a general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids the overlapping covolume case we present a general framework for constructing and analyzing finite volume methods applied to the mixed formulation of second order elliptic problems on quadrilateral grids the control volumes or covolumes in the grids overlap an overlapping finite volume method of this type was first introduced by russell in t f russell tech report 3 reservoir simulation research corp tulsa ok 1995 and was tested for a variety of problems on rectangular and quadrilateral grids in z cai et al comput geosci 1 1997 pp 289 315 later in s h chou and d y kwak siam j numer anal 37 2000 pp 758 771 chou and kwak reformulated it as their mixed covolume method and proved optimal order error estimates using the covolume methodology from s h chou math comp 66 1997 pp 85 104 and s h chou and d y kwak siam j numer anal 35 1998 pp 494 507 however their treatment was restricted to the case of diagonal coefficient tensor and rectangular grids since a different approach was needed for the quadrilateral distorted rectangular case in this paper we give a new framework which can handle not only the rectangular anisotropic case but also the anisotropic and irregular grid cases in which the locally supported test functions are images of the natural unit coordinate vectors under the piola transformation our theory sheds light on how to create new test functions using quadratures and now covers russell s quadrilateral case

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

the accuracy of cell vertex finite volume methods on quadrilateral meshes for linear first order hyperbolic equations in two dimensions we restate the cell vertex finite volume scheme as a finite element method on structured meshes consisting of distorted quadrilaterals the global error is shown to be of second order in various mesh dependent norms provided that the quadrilaterals are close to parallelograms in the sense that the distance between the midpoints of the diagonals is of the same order as the measure of the guadrilateral on tensor product nonuniform meshes the cell vertex scheme coincides with the familiar box scheme in this case second order accuracy is shown without any additional assumption on the regularity of the mesh which explains the insensitivity of the cell vertex scheme to mesh stretching in the coordinate directions observed in practice 17 refs

convergence of the nonconforming wilson element for a class of nonlinear parabolic problems this paper deals with the convergence properties of the nonconforming quadrilateral wilson element for a class of nonlinear parabolic problems in two space dimensions optimal h1 and l2 error estimates for the continuous time galerkin approximations are derived it is also shown for rectangular meshes that the gradient of the wilson element solution possesses superconvergence and that the l error on the gradient is of order h log 1 h

superconvergence and extrapolation for mixed finite element methods on rectangular domains asymptotic expansions for the rt raviart thomas mixed finite element approximation by the lowest order rectangular element associated with a second order elliptic equation on a rectangular domain are derived superconvergence for the vector field along the gauss lines is obtained as a result of the expansion a procedure of postprocessed extrapolation is presented for the scalar field as well as procedures of pure richardson extrapolation for both the vector and the scalar fields

superconvergence of both the crouzeix raviart and morley elements in this paper a new method is proposed to prove the superconvergence of both the crouzeix raviart and morley elements the main idea is to fully employ equivalences with the first order raviart thomas element and the first order hellan herrmann johnson element respectively in this way some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements finally the superconvergence of one and a half order by postprocessing is proved for both nonconforming elements

mixed finite element methods for general quadrilateral grids abstract we study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the h div norm this new element is designed so that the h div projection h satisfies h p h div a rigorous optimal order error estimate is carried out by proving a modified version of the bramble hilbert lemma for vector variables we show that a local h div projection reproducing certain polynomials suffices to yield an optimal l 2 error estimate for the velocity and hence our approach also provides an improved error estimate for original raviart thomas element of lowest order numerical experiments are presented to verify our theory

mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems we construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order pdes in the state variable e g pressure and its flux e g darcy velocity an important point is that no staggered grids or covolumes are used to stabilize the system only a single primary grid system is adopted and the degrees of freedom are imposed on the interfaces the approximate flux is sought in the lowest order raviart thomas space and the pressure field in the rotated q1 nonconforming space furthermore we demonstrate that the present finite volume method can be interpreted as a rotated q1 nonconforming finite element method for the pressure with a simple local recovery of flux numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

a convergence condition for the quadrilateral wilson element the paper deals with the convergence properties of the nonconforming quadrilateral wilson element which violates the patch test the convergence of the element is proved under a certain condition on mesh subdivisions without any modifications of the variational formulation this result extends the range of applicability of wilson s element the necessity of the proposed condition is also discussed

the fractal dimension of the two well potential strange attractor abstract the dimension of the duffing holmes two well potential strange attractor has been measured using both numerical solutions and experimental data from chaotic vibrations of a buckled beam using numerical simulation the fractal dimension of the poincare section of the strange attractor has been calculated as a function of damping using a correlation function technique the dimension of the poincare section is between one and two with higher damping resulting in a smaller dimensional attractor an equivalent cantor set for the attractor is defined the dimension seems to be relatively insensitive to the phase of the poincare section an analytical expression for the dimension as a function of damping is proposed the dimension algorithm is applied to experimentally generated poincare maps from chaotic vibrations of a buckled beam the calculation yields a dimension of the poincare map between one and two and seems to be insensitive to the phase of the poincare section

a priori and a posteriori analysis for a locking free low order quadrilateral hybrid finite element for reissner mindlin plates this paper proposes a quadrilateral finite element method of the lowest order for reissner mindlin r m plates on the basis of hellinger reissner variational principle which includes variables of displacements shear stresses and bending moments this method uses continuous piecewise isoparametric bilinear interpolation for the approximation of transverse displacement and rotation the piecewiseindependent shear stress bending moment approximation is constructed by following a self equilibrium criterion and a shear stress enhanced condition a priori and reliable a posteriori error estimates are derived and shown to be uniform with respect to the plate thickness t numerical experiments confirm the theoretical results

