existence and features of similarity solutions for non boussinesq gravity currents in this paper the flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions the current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of much lower density unlike earlier investigators we assume that the ambient fluid exerts no significant resisting action on the current and therefore the flow depth is expected to drop to zero at the front in the absence of friction in this context the shallow water equations are highly appropriate for computing the mean velocity and flow depth of the current the boundary condition imposed at the front leads to technical mathematical difficulties indeed unlike in the boussinesq case no regular solution to the shallow water equations satisfies the downstream condition but when the flow is supercritical at the channel inlet it is possible to construct a piecewise solution by patching a regular solution to an exceptional solution which represents the head behavior to better understand this result and make sure that the result is physically relevant we consider the navier stokes equations within the high reynolds number limit approximate similarity solutions can be worked out which support our earlier analysis on the shallow water equations while the flow body is self similar and weakly rotational the head is not self similar but tends toward a self similarity shape at long times it is characterized by a strong vorticity a straight free surface and a nonuniform velocity profile which becomes flatter and flatter with time

a stream virtual element formulation of the stokes problem on polygonal meshes in this paper we propose and analyze a novel stream formulation of the virtual element method vem for the solution of the stokes problem the new formulation hinges upon the introduction of a suitable stream function space characterizing the divergence free subspace of discrete velocities and it is equivalent to the velocity pressure inf sup stable mimetic scheme presented in l beirao da veiga et al j comput phys 228 2009 pp 7215 7232 up to a suitable reformulation into the vem framework both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

variable window width kernel estimates of probability densities kernel density estimators which allow different amounts of smoothing at different locations are studied modifications of estimators proposed by breiman meisel and purcell 1977 and abramson 1982a which have variable window widths are seen to have very fast rates of convergence these rates have traditionally been obtained using a less natural higher order kernel which has the disadvantage of allowing an estimator which takes on negative values

nonlinear moving boundary problems and a keller box scheme a modification of the original kelley box scheme is used to approximate nonlinear parabolic partial differential equations with moving boundaries the proposed scheme solves two systems of linear algebraic equations at each time step to produce second order approximations to the solution its derivative the flux and the position of the moving boundary the scheme is applied to a physical problem which has a known similarity solution a comparison of numerical results with the similarity solution gives evidence of stability and convergence

virtual element method for a nonlocal elliptic problem of kirchhoff type on polygonal meshes abstract we consider the discretization of the nonlocal elliptic problem of kirchhoff type using the virtual element method vem over polygonal meshes the nonlocal diffusion coefficient is approximated by using the l 2 projection operator which is directly computable from the degrees of freedom however the presence of the nonlocal term reduces the sparsity of the jacobian matrix which would increase the computational burden to avoid this and to retain the sparsity of the jacobian the nonlinear system is replaced by an equivalent system inspired by the work on the fem by gudi 2012 the numerical results show that the proposed methodology yields optimal convergence rate in the l 2 norm theoretical estimates are derived for h 1 norms and are verified by solving a numerical example

an analysis of some mixed enhanced finite element for plane linear elasticity the paper investigates mixed enhanced strain finite elements developed within the context of the u p formulation for nearly incompressible linear elasticity problems a rigorous convergence and stability analysis is detailed providing also l2 error estimates for the displacement field extensive numerical tests are developed showing in particular the accordance of the computational results with the theoretical predictions

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

on bandwidth choice for spatial data density estimation bandwidth choice is crucial in spatial kernel estimation in exploring non gaussian complex spatial data the paper investigates the choice of adaptive and non adaptive bandwidths for density estimation given data on a spatial lattice an adaptive bandwidth depends on local data and hence adaptively conforms with local features of the spatial data we propose a spatial cross validation scv choice of a global bandwidth this is done first with a pilot density involved in the expression for the adaptive bandwidth the optimality of the procedure is established and it is shown that a non adaptive bandwidth choice comes out as a special case although the cross validation idea has been popular for choosing a non adaptive bandwidth in data driven smoothing of independent and time series data its theory and application have not been much investigated for spatial data for the adaptive case there is little theory even for independent data conditions that ensure asymptotic optimality of the scv selected bandwidth are derived actually also extending time series and independent data optimality results further for the adaptive bandwidth with an estimated pilot density oracle properties of the resultant density estimator are obtained asymptotically as if the true pilot were known numerical simulations show that finite sample performance of the scv adaptive bandwidth choice works quite well it outperforms the existing r routines such as the rule of thumb and the so called second generation sheather jones bandwidths for moderate and big data sets an empirical application to a set of spatial soil data is further implemented with non gaussian features significantly identified

