effect of symmetry to the structure of positive solutions in nonlinear elliptic problems ii abstract we consider the problem u hu f u 0 in r u 0 on r u 0 in r where r x r n r 1

multi peak solutions for the henon equation with slightly subcritical growth we study the dirichlet problem for the henon equation left begin array ll delta u x alpha u frac n 2 n 2 varepsilon hbox in omega u 0 hbox in omega u 0 hbox on partial omega end array right where is the unit ball in mathbb r n with n 3 the power is positive and varepsilon is a small positive parameter we prove that for every integer k 1 the above problem has a solution which blows up at k different points of as varepsilon goes to zero we also show that the ground state solution which blows up at one point is unique

multiple solutions for a henon like equation on the annulus abstract for the equation u x 2 u p 1 1 x 3 we prove the existence of two solutions for large and of two additional solutions when p is close to the critical sobolev exponent 2 2 n n 2 a symmetry breaking phenomenon appears showing that the least energy solutions cannot be radial functions

a note on the radial solutions for the supercritical henon equation we prove the existence of a positive radial solution for the henon equation with arbitrary growth the solution is found by means of a shooting method and turns out to be an increasing function of the radial variable some numerical experiments suggest the existence of many positive oscillating solutions

a mixed variable continuously deforming finite element method for parabolic evolution problems part i the variational formulation for a single evolution equation the objective of the research presented here was to develop a generic adaptive computational method for porous media evolution problems that involve coupled heat flow fluid flow and species transport processes with sharply defined phase change interfaces in this paper we examine the general least squares variational approach and develop the conceptual framework for a rate least squares variational formulation of a continuously deforming mixed variable finite element method for solving highly non linear time dependent partial differential equations in part ii of this paper1 we extend the formulation given here for a single evolution equation to a system of coupled evolution equations in part iii2 we discuss in detail the numerical procedures that were implemented in a computer program and present several numerical examples that demonstrate the performance of this computational method

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

duality in fuzzy linear systems according to fuzzy arithmetic dual fuzzy linear system can not be replaced by a fuzzy linear system in this paper we investigate the existence of a solution of duality fuzzy linear equation systems two necessary and sufficient conditions for the solution existence are given

new type of solutions to a slightly subcritical henon type problem on general domains abstract we consider the following slightly subcritical problem e u x u p 1 e u in u 0 on where is a smooth bounded domain in r n 3 n 6 p n 2 n 2 is the sobolev critical exponent e is a small positive parameter and c 2 is a positive function we assume that there exists a nondegenerate critical point of the restriction of to the boundary such that 2 p 1 0 where denotes the inner normal unit vector on given any integer k 1 we show that for e 0 small enough problem e has a positive solution which is a sum of k bubbles which accumulate at as e tends to zero we also prove the existence of a sign changing solution whose shape resembles a sum of a positive bubble and a negative bubble near the point

new type of solutions to a slightly subcritical henon type problem in dimensions 7 8 and 9 abstract inspired from the constructive method of davilla et al 10 with new ingredients we extend their existence results to dimensions 7 n 9 concerning the following henon type problem u k u p 1 e u in u 0 on where is a smooth bounded domain in r n e is a positive real parameter p 1 2 n n 2 is the critical sobolev exponent and the function k c 2 is positive satisfying condition 1 1

symmetry results for solutions of semilinear elliptic equations with convex nonlinearities abstract in this paper we study the symmetry properties of the solutions of the semilinear elliptic problem u f x u in u g x on where is a bounded symmetric domain in r n n 2 and f r r is a continuous function of class c1 in the second variable g is continuous and f and g are somehow symmetric in x our main result is to show that all solutions of the above problem of index one are axially symmetric when is an annulus or a ball g 0 and f is strictly convex in the second variable to do this we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of is a sufficient condition for the symmetry of the solution when f is a convex function

least energy solutions of the generalized henon equation in reflectionally symmetric or point symmetric domains abstract we study the generalized henon equation in reflectionally symmetric or point symmetric domains and prove that a least energy solution is neither reflectionally symmetric nor even moreover we prove the existence of a positive solution with prescribed exact symmetry

infinitely many spike solutions for the henon equation with critical growth following the constructive method of wei and yan 22 with new ingredients to take care of n 3 n 3 we prove the existence of infinitely many solutions of the henon equation u x u n 2 n 2 in the unit ball of r n r n n 3 n 3 0 0 with the dirichlet boundary condition

p version least squares finite element formulation for convection diffusion problems this paper presents a p version least squares finite element formulation of the convection diffusion equation the second order differential equation describing convection diffusion is reduced to a series of equivalent first order differential equations for which the least squares formulation is constructed using the same order of approximation for each of the dependent variables the hierarchical approximation functions and the nodal variable operators are established by first constructing the one dimensional hierarchical approximation functions of orders p and p and the corresponding nodal variable operators in and direction and then taking their products numerical results are presented and compared with analytical and numerical solutions for a two dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation the gaussian quadrature rule used to calculate the numerical values of the element matrices vectors as well as the error functional i e is established based on the highest degree of the polynomial in the integrands it is demonstrated that this quadrature rule with the present p version formulation produces excellent results for very low as well as extremely high peclet numbers 10 106 and furthermore the error functional i sum of the integrals of e2 is a monotonically decreasing function of the number of degrees of freedom as the p levels are increased for a fixed mesh it is shown that exact integration with the h version linear and parabolic elements produces inaccurate solutions at high peclet numbers results are also presented using reduced integration it is shown that the reduced integration with p version produces accurate values of the primary variable even for relatively low p levels but the error functional i when calculated using the proper integration rule has a much higher value due to errors in the derivatives of the primary variable and is a non monotonic function of the degrees of freedom as p levels are increased for a fixed mesh similar behaviour of the error functional i is also observed for the h models using linear elements when reduced integration is used although the h models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased the error values are inferior to the p version results using exact integration

infinitely many double boundary peak solutions for a henon like equation with critical nonlinearity abstract in this paper we study the following henon like equation u y 2 u p u 0 in u 0 on where 0 p n 2 n 2 y r n 1 y 3 n 4 we show that for 0 the above problem has infinitely many positive solutions concentrating simultaneously near the interior boundary x r n x 1 and the outward boundary x r n x 3 whose energy can be made arbitrarily large

