grid generation techniques in computational fluid dynamics revue des differentes methodes on montre que les regions tridimensionnelles sont mieux traitees par fragmentation du champ en sous regions avec generation de grilles dans chaque sous region interet des grilles adaptatives couplees a des solutions physiques

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

coordinate generation with precise controls over mesh properties abstract coordinate generation techniques with precise controls over mesh properties are mathematically developed the controls are precise because mesh properties can be explicitly specified within a local region independent of the mesh elsewhere the local regions can adjoin boundaries where a particular mesh form can greatly simplify a problem or can be used for a smooth juncture between distinct coordinate systems where in effect a branch cut with a prescribed geometry and mesh distribution can be obtained away from boundaries a local region can be given a particular mesh form to model internal objects or to simplify a problem

a chebyshev rational chebyshev spectral method for the helmholtz equation in a sector on the surface of a sphere defeating corner singularities when the boundaries of a domain meet at an angle the solutions to an elliptic partial differential equation will usually be singular at the corner using the example of the helmholtz equation on the surface of a sphere in a domain bounded by meridians we show how corner singularities can be defeated by mapping the corner to infinity by applying a chebyshev series in longitude and a rational chebyshev series in the mercator coordinate y arctanh cos colatitude we obtain an exponential rate of convergence despite the corner singularities

the obstacle problem for the p laplacian via optimal stopping of tug of war games we present a probabilistic approach to the obstacle problem for the p laplace operator the solutions are approximated by running processes determined by tug of war games plus noise and letting the step size go to zero not unlike the case when brownian motion is approximated by random walks rather than stopping the process when the boundary is reached the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

superposition in the p laplace equation abstract that a superposition of fundamental solutions to the p laplace equation is p superharmonic even in the non linear cases p 2 has been known since m crandall and j zhang published their paper another way to say harmonic in 2003 we give a simple proof and extend the result by means of an explicit formula for the p laplacian of the superposition

another way to say harmonic it is known that solutions of u n i j 1 u xi u xj u xjxj 0 that is the harmonic functions are exactly those functions having a comparison property with respect to the family of translates of the radial solutions g x a x we establish a more difficult linear result a function in r n is harmonic if it has the comparison property with respect to sums of n translates of the radial harmonic functions g x a x 2 n for n 2 and g x b ln x for n 2 an attempt to generalize these results for u 0 p and u 0 p 2 to the general p laplacian leads to the fascinating discovery that certain sums of translates of radial p superharmonic functions are again p superharmonic mystery remains the class of p superharmonic functions so constructed for p 2 does not suffice to characterize p subharmonic functions

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

rational chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial growth special basis functions abstract in the method of matched asymptotic expansions one is often forced to compute solutions which grow as a polynomial in y as y similarly the integral or repeated integral of a bounded function f y is generally unbounded also the k th integral of a function f y solves we describe a two part algorithm for solving linear differential equations on y where u y grows as a polynomial as y first perform an explicit analytic transformation to a new unknown v so that v is bounded second expand v as a rational chebyshev series and apply a pseudospectral or galerkin discretization for our examples it is convenient to perform a preliminary step of splitting the problem into uncoupled equations for the parts of u which are symmetric and antisymmetric with respect to y 0 but although this is very helpful when applicable it is not necessary for the integral and interated integrals and for constant coefficient differential equations in general the galerkin matrices are banded with very low bandwidth we derive an improvement on the last coefficient error estimate of the author s book which applies to series with a subgeometric rate of convergence as is normally true of rational chebyshev expansions

automatic algebraic coordinate generation a computer software system has been developed to automatically generate two dimensional coordinates from algebraic transformations for topologically complex regions a smooth assembly of the transformations can be used to automatically produce a composite mesh where a general gridded format is retained the algebraic mesh generation system consists of a collection of operator subroutines which are applied to an established data structure and which automatically perform the necessary parts of mesh construction from a sequence of multisurface transformations the system operators are discussed taking into account the data base the order of application direct surface generators geometric surface operators surface generators from existing surfaces transverse operators mesh operators assembly operators and data visualization operators attention is given to applications related to airfoils

the geometrical construction of pointwise distributions on curves a method is developed to generate desirable pointwise distributions along curves this is accomplished with a simple geometrical construction which provides a global parameter for curvature clustering together with other parameters for arbitrary local clustering specifications the level of available precision is considerable in that exact numbers of points can be assigned to both local clusters and to curvature simultaneously with specified spacing from the endpoints the basic construction simply involves the generation of an auxiliary curve along outward normal directions from the given one the distribution results when uniform arc length increments are taken along the auxiliary curve and are projected back along the normals to our given curve this construction can be applied either directly or in the form of equivalent weight functions moreover it is valid regardless of whether the curve lies in euclidian space or in surfaces and regardless of the dimensionality of the space in which the curve lies

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

optimal error estimates of the chebyshev legendre spectral method for solving the generalized burgers equation in this paper the chebyshev legendre collocation method is applied to the generalized burgers equation optimal error estimate of the method is proved for the problem with the dirichlet boundary conditions also a legendre galerkin chebyshev collocation method is given for the generalized burgers equation the scheme is basically formulated in the legendre spectral form but with the nonlinear term being treated by the chebyshev collocation method so that the scheme can be implemented at chebyshev gauss lobatto points efficiently optimal order convergence is also obtained through coupling estimates in the l2 norm and the h1 norm

some notes on supersolutions of fractional p laplace equation abstract this paper gives a further understanding of supersolutions to the regional fractional p laplace equation to do so the obstacle problem and the fractional superharmonic functions associated with this equation are further studied in particular we clarify the connections among supersolutions of the fractional p laplace equation the fractional superharmonic functions and solutions to the obstacle problem

a spectral method for unbounded domains a spectral method for an unbounded domain is presented rational basis functions which are algebraically mapped legendre functions are used for expansion in the radial direction of polar coordinates r or r z they satisfy the pole condition exactly at the coordinate singularity and their behavior asr is suitable for expanding smooth functions which decay algebraically or exponentially asr the method is not stiff when it is applied to initial value problems despite the presence of the coordinate singularity solenoidal vector fields are treated efficiently by the toroidal and poloidal decomposition which reduces the number of dependent variables from 3 to 2 examples include the computation of vortex dynamics in two and three dimensions

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

chebyshev rational spectral and pseudospectral methods on a semi infinite interval a weighted orthogonal system on the half line based on the chebyshev rational functions is introduced basic results on chebyshev rational approximations of several orthogonal projections and interpolations are established to illustrate the potential of the chebyshev rational spectral method a model problem is considered both theoretically and numerically error estimates for the chebyshev rational spectral and pseudospectral methods are established preliminary numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach

maximum principle for pucci equations with sublinear growth in du and its applications abstract it is obtained that there exist strong solutions of pucci extremal equations with sublinear growth in d u and measurable ingredients it is proved that a strong maximum principle holds in a local sense in lemma 4 1 although even the weak maximum principle fails by using this existence result it is shown that the abp type maximum principle and the weak harnack inequality for viscosity solutions hold true as an application the holder continuity for viscosity solutions of possibly singular quasilinear equations is established