wave propagation in non homogeneous thin elastic rods subjected to time dependent stress impact abstract the problem of wave propagation in a non homogeneous elastic rod subjected to time dependent stress impact is considered similarity transformations are applied to the equation of motion of the rod and its boundary condition along with the similarity characteristic relationship on the moving front these are used to obtain the similarity representation as a boundary value problem a closed form solution of the similarity representation is obtained and in addition restrictions on the parameters and relations among them are also obtained results for the stress distribution in the rod at a specific time are graphically represented in non dimensional co ordinates

estimation of relative risk functions the risk associated with different levels of a quantitative factor x is often measured relative to the level corresponding to x 0 there are situations however where there is no natural zero for x for example where the risk factor is the age of an individual in this case it is more natural to measure risk relative to an overall average for the study population to use the whole population in this way also raises the possibility of regarding x as truly continuous rather than as a grouped variable this gives rise to the concept of a relative risk function methods for estimating such functions are discussed concentrating for the most part on the discrete case the extension to higher dimensions permits the investigation of joint effects of several factors while the problem of controlling for confounding variables can be handled by fitting multiplicative risk models relating the latter to the log linear model permits the estimation of adjusted relative risk functions the method is illustrated using data on childhood cancer the continuous case can in principle be handled in a similar way using density estimation techniques

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

a stress displacement virtual element method for plane elasticity problems abstract the numerical approximation of 2d elasticity problems is considered in the framework of the small strain theory and in connection with the mixed hellinger reissner variational formulation a low order virtual element method vem with a priori symmetric stresses is proposed several numerical tests are provided along with a rigorous stability and convergence analysis

the nonconforming virtual element method for elasticity problems abstract we present the nonconforming virtual element method for linear elasticity problems in the pure displacement formulation this method is uniformly convergent for the nearly incompressible case the optimal convergence in the h 1 norm is proved under some regularity assumptions we mention that for the lowest order case on triangular meshes the nonconforming virtual element method coincides with the nonconforming finite element method presented by brenner and sung 1992 19 up to an approximation of the right hand side besides we also present the nonconforming virtual element method for the pure traction problem which is proved to be uniformly convergent with respect to the lame constant finally we demonstrate the optimal convergence and stability of the nonconforming virtual element method by some numerical results

the maximal smoothing principle in density estimation abstract we propose a widely applicable method for choosing the smoothing parameters for nonparametric density estimators it has come to be realized in recent years e g see hall and marron 1987 scott and terrell 1987 that cross validation methods for finding reasonable smoothing parameters from raw data are of very limited practical value their sampling variability is simply too large the alternative discussed here the maximal smoothing principle suggests that we consider using the most smoothing that is consistent with the estimated scale of our data this greatly generalizes and exploits a phenomenon noted in terrell and scott 1985 that measures of scale tend to place upper bounds on the smoothing parameters that minimize asymptotic mean integrated squared error of density estimates such as histograms and frequency polygons the method avoids the extreme sampling variability of cross validation by using ordinary scale estimators such as the standard deviation and interquartile range which

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

a survey on free boundary identification of the truncated boundary in numerical bvps on infinite intervals a free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in fazio siam j numer anal 33 1996 1473 we consider here a survey on recent developments related to the free boundary identification of the truncated boundary the goals of this survey are to recall the reasoning for a free boundary identification of the truncated boundary to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line

the choice of test for detecting raised disease risk near a point source this paper considers the problem of testing for excess risk near a point source of risk s as might be postulated to exist near a nuclear installation for example the data are assumed to be in the form of counts in small areas for which expected numbers of cases have also been calculated by reference to national rates it is shown how to derive the most powerful test against any given alternative hypothesis such linear risk score tests are further considered particularly that which scores each case according to the reciprocal of the rank of the distance from s these tests are compared with two tests due to stone for general ordered alternatives and the important distinction is drawn between conditional and unconditional versions of the tests their behaviour is illustrated by application to data on childhood leukaemia in relation to nuclear installations in britain

kernel estimation of risk surfaces without the need for edge correction kernel estimates of relative risk surfaces can be used to examine the geographical variation of disease risk these surfaces can be expressed as ratios of bivariate kernel density estimates constructed from case and control data but care must be taken to avoid excessive bias at the boundaries of the region under study it is possible to correct this bias without the complications of explicit edge correction through the use of a specific smoothing regimen

extended finite element method on polygonal and quadtree meshes in this paper we present mesh independent modeling of discontinuous flelds on polygonal and quadtree flnite element meshes this approach falls within the class of extended and generalized flnite element methods where the partition of unity framework is used to introduce additional enrichment functions within the classical displacement based flnite element approximation for crack modeling a discontinuous function and the two dimensional asymptotic crack tip flelds are used as enrichment functions linearly complete partition of unity approximations are adopted on polygonal convex and nonconvex elements and quadtree meshes excellent agreement with reference solution results is obtained for mixed mode stress intensity factors on benchmark crack problems and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

the nonconforming virtual element method for the stokes equations we present the nonconforming virtual element method vem for the numerical approximation of velocity and pressure in the steady stokes problem the pressure is approximated using discontinuous piecewise polynomials while each component of the velocity is approximated using the nonconforming virtual element space on each mesh element the local virtual space contains the space of polynomials of up to a given degree plus suitable nonpolynomial functions the virtual element functions are implicitly defined as the solution of local poisson problems with polynomial neumann boundary conditions as typical in vem approaches the explicit evaluation of the non polynomial functions is not required this approach makes it possible to construct nonconforming virtual spaces for any polynomial degree regardless of the parity for two and three dimensional problems and for meshes with very general polygonal and polyhedral elements we show that the nonconforming vem is inf sup stable and establish optimal a pri

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

the analysis of contingency tables preface contingency tables and the chi square test 2 x 2 contingency tables r x c contingency tables multidimensional tables log linear models for contingency table linear logistic models contingency tables with ordered categories some special types of contingency tables appendices percentage points of the chi squared distribution computer software for fitting log linear and logistic models reference index

group theoretic and similarity analysis of hyperbolic partial differential equations abstract multiparameter dimensional groups of transformations are applied to a system of quasilinear hyperbolic partial differential equations along with their auxiliary conditions it is shown that for such a class of problems the similarity characteristic relationship can be stated and the location of the wavefront in terms of similarity variable can be determined two theorems are stated and their proofs given furthermore it is shown that similarity transformation is one to one and onto similarity analysis of two problems arising in wave propagation in nonlinear nonhomogeneous rods is presented in the light of theory presented in the paper

a mimetic discretization of the stokes problem with selected edge bubbles a new mimetic finite difference method for the stokes problem is proposed and analyzed the mimetic discretization methodology can be understood as a generalization of the finite element method to meshes with general polygons polyhedrons in this paper the mimetic generalization of the unstable p 1 p 0 and the conditionally stable q1 p0 finite element is shown to be fully stable when applied to a large range of polygonal meshes moreover we show how to stabilize the remaining cases by adding a small number of bubble functions to selected mesh edges a simple strategy for selecting such edges is proposed and verified with numerical experiments

the iterative transformation method and length estimation for tubular flow reactors in this note we apply an iterative transformation method in order to find the numerical solution and more important to provide the numerical evidence of the existence and uniqueness for the free boundary value problem describing the optimal length determination for tabular flow reactors the main idea is to introduce in the mathematical model a nonphysical parameter and then require partial invariance under an extended stretching group the solution of the original problem can be obtained by requiring that the parameter transforms in the unity finally we report numerical results that show the existence and uniqueness of the solution and give us an approximation to the free boundary value involved

a mimetic discretization of the reissner mindlin plate bending problem we present a mimetic approximation of the reissner mindlin plate bending problem which uses deflections and rotations as discrete variables the method applies to very general polygonal meshes even with non matching or non convex elements we prove linear convergence for the method uniformly in the plate thickness

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

high order discontinuous galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes abstract we propose and analyze a high order discontinuous galerkin finite element method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements we analyze the well posedness of the resulting formulation prove h p version error a priori estimates and present a dispersion analysis showing that polygonal meshes behave as classical simplicial quadrilateral grids in terms of dispersion properties the theoretical estimates are confirmed through various two dimensional numerical verifications