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

least squares finite element method and preconditioned conjugate gradient solution a least squares variational procedure for first order systems of differential equations and an approximate formulation based on finite elements are developed error estimates a condition number bound and analysis of weighting factors are given steepest descent and conjugate gradient solution procedures are examined and an appropriate preconditioner constructed which is demonstrated to yield rapid convergence and to be insensitive to problem size numerical studies of rates of convergence for a test problem are given

p version least squares finite element formulation for axisymmetric incompressible non newtonian fluid flow abstract the paper presents a p version least squares finite element formulation lsfef for axisymmetric incompressible non newtonian fluid flow the dimensionless form of the equations describing the fluid motion are cast into a set of first order coupled partial differential equations involving non newtonian stresses as auxiliary variables the pressure velocities primary variables and the stresses auxiliary variables are interpolated using equal order c0 continuity p version hierarchical approximation functions the least squares functional or error functional is constructed using the system of coupled first order non linear partial differential equations integrated sum of squares of the errors resulting from the individual equations for the entire discretization without linearization approximations or assumptions the minimization of this least squares error functional results in finding a solution vector for which the partial derivatives of the error functional with respect to the nodal degrees of freedom becomes zero this is accomplished by using newton s method with a line search the paper presents an implementation of the power law model for non newtonian viscosity numerical examples are presented to demonstrate the convergence characteristics and the accuracy of the proposed formulation

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

fuzzy linear systems a general fuzzy linear system is investigated using the embedding approach conditions for the existence of a unique fuzzy solution to n n linear system are derived and a numerical procedure for calculating the solution is designed the applicability of the proposed model is illustrated with examples

a mixed variable continuously deforming finite element method for parabolic evolution problems part ii the coupled problem of phase change in porous media the conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed variable finite element computational scheme for a single evolution equation was presented in part i 1 in this paper part ii we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two dimensional transport problems governed by a system of coupled non linear partial differential evolution equations in particular we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts and abrupt discontinuous moving phase change interfaces in this method the potentials such as the temperature pressure and species concentration and the corresponding fluxes are permitted to jump in value across the phase change interfaces the equations and the jump conditions governing the physical phenomena which were specialized from a general multiphase multiconstituent mixture theory provided the basis for the development and implementation of a two dimensional numerical simulator this simulator can effectively resolve steep continuous fronts i e shock capturing without oscillations or numerical dispersion and can accurately represent and track discontinuous fronts i e shock fitting through adaptive grid deformation and redistribution the numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in part iii2 of this paper

solving systems of linear fuzzy equations abstract in this paper we wish to construct solutions for x to the fuzzy matrix equation ax b when the elements in a and b are triangular fuzzy numbers a is square and always non singular we argue that the previous method of solving for x based on the extension principle and regular fuzzy arithmetic should be abandoned since it too often fails to produce a solution we present six new solutions of which we show that five are identical we then adopt the common value x of these five new solutions as our solution to ax b we show that x is a fuzzy vector which is the generalization to rn of real fuzzy numbers we also show how these results pertain to and extend previous research on this problem within the area of interval analysis

solving fuzzy linear systems using a block representation of generalized inverses the moore penrose inverse abstract in this paper we present a new method for solving an m n fuzzy linear system fls a x y where the coefficient matrix a is real using the block representation of generalized inverses a necessary and sufficient condition for a block matrix to be the moore penrose inverse of the full rank matrix associated to a fls is given we obtain a necessary and sufficient condition for the existence of solutions of a fls with arbitrary real coefficient matrix the exact algebraic form with respect to the moore penrose inverse of any solution of fls of this type is established a general efficient and universal method for obtaining the exact solutions is introduced some numerical examples are presented to illustrate the proposed method

non radial solutions for a non homogeneous henon equation abstract in this paper we study a henon like equation see eq 1 where the nonlinearity f t is not homogeneous i e it is not a power by minimization on the nehari manifold we prove that for large values of the parameter there is a breaking of symmetry and non radial solutions appear this holds for sub and super critical growth of the nonlinearity f

a space time least square finite element scheme for advection diffusion equations abstract a space time least square finite element scheme is presented for the advection diffusion problems at moderate to high peclet numbers this scheme is designed to eliminate spurious oscillations and can be used to define the steady state solution as the asymptotic transient solution for large time numerical results using linear elements in a 1d space and bilinear elements in a 2d space demonstrate the accuracy and the stability of the new scheme

least energy solutions of the emden fowler equation in hollow thin symmetric domains abstract in this paper the emden fowler equation is studied in a hollow thin domain which is invariant under the action of a closed subgroup of the orthogonal group then it is proved that if the domain is thin enough a least energy solution is not group invariant

on a class of nonhomogeneous equations of henon type symmetry breaking and