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

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

a q1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh abstract in this paper we study a q 1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh it is known that the coercivity is the basement for some other theoretical results stability h 1 and l 2 error estimates etc and the existing results were mainly obtained on h 1 parallelogram meshes with scalar diffusion coefficients for the cases of full diffusion tensors and arbitrary convex quadrilateral meshes we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form based on this result a sufficient condition is suggested to guarantee the coercivity of the scheme more interesting is that this sufficient condition covers the traditional h 1 parallelogram mesh assumption and has an explicit expression by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h 0 moreover an h 1 error estimate is obtained without the h 1 parallelogram assumption and some numerical results are also provided to validate the theoretical results

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

the lowest order weak galerkin finite element method for the darcy equation on quadrilateral and hybrid meshes abstract this paper investigates the lowest order weak galerkin finite element method for solving the darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles in this approach the pressure is approximated by constants in element interiors and on edges the discrete weak gradients of these constant basis functions are specified in local raviart thomas spaces specifically r t 0 for triangles and unmapped r t 0 for quadrilaterals these discrete weak gradients are used to approximate the classical gradient when solving the darcy equation the method produces continuous normal fluxes and is locally mass conservative regardless of mesh quality and has optimal order convergence in pressure velocity and normal flux when the quadrilaterals are asymptotically parallelograms implementation is straightforward and results in symmetric positive definite discrete linear systems we present numerical experiments and comparisons with other existing methods

effects of element distortions on the performance of isoparametric elements we discuss the effects of element distortions on the performance of displacement based isoparametric quadrilateral finite elements suitable example problems for both the lagrangian and the serendipity types of elements are used to show numerically the effects of element distortions the effects of angular and curved edge distortions are evaluated analytically in terms of the abilities of the elements to represent exactly various polynomial displacement fields we then solve a plane stress problem adaptively to demonstrate the effect of element distortions on the accuracy of the stress distribution the conclusion is that the lagrangian elements are not affected by angular distortions and are therefore the more reliable elements for general use

homoclinic and chaotic transitions in the rf squid abstract the results presented here are a continuation of recent work investigating the onset of homoclinic behavior in the rf squid the transition from periodic to chaotic behavior is studied through a calculation of the lyapunov exponents for the system and the threshold condition for the onset of chaotic behavior is compared with the melnikov condition for the homoclinic threshold the dimension calculated from the poincare section of the strange attracting set by means of the correlation function technique and the box counting technique is compared with the dimension derived from the lyapunov exponents

on the regularity and uniformness conditions on quadrilateral grids in the literature there are various regularity conditions as well as many different so called almost parallelogram conditions on the subdivisions of quadrilateral grids in this paper we investigate and clarify their relations

spectral loudness summation and simple reaction time the present study evaluates the relation between loudness and simple reaction time rt loudness matches between a narrowband noise 125 hz wide and a broadband noise 1500 hz were made at levels from near threshold to near 100 db spl over a similarly wide range of levels rt to each of the noise bands was also measured as reported often in previous loudness matching studies loudness summation depended strongly on level with increasing spl the level difference between the noises needed to keep them equally loud first increased to around 10 db at moderate levels and then decreased except for one listener the rt data show the same pattern the level difference needed to keep rt to the two noises the same first increased and then decreased these results show that rt is closely related to loudness but not to sensation level if rt depended on sensation level the level difference between the two noises needed to achieve equal rt would not change with spl because the difference in sensation level between two sounds is a constant overall the average results provide strong support for the contention that simple rt and loudness are closely related

equal loudness level contours for pure tones equal loudness level contours provide the foundation for theoretical and practical analyses of intensity frequency characteristics of auditory systems since 1956 equal loudness level contours based on the free field measurements of robinson and dadson br j appl phys 7 166 181 1956 have been widely accepted however in 1987 some questions about the general applicability of these contours were published h fastl and e zwicker fortschritte der akustik daga 87 pp 189 193 1987 as a result a new international effort to measure equal loudness level contours was undertaken the present paper brings together the results of 12 studies starting in the mid 1980s to arrive at a new set of contours the new contours estimated in this study are compared with four sets of classic contours taken from the available literature the contours described by fletcher and munson j acoust soc am 5 82 108 1933 exhibit some overall similarity to our proposed estimated contours in the mid frequency range up to 60 phons the contours described by robinson and dadson exhibit clear differences from the new contours these differences are most pronounced below 500 hz and the discrepancy is often as large as 14 db

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

near threshold equal loudness contours for harbor seals phoca vitulina derived from reaction times during underwater audiometry a preliminary study equal loudness functions describe relationships between the frequencies of sounds and their perceived loudness this pilot study investigated the possibility of deriving equal loudness contours based on the assumption that sounds of equal perceived loudness elicit equal reaction times rts during a psychoacoustic underwater hearing study the responses of two young female harbor seals to tonal signals between 0 125 and 100 khz were filmed frame by frame analysis was used to quantify rt the time between the onset of the sound stimulus and the onset of movement of the seal away from the listening station near threshold equal latency contours as surrogates for equal loudness contours were estimated from rt level functions fitted to mean rt data the closer the received sound pressure level was to the 50 detection hearing threshold the more slowly the animals reacted to the signal rt range 188 982 ms equal latency contours were calculated relative to the rts shown by each seal at sound levels of 0 10 and 20 db above the detection threshold at 1 khz fifty percent detection thresholds are obtained with well trained subjects actively listening for faint familiar sounds when calculating audibility ranges of sounds for harbor seals in nature it may be appropriate to consider levels 20 db above this threshold 2011 acoustical society of america