testing for changes in spatial relative risk the spatial relative risk function is a useful tool for describing geographical variation in disease incidence we consider the problem of comparing relative risk functions between two time periods with the idea of detecting alterations in the spatial pattern of disease risk irrespective of whether there has been a change in the overall incidence rate using case control datasets for each period we use kernel smoothing methods to derive a test statistic based on the difference between the log relative risk functions which we term the log relative risk ratio for testing a null hypothesis of an unchanging spatial pattern of risk we show how p values can be computed using both randomization methods and an asymptotic normal approximation the methodology is applied to data on campylobacteriosis from 2006 to 2013 in a region of new zealand we find clear evidence of a change in the spatial pattern of risk between those years which can be explained in differences by response to a public health initiative between urban and rural communities copyright 2017 john wiley sons ltd

mixed virtual element methods for elastodynamics with weak symmetry abstract we propose and analyze a mixed virtual element method for linear elastodynamics in velocity stress formulation with weak symmetry in this formulation the symmetry of the stress is relaxed by the rotation of the displacement and the system of second order differential equation in time is reduced to first order differential equations in time by introducing velocity the proposed method uses h div conforming virtual element space of order k k 1 for the stress and discontinuous piecewise polynomial spaces of degree k for the velocity and rotation for time discretization we use the crank nicolson scheme both semidiscrete and fully discrete error estimates are robust for nearly incompressible materials numerical experiments confirm our theoretical predictions

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

h text div h div and h mathbf curl h curl conforming virtual element methods in the present paper we construct virtual element spaces that are h mathrm div h div conforming and h mathbf curl h curl conforming on general polygonal and polyhedral elements these spaces can be interpreted as a generalization of well known finite elements we moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations finally we discuss the construction of exact sequences of vem spaces

numerical transformation methods a constructive approach abstract within group invariance theory we consider a constructive approach in order to define numerical transformation methods these are initial value methods for the solution of boundary value problems governed by ordinary differential equations here we consider the class of free boundary value problems governed by the most general second order equation in normal form for this class of problems the main theorem is concerned with the definition of an iterative transformation method the definition of a noniterative method applicable to a subclass of the original class of problems follows as a corollary therefore the proposed constructive approach allows us to establish a unifying framework for noniterative and iterative transformation methods

conforming polygonal finite elements summary in this paper conforming finite elements on polygon meshes are developed polygonal finite elements provide greater flexibility in mesh generation and are better suited for applications in solid mechanics which involve a significant change in the topology of the material domain in this study recent advances in meshfree approximations computational geometry and computer graphics are used to construct different trial and test approximations on polygonal elements a particular and notable contribution is the use of meshfree natural neighbour nn basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons this numerical formulation enables the construction of conforming approximation on n gons n 3 and hence extends the potential applications of finite elements to convex polygons of arbitrary order numerical experiments on second order elliptic boundary value problems are presented to demonstrate the accuracy and convergence of the proposed method copyright 2004 john wiley sons ltd

polygonal finite elements for topology optimization a unifying paradigm in topology optimization literature the parameterization of design is commonly carried out on uniform grids consisting of lagrangian type finite elements e g linear quads these formulations however suffer from numerical anomalies such as checkerboard patterns and one node connections which has prompted extensive research on these topics a problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members leading to mesh dependent sub optimal designs thus to address the geometric features of the spatial discretization we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions more specifically we consider polygonal meshes constructed from voronoi tessellations which in addition to possessing higher degree of geometric isotropy allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities copyright 2009 john wiley sons ltd

a virtual element method for the acoustic vibration problem we analyze in this paper a virtual element approximation for the acoustic vibration problem we consider a variational formulation relying only on the fluid displacement and propose a discretization by means of mathrm h mathrm div h div virtual elements with vanishing rotor under standard assumptions on the meshes we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates with this end we prove approximation properties of the proposed virtual elements we also report some numerical tests supporting our theoretical results

numerical length estimation for tubular flow reactors abstract this work was motivated by the engineering problem of determining the optimal length for tubular flow reactors we considered an isothermal nth order chemical reaction occuring in a tubular flow reactor with axial missing the mathematical model describing the problem was a free boundary value one the free boundary represented the reactor length in this context we proved a theorem characterizing an upper bound for the free boundary in order to solve the problem in point we introduced two different numerical methods first a noniterative method was considered in order to obtain an upper bound of the free boundary then we used this upper bound in order to start an iterative method that allowed us to find an approximation of the free boundary in closing we obtained numerical results either for an even or an odd order reaction

a c1 virtual element method on polyhedral meshes abstract the purpose of the present paper is to develop c 1 virtual elements in three dimensions for linear elliptic fourth order problems motivated by the difficulties that standard conforming finite elements encounter in this framework we focus the presentation on the lowest order case the generalization to higher orders being briefly provided in the appendix the degrees of freedom of the proposed scheme are only 4 per mesh vertex representing function values and gradient values interpolation error estimates for the proposed space are provided together with a set of numerical tests to validate the method at the practical level

