surface crouzeix raviart element for the laplace beltrami equation in this paper we are concerned with the nonconforming finite element discretization of geometric partial differential equations we construct a surface crouzeix raviart element on the linear approximated surface analogous to a flat surface the optimal convergence theory for the new nonconforming surface finite element method is developed even though the geometric error exists by taking an intrinsic viewpoint of manifolds we introduce a new superconvergent gradient recovery method for the surface crouzeix raviart element using only the information of discretized surface the potential of serving as an asymptotically exact a posteriori error estimator is also exploited a series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method

branched polymers and dimensional reduction we establish an exact relation between self avoiding branched polymers in d 2 continuum dimensions and the hard core continuum gas at negative activity in d dimensions we review conjectures and results on critical exponents for d 2 2 3 4 and show that they are corollaries of our result we explain the connection first proposed by parisi and sourlas between branched polymers in d 2 dimensions and the yang lee edge singularity in d dimensions

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

an adaptive finite element method for the laplace beltrami operator on implicitly defined surfaces we present an adaptive finite element method for approximating solutions to the laplace beltrami equation on surfaces in mathbb r 3 which may be implicitly represented as level sets of smooth functions residual type a posteriori error bounds which show that the error may be split into a residual part and a geometric part are established in addition implementation issues are discussed and several computational examples are given

high order discontinuous galerkin methods for elliptic problems on surfaces we derive and analyze high order discontinuous galerkin methods for second order elliptic problems on implicitly defined surfaces in mathbb r 3 this is done by carefully adapting the unified discontinuous galerkin framework of d n arnold et al siam j numer anal 39 2002 pp 1749 1779 on a triangulated surface approximating the smooth surface we prove optimal error estimates in both a mesh dependent energy and l 2 norms numerical results validating our theoretical estimates are also presented

adaptive discontinuous galerkin methods on surfaces we present a dual weighted residual based a posteriori error estimate for a discontinuous galerkin approximation of a surface partial differential equation we restrict our analysis to a linear second order elliptic problem posed on hypersurfaces in mathbb r 3 r3 which are implicitly represented as level sets of smooth functions we show that the error in the energy norm may be split into a residual part and a higher order geometric part upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator we also present a novel geometric driven refinement strategy for pdes on surfaces which considerably improves the performance of the method on complex surfaces

dimensional reduction formulas for branched polymer correlation functions in bi01 we have proven that the generating function for self avoiding branched polymers in d 2 continuum dimensions is proportional to the pressure of the hard core continuum gas at negative activity in d dimensions this result explains why the critical behavior of branched polymers should be the same as that of the i 3 or yang lee edge field theory in two fewer dimensions as proposed by parisi and sourlas in 1981 in this article we review and generalize the results of bi01 we show that the generating functions for several branched polymers are proportional to correlation functions of the hard core gas we derive ward identities for certain branched polymer correlations we give reduction formulae for multi species branched polymers and the corresponding repulsive gases finally we derive the massive scaling limit for the 2 point function of the one dimensional hard core gas and thereby obtain the scaling form of the 2 point function for branched polymers in three dimensions

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

a higher order finite element method for partial differential equations on surfaces a new higher order finite element method for elliptic partial differential equations on a stationary smooth surface gamma is introduced and analyzed we assume that gamma is characterized as the zero level of a level set function phi and only a finite element approximation phi h of degree k geq 1 of phi is known for the discretization of the partial differential equation finite elements of degree m geq 1 on a piecewise linear approximation of gamma are used the discretization is lifted to gamma h which denotes the zero level of phi h using a quasi orthogonal coordinate system that is constructed by applying a gradient recovery technique to phi h a complete discretization error analysis is presented in which the error is split into a geometric error a quadrature error and a finite element approximation error the main result is a h 1 gamma error bound of the form c h m h k 1 results of numerical experiments illustrate the higher order convergence of t

afem for the laplace beltrami operator on graphs design and conditional contraction property we present an adaptive finite element method afem of any polynomial degree for the laplace beltrami operator on c 1 graphs in r d d 2 we first derive residual type a posteriori error estimates that account for the interaction of both the energy error in h 1 and the surface error in w 1 1 due to approximation of we devise a marking strategy to reduce the total error estimator namely a suitably scaled sum of the energy geometric and inconsistency error estimators we prove a conditional contraction property for the sum of the energy error and the total estimator the conditional statement encodes resolution of in w 1 1 we conclude with one numerical experiment that illustrates the theory

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a stable numerical method for the dynamics of fluidic membranes we develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in navier stokes flow local inextensibility of the membrane is ensured by solving a tangential navier stokes equation taking surface viscosity effects of boussinesq scriven type into account in our approach the bulk and surface degrees of freedom are discretized independently which leads to an unfitted finite element approximation of the underlying free boundary problem bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by dziuk numer math 111 55 80 2008 the obtained numerical scheme can be shown to be stable and to have good mesh properties finally the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions the numerical results demonstrate the robustness of the method and it is observed that the conservation properties are fulfilled to a high precision

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part i low order conforming nonconforming and mixed fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper their reliablility is shown for conforming nonconforming and mixed low order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

on the universality of the nonphase transition singularity in hard particle systems activity series for hard particle lattice gases and hard particles in continuous space are examined with respect to the singularity on the negative activity axis for approximately spherical particles it is found that the nature of the singularity depends only on the dimensionality of space

superconvergence and gradient recovery of linear finite elements for the laplace beltrami operator on general surfaces superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the laplace beltrami equation on general surfaces with mildly structured triangular meshes for a large class of practically useful grids the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the laplace beltrami equation and as a result various postprocessing gradient recovery including simple and weighted averaging local and global l 2 projections and zienkiewicz and zhu z z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution numerical experiments are presented to confirm the theoretical results

finite element approximation of the laplace beltrami operator on a surface with boundary we develop a finite element method for the laplace beltrami operator on a surface with boundary and nonhomogeneous dirichlet boundary conditions the method is based on a triangulation of the surfa

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality