gradient estimates in orlicz space for nonlinear elliptic equations abstract in this paper we generalize gradient estimates in l p space to orlicz space for weak solutions of elliptic equations of p laplacian type with small bmo coefficients in reifenberg flat domains our results improve the known results for such equations using a harmonic analysis free technique

product integration rules for chebyshev weight functions with chebyshev abscissae we study two product integration rules one for the chebyshev weight of the first kind based on the chebyshev abscissae of the second kind and another one constructed the other way around i e relative to the chebyshev weight of the second kind and based on the chebyshev abscissae of the first kind the new rules are shown to have positive weights given by explicit formulae furthermore we determine the precise degree of exactness and we compute the variance of the quadrature formulae we examine their definiteness or nondefiniteness and we obtain error bounds for these formulae either asymptotically optimal by peano kernel methods or for analytic functions by hilbert space techniques in addition the convergence of the quadrature formulae is shown not only for riemann integrable functions on 1 1 but also by generalizing a result of rabinowitz for functions having a monotonic singularity at one or both endpoints of 1 1

multiple solutions for a class of p x laplacian equations in involving the critical exponent in this paper we first establish a principle of concentration compactness in then based on this concentration compactness principle we study the existence of solutions for a class of p x lapl

nodal domain distribution for a nonintegrable two dimensional anharmonic oscillator we investigate the transition from integrable to chaotic dynamics in the quantum mechanical wave functions from the point of view of the nodal structure by employing a two dimensional quartic oscillator we find that the number of nodal domains is drastically reduced as the dynamics of the system changes from integrable to nonintegrable and then gradually increases as the system becomes chaotic the number of nodal intersections with the classical boundary as a function of the level number shows a characteristic dependence on the dynamics of the system too we also calculate the area distribution of nodal domains and study the emergence of the power law behavior with the fisher exponent in the chaotic limit

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and l1 data abstract in this paper we prove the existence and uniqueness of both renormalized solutions and entropy solutions for nonlinear parabolic equations with variable exponents and l 1 data and moreover we obtain the equivalence of renormalized solutions and entropy solutions

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

properties of nodal domains in a pseudointegrable barrier billiard we investigate experimentally the scaling properties of the number of nodal domains of the wave functions and the distribution of the nodal domain areas in a pseudointegrable barrier billiard the number of nodal domains is smaller than expected for billiards whose dynamics is chaotic or regular this reduction is explained by the appearance of superscars in the barrier billiard i e regular structures which are embedded into the underlying chaotic wave functions furthermore the area distribution of nodal domains follows the predictions of a percolation model for chaotic systems

recent developments in numerical methods for fully nonlinear second order partial differential equations this article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations pdes a relatively new subarea within numerical pdes due to their ever increasing importance in mathematics itself e g differential geometry and pdes and in many scientific and engineering fields e g astrophysics geostrophic fluid dynamics grid generation image processing optimal transport meteorology mathematical finance and optimal control numerical solutions to fully nonlinear second order pdes have garnered a great deal of interest from the numerical pde and scientific communities significant progress has been made for this class of problems in the past few years but many problems still remain open this article intends to introduce these current advancements and new results to the siam community and generate more interest in numerical methods for fully nonlinear pdes

nodal domains in open microwave systems nodal domains are studied both for real psir and imaginary part psii of the wave functions of an open microwave cavity and found to show the same behavior as wave functions in closed billiards in addition we investigate the variation of the number of nodal domains and the signed area correlation by changing the global phase phig according to psir ipsii eiphig psir ipsii this variation can be qualitatively and the correlation quantitatively explained in terms of the phase rigidity characterizing the openness of the billiard

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

numerical integration in the presence of an interior singularity abstract the convergence of sequences of integration rules applied to an improperly integrable function with an interior singularity is investigated the rules studied include the gauss jacobi rules and the interpolatory integration rules based on the zeros of 1 x 2 m p n 2m x m 0 1 for certain values of and the results are then applied to the study of the convergence of hunter s method for cauchy principal value integrals

gradient estimates for higher order elliptic equations on nonsmooth domains abstract we establish optimal gradient estimates in orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane as a consequence we prove the classical w m p m 1 2 1 p estimates for such a higher order equation our results easily extend to higher order elliptic and parabolic systems

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

integral formulas for chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions we evaluate explicitly the integrals 1 1 n t r t dt r 1 with the n being any one of the four chebyshev polynomials of degree n these integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with chebyshev abscissae when the function to be integrated is analytic in a domain containing 1 1 in its interior

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

geometric characterization of nodal domains the area to perimeter ratio in an attempt to characterize the distribution of forms and shapes of nodal domains in wavefunctions we define a geometric parameter the ratio between the area of a domain and its perimeter measured in units of the wavelength we show that the distribution function can distinguish between domains in which the classical dynamics is regular or chaotic for separable surfaces we compute the limiting distribution and show that it is supported on a compact interval which is independent of the properties of the surface in systems which are chaotic or in random waves the area to perimeter distribution has substantially different features which we study numerically we compare the features of the distribution for chaotic wavefunctions with the predictions of the percolation model to find agreement but only for nodal domains which are big with respect to the wavelength scale this work is also closely related to and provides a new point of view on isoperimetric inequalities

on a corrected fejer quadrature formula of the second kind we consider an interpolatory quadrature formula having as nodes the zeros of the nth degree chebyshev polynomial of the second kind on which the fejer formula of the second kind is based and the additional points pm tau c pm cos frac pi 2 n 1 c cos 2 n 1 the new formula is shown to have positive weights given by explicit formulae furthermore we determine the precise degree of exactness and we obtain optimal error bounds for this formula either by peano kernel methods or by hilbert space techniques for analytic functions and 1 le n le 40 1 n 40 in addition the convergence of the quadrature formula is shown not only for riemann integrable functions on 1 1 1 1 but also for functions having monotonic singularities at pm 1 the new formula has essentially the same rate of convergence as and it is therefore an alternative to the well known clenshaw curtis formula

interpolatory quadrature formulae with chebyshev abscissae of the third or fourth kind abstract we consider interpolatory quadrature formulae relative to the legendre weight function on 1 1 having as nodes the zeros of the nth degree chebyshev polynomial of the third or fourth kind szego has shown that the weights of these formulae are all positive we derive explicit formulae for the weights and subsequently use them to establish the convergence of the quadrature formulae for functions having a monotonic singularity at one or both endpoints of 1 1 moreover we generate two new quadrature formulae by adding 1 1 to the sets of nodes considered previously and show that these new formulae have almost all weights positive exceptions occurring only among the weights corresponding to 1 1 also we determine the precise degree of exactness of all the quadrature formulae in consideration we obtain asymptotically optimal error bounds for these formulae and show that almost all of them are nondefinite exceptions occurring only among the formulae with a small number of nodes

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients abstract in this paper based on the theory of variable exponent spaces we study the higher integrability for a class of nonlinear elliptic equations with variable growth and discontinuous coefficients under suitable assumptions we obtain a local gradient estimate in orlicz space for weak solution

the kink phenomenon in fejer and clenshaw curtis quadrature the fejer and clenshawcurtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions when n the number of points in the integration rule increases the error does not decay to zero evenly but does so in two distinct stages for n less than a critical value the error behaves like o 2n where is a constant greater than 1 for these values of n the accuracy of both the fejer and clenshawcurtis rules is almost indistinguishable from that of the more celebrated gausslegendre quadrature rule for larger n however the error decreases at the rate o n i e only half as fast as before convergence curves typically display a kink where the convergence rate cuts in half in this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon

error estimate for a corrected clenshaw curtis quadrature rule a new endpoint corrected rule for the clenshaw curtis c c quadrature is proposed to improve the convergence rate the error behavior is compared analytically and numerically to the c c rule and related quadrature rules the fejer rules of the first and second kind and the basu rule

harnack s inequality and the strong p laplacian abstract we study solutions of the strong p laplace equation we show that in contrast to p laplace solutions these solutions satisfy the ordinary scale invariant harnack inequality as consequences we derive the strong maximum principle and global integrability of solutions

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

a convergent difference scheme for the infinity laplacian construction of absolutely minimizing lipschitz extensions this article considers the problem of building absolutely minimizing lipschitz extensions to a given function these extensions can be characterized as being the solution of a degenerate elliptic partial differential equation the infinity laplacian for which there exist unique viscosity solutions a convergent difference scheme for the infinity laplacian equation is introduced which arises by minimizing the discrete lipschitz constant of the solution at every grid point existence and uniqueness of solutions to the scheme is shown directly solutions are also shown to satisfy a discrete comparison principle solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation

error estimates for interpolatory quadrature formulae in this paper we study the remainder of interpolatory quadrature formulae for this purpose we develop a simple but quite general comparison technique for linear functionals applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one by repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all with the help of this method we obtain asymptotically best possible error bounds for the clenshaw curtis quadrature and other polya type formulae our comparison technique can also be applied to the problem of definiteness i e the question whether the remainderr f of a formula of orderm can be represented asc f m by successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases finally we ask for good and worst quadrature formulae within certain classes we shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the gauss type formulae are worst interpreted in terms of peano kernels our theorem yields results on monosplines which may be of interest in themselves

on the convergence of closed interpolatory integration rules based on the zeros of gegenbauer polynomials abstract the gegenbauer polynomials c n x are orthogonal with respect to the weight function 1 x2 1 2 it is known that the interpolatory integration rules based on the zeros of 1 x2 c n 2 x converge for all riemann integrable functions for 1 2 4 this is shown to hold also for 1 2 1 2

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

convergent finite difference methods for one dimensional fully nonlinear second order partial differential equations this paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations pdes in 1 d the goal of the paper is to extend the successful framework of monotone consistent and stable finite difference methods for first order fully nonlinear hamilton jacobi equations to second order fully nonlinear pdes such as monge ampere and bellman type equations new concepts of consistency generalized monotonicity and stability are introduced among them the generalized monotonicity and consistency which are easier to verify in practice are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear hamilton jacobi equations the main component of the proposed framework is the concept of a numerical operator and the main idea used to design consistent generalized monotone and stable finite difference methods is the concept of a numerical moment these two new concepts play the same roles the numerical hamiltonian and the numerical viscosity play in the finite difference framework for first order fully nonlinear hamilton jacobi equations in the paper two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear pdes the first class contains lax friedrichs like methods which also are proved to be stable and the second class contains godunov like methods numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

new interpolatory quadrature formulae with gegenbauer abscissae we study interpolatory quadrature formulae relative to the legendre weight function on 1 1 having as nodes the zeros of the nth degree gegenbauer polynomial plus one of the points 1 or 1 in particular we establish the convergence or nonconvergence for continuous and riemann integrable functions on 1 1 we determine the precise degree of exactness we obtain asymptotically optimal error bounds and we examine the definiteness or nondefiniteness of these formulae in addition we try numerically to investigate the question of positivity of all quadrature weights and to fill a gap regarding the definiteness or nondefiniteness property the paper concludes by comparing the results derived here for the quadrature formulae in question with those that have been previously obtained for the corresponding open and closed interpolatory formulae with gegenbauer abscissae

on the nodal count statistics for separable systems in any dimension we consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in an arbitrary dimension we give an explicit expression for the limiting distribution of normalized nodal counts and analyse some of its universal properties our results are illustrated by the detailed discussion of simple examples and numerical nodal count distributions

elliptic equations with bmo coefficients in lipschitz domains in this paper we study inhomogeneous dirichlet problems for elliptic equations in divergence form optimal regularity requirements on the coefficients and domains for the w 1 p 1 p estimates are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms the domain is supposed to have lipschitz boundary with small lipschitz constant these conditions for the w 1 p theory do not just weaken the requirements on the coefficients they also lead to a more general geometric condition on the domain

on finite element methods for fully nonlinear elliptic equations of second order for the first time we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard c 1 finite element method fem we consider throughout the paper two cases in parallel for convex bounded polyhedral domains in bf r n or for c 2 bounded domains in bf r 2 we prove stability and convergence for the corresponding conforming or nonconforming c 1 fem respectively the results for equations and systems of orders 2 and 2m and quadrature approximations appear elsewhere the classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator the consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated the stability has to be proved in an unusual way this is the hard core of the paper essential tools are linearization a compactness argument the interplay between the weak and strong form of the linearized operator and a new regularity result for solutions of finite element equations an essential basis for our proofs are davydov s results for c 1 fes on polyhedral domains in bf r n or of local degree 5 for c 2 domains in bf r 2 better convergence and extensions to bf r n for c 2 domains are to be expected from his forthcoming results on curved domains our proof for the second case in bf r n includes the first essentially as a special case the method applies to quasi linear elliptic problems not in divergence form as well a discrete newton method is shown to converge locally quadratically essentially independently of the actual grid size by the mesh independence principle

new interpolatory quadrature formulae with chebyshev abscissae abstract we study interpolatory quadrature formulae relative to the legendre weight function on 1 1 having as nodes the zeros of any one of the four chebyshev polynomials of degree n plus one of the points 1 or 1 in particular we derive explicit formulae for the weights and examine their positivity we determine the precise degree of exactness we obtain asymptotically optimal error bounds and we examine the definiteness f these quadrature formulae in addition we establish their convergence for riemann integrable functions on 1 1 as well as for functions having a monotonic singularity at 1 or 1

the use of rational functions in numerical quadrature quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree but also for rational functions having the same or the most important poles as the function to be integrated constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts the superiority of such rational polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

nodal domains of a non separable problem the right angled isosceles triangle we study the nodal set of eigenfunctions of the laplace operator on the right angled isosceles triangle a local analysis of the nodal pattern provides an algorithm for computing the number n of nodal domains for any eigenfunction in addition an exact recursive formula for the number of nodal domains is found to reproduce all existing data eventually we use the recursion formula to analyse a large sequence of nodal counts statistically our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes furthermore we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics

the error bounds of gauss quadrature formulae for the modified weight functions of chebyshev type abstract in this paper we consider the gauss quadrature formulae corresponding to some modifications of each of the four chebyshev weights considered by gautschi and li in 4 as it is well known in the case of analytic integrands the error of these quadrature formulas can be represented as a contour integral with a complex kernel we study the kernel of the mentioned quadrature formulas on suitable elliptic contours in such a way that the behavior of its modulus is analyzed in a rather simple manner allowing us to derive some effective error bounds in addition some numerical examples checking the accuracy of such error bounds are included

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

an evaluation of clenshaw curtis quadrature rule for integration w r t singular measures this work is devoted to the study of quadrature rules for integration with respect to w r t general probability measures with known moments automatic calculation of the clenshaw curtis rules is considered and analyzed it is shown that it is possible to construct these rules in a stable manner for quadrature w r t balanced measures in order to make a comparison gauss rules and their stable implementation for integration w r t balanced measures are recalled convergence rates are tested in the case of binomial measures

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved

a multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data