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

a similarity approach to the numerical solution of free boundary problems the aim of this work is to point out that within a similarity approach some classes of free boundary value problems governed by ordinary differential equations can be transformed to initial value problems the interest in the numerical solution of free boundary problems arises because these are always nonlinear problems furthermore we show that free boundary problems arise also via a similarity analysis of moving boundary hyperbolic problems and they can be obtained as approximations of boundary value problems defined on infinite intervals most of the theoretical content of this survey is original it generalizes and unifies results already available in literature as far as applications of the proposed approach are concerned three problems of interest are considered and numerical results for each of them are reported

high order virtual element method on polyhedral meshes abstract we develop a numerical assessment of the virtual element method for the discretization of a diffusion reaction model problem for higher polynomial order k and three space dimensions although the main focus of the present study is to illustrate some h convergence tests for different orders k we also hint on other interesting aspects such as structured polyhedral voronoi meshing robustness in the presence of irregular grids sensibility to the stabilization parameter and convergence with respect to the order k

hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems we present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems under a saturation assumption the global reliability and efficiency of the proposed a posteriori estimator are proved several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies finally we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures

non parametric estimation of spatial variation in relative risk we consider the problem of estimating the spatial variation in relative risks of two diseases say over a geographical region using an underlying poisson point process model we approach the problem as one of density ratio estimation implemented with a non parametric kernel smoothing method in order to assess the significance of any local peaks or troughs in the estimated risk surface we introduce pointwise tolerance contours which can enhance a greyscale image plot of the estimate we also propose a monte carlo test of the null hypothesis of constant risk over the whole region to avoid possible over interpretation of the estimated risk surface we illustrate the capabilities of the methodology with two epidemiological examples

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

exploring high order three dimensional virtual elements bases and stabilizations abstract we present numerical tests of the virtual element method vem tailored for the discretization of a three dimensional poisson problem with high order polynomial degree up to p 10 besides we discuss possible reasons for which the method could return suboptimal wrong error convergence curves among these motivations we highlight ill conditioning of the stiffness matrix and not particularly clever choices of the stabilizations we propose variants of the definition of face bulk degrees of freedom as well as of stabilizations which lead to methods that are much more robust in terms of numerical performances

a robust numerical method for stokes equations based on divergence free h div finite element methods a computational method based on a divergence free h div approach is presented for the stokes equations in this article this method is designed to find velocity approximation in an exact divergence free subspace of the corresponding h div finite element space that is the continuity equation is strongly enforced a priori and the pressure is eliminated from the linear system in calculation a strength of this approach is that the saddle point problem for stokes equations is reduced to a symmetric positive definite problem in a subspace for which basis functions are readily available the resulting discrete system can then be solved by using existing sophisticated solvers the aim of this article is to demonstrate the efficiency and robustness of h div finite element methods for stokes equations the results not only confirm the existing theoretical results but also reveal additional advantages of the method in dealing with discontinuous boundary conditions

adaptive kernel estimation of spatial relative risk kernel smoothing is routinely used for the estimation of relative risk based on point locations of disease cases and sampled controls over a geographical region typically fixed bandwidth kernel estimation has been employed despite the widely recognized problems experienced with this methodology when the underlying densities exhibit the type of spatial inhomogeneity frequently seen in geographical epidemiology a more intuitive approach is to utilize a spatially adaptive variable smoothing parameter in this paper we examine the properties of the adaptive kernel estimator by both asymptotic analysis and a simulation study finding advantages over the fixed kernel approach in both the cases we also look at practical issues with implementation of the adaptive relative risk estimator including bandwidth choice and boundary correction and develop a computationally inexpensive method for generating tolerance contours to highlight areas of significantly elevated risk

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

a low order locking free virtual element for linear elasticity problems abstract we propose and analyze a low order virtual element for linear elasticity problems it can be seen as an evolution of the bernardi raugel element to general polygonal meshes like the bernardi raugel element the local degrees of freedoms for the new virtual element are the values of two components of v at vertexes and the values of the lowest moment of normal component of v on each edge the proposed method is well posed and optimal error estimates are obtained in the l 2 and h 1 norms moreover the generic constants in these estimates are shown to be uniform with respect to the lame coefficient numerical tests are provided to illustrate the good performance of the method and confirm our theoretical predictions

some basic formulations of the virtual element method vem for finite deformations abstract we present a general virtual element method vem framework for finite elasticity which emphasizes two issues element level volume change volume average of the determinant of the deformation gradient and stabilization to address the former issue we provide exact evaluation of the average volume change in both 2d and 3d on properly constructed local displacement spaces for the later issue we provide a new stabilization scheme that is based on the trace of the material tangent modulus tensor which captures highly heterogeneous and localized deformations two vem formulations are presented a two field mixed and an equivalent displacement based which is free of volumetric locking convergence and accuracy of the vem formulations are verified by means of numerical examples and engineering applications are demonstrated

virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation abstract we propose and analyze a virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation that is to say stresses are not require to be symmetric but only to satisfy a weaker condition based on lagrange multipliers the proposed method is well posed and the error bounds are shown to be uniform with the incompressibility parameter numerical tests confirm the convergence rate that is expected from the theory

on bandwidth variation in kernel estimates a square root law we consider kernel estimation of a smooth density f at a point but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density proportionally varying the bandwidths like f 1 2 at the contributing readings lowers the bias to a vanishing fraction of the usual value and makes for performance seen in well known estimators that force moment conditions on the kernel and so sacrifice positivity of the curve estimate issues of equivariance and variance stabilitization are treated

modelling dichotomously marked muscle fibre configurations human skeletal muscle consists of contractile elements fibres that may be differentiated according to their physiological and biochemical properties the different types of fibre are distributed throughout each muscle with the pattern when viewed as a cross section of cell distribution being an important determinant of the functional properties of each muscle it is well known that the proportions and distributions of muscle fibre types change with advancing age or disease but few studies have quantitatively investigated these changes a better knowledge of the nature of changes in muscle fibre distributions is an essential requirement for future development of therapies and interventions directed at maintaining or restoring good muscle function in this work we examine several statistical methods designed to gauge the departure of a dichotomously labelled muscle fibre distribution from that of a random fibre type dispersal these methods are also applicable to a wide range of biological investigations in which the spatial distribution of cells or specimens underpins an important biological principle this work includes the proposal of a novel technique based on weighted kernel smoothed density ratios which can account for the variable areas of the individual fibres we illustrated the methodology by using a number of real data examples and we employed a comprehensive set of simulations to assess the empirical power and false positive rates of these tests copyright 2013 john wiley sons ltd

numerical transformation methods blasius problem and its variants blasius problem is the simplest nonlinear boundary layer problem we hope that any approach developed for this epitome can be extended to more difficult hydrodynamics problems with this motivation we review the so called topfer transformation which allows us to find a non iterative numerical solution of the blasius problem by solving a related initial value problem and applying a scaling transformation the applicability of a non iterative transformation method to the blasius problem is a consequence of its partial invariance with respect to a scaling group several problems in boundary layer theory lack this kind of invariance and cannot be solved by non iterative transformation methods to overcome this drawback we can modify the problem under study by introducing a numerical parameter and require the invariance of the modified problem with respect to an extended scaling group involving this parameter then we apply initial value methods to the most recent developments involving variants and extensions of the blasius problem

reformulation of boundary value problems into standard form boundary value problems in ode s arising in various applications are frequently not in the standard form required by the currently existing software however many problems can be converted to such a form thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software here various conversion devices are surveyed

a novel approach to the numerical solution of boundary value problems on infinite intervals the classical numerical treatment of two point boundary value problems on infinite intervals is based on the introduction of a truncated boundary instead of infinity where appropriate boundary conditions are imposed then the truncated boundary allowing for a satisfactory accuracy is computed by trial motivated by several problems of interest in boundary layer theory here we consider boundary value problems on infinite intervals governed by a third order ordinary differential equation we highlight a novel approach to define the truncated boundary the main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero in the proposed formulation the truncated boundary is an unknown free boundary and has to be determined as part of the solution for the numerical solution of the free boundary formulation a noniterative and an iterative transformation method are introduced furthermore we characterize the class of free boundary value problems that can be solved noniteratively a nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved moreover the falkner skan equation with relevant boundary conditions is considered and representative results obtained by the iterative transformation method are listed for the homann flow all the obtained numerical results clearly indicate the effectiveness of our approach finally we discuss the possible extensions of the proposed approach and for the question of a priori error analysis

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity