest because we can use the corresponding hierarchical basis for preconditioning fourth order elliptic equations leading to uniformly well conditioned stiffness matrices compared to the hierarchical riesz bases by davydov and stevenson hierarchical riesz bases for hs 1 s 5 2 constructive approximation to appear our construction is simpler

on super splines and finite elements super splines are spaces of multivariate splines consisting of piecewise polynomials defined on triangulations where the degree of smoothness enforced at the vertices is larger than the degree of smoothness enforced across the edges dimension formulae for super spline spaces are established and the construction of explicit bases of locally supported splines is presented in addition it is shown that many classical finite element spaces can be interpreted as super spline spaces providing a link between spline theory and the highly applicable finite element theory

a local regularization operator for triangular and quadrilateral finite elements this paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes this operator is a variant of the regularization operator of clement however ours is constructed via a local projection in a reference domain we prove in this paper that it has the same optimal approximation properties as the standard interpolation operator and we present some applications

a local lagrange interpolation method based on c 1 cubic splines on freudenthal partitions a trivariate lagrange interpolation method based on c 1 cubic splines is described the splines are defined over a special refinement of the freudenthal partition of a cube partition the interpolating splines are uniquely determined by data values but no derivatives are needed the interpolation method is local and stable provides optimal order approximation and has linear complexity

error estimates in w 2 semi norms for discrete interpolating d 2 splines we derive error estimates in w 2 semi norms for multivariate discrete d 2 splines that interpolate an unknown function at the vertices of given triangulations these results are widely based on the construction of approximation operators and linear projectors onto piecewise polynomial spaces having weakly stable local bases

cr rational splines over a triangulation abstract we propose a construction of c r bivariate rational splines over a triangulation via a finite element approach for r 2 the usual bivariate polynomial splines require some supersmoothness at the vertices of the triangles even when dealing with macro elements this is no more a requirement for our rational approach in this paper we give a constructive method for rational c r finite elements in r 2 defined over triangles beside the general case the process of construction is illustrated with two examples namely the c 1 case and the c 2 case

some new formulations of smoothness conditions and conformality conditions for bivariate splines abstract this paper is concerned with a study of some new formulations of smoothness conditions and conformality conditions for multivariate splines in terms of b net representation in bivariate setting a group of new parameters of bivariate cubic polynomials over a planar simplex is introduced and smoothness conditions and conformality conditions of bivariate cubic c 1 splines are simplified

a topological reconstruction theorem for d infty modules in this paper we prove that any perfect complex of d infty modules may be reconstructed from its holomorphic solution complex provided that we keep track of the natural topology of this last complex this is to be compared with the reconstruction theorem for regular holonomic d modules which follows from the well known riemann hilbert correspondence to obtain our result we consider sheaves of holomorphic functions as sheaves with values in the category of ind banach spaces and study some of their homological properties in particular we prove that a k quot u nneth formula holds for them and we compute their poincar x27 e verdier duals as a corollary we obtain the form of the kernels of continuous cohomological correspondences between sheaves of holomorphic forms this allows us to prove a kind of holomorphic schwartz kernel theorem and to show that d infty rhomtop o o our reconstruction theorem is a direct consequence of this last isomorphism note that the main problem is the vanishing of the topological ext s and that this vanishing is a consequence of the acyclicity theorems for dfn spaces which are established in the paper

norms of projectors onto spaces with riesz bases abstract many important linear operators p x s of a linear space x onto a subspace s of x are defined by the minimum problem f p f min f u u s f x where the norm on x is induced by an inner product we study the l norm of such operators p when s has a riesz basis b then we apply the results in detail to the best l2 and l2 approximations for univariate polynomial splines spaces and to box spline series

on the approximation power of splines on triangulated quadrangulations we study the approximation properties of the bivariate spline spaces srt cal s r 3r hbox diamondsuit kern 0 78em raise0 10786ex hbox of smoothness r and degree 3r defined on triangulations lai hbox diamondsuit kern 0 78em raise0 10786ex hbox which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral

instability in the dimension of spaces of bivariate piecewise polynomials of degree 2 r and smoothness order r the spaces of bivariate splines on a particular triangulation are considered the dimension of the space of splines with degree 2 and with smoothness order 1 on this triangulation has long been known to depend on the geometry of the triangulation this result is extended to the space of splines of degree 2r and smoothness order r the dimension of the spline space is shown to have a similar dependence on geometry for all values of r geqq 1

isogeometric analysis with powell sabin splines for advection diffusion reaction problems abstract this paper presents the use of powell sabin splines in the context of isogeometric analysis for the numerical solution of advection diffusion reaction equations powell sabin splines are piecewise quadratic c 1 functions defined on a given triangulation with a particular macro structure we discuss the galerkin discretization based on a normalized powell sabin b spline basis we focus on the accurate detection of internal and boundary layers and on local refinements we apply the approach to several test problems and we illustrate its effectiveness by a comparison with classical finite element and recent isogeometric analysis procedures

c1 quintic interpolation over triangles two explicit representations two explicit representation of a c1 quintic interpolant over triangles are given these representations are derived by generalization of coons methods and bernstein bezier methods respectively

monotonicity preserving surface interpolation c 1 monotone quadratic splines are analyzed this motivates an algorithm for interpolating monotone data given on a rectangular grid with a c 1 monotone quadratic spline surface error estimates an operation count and numerical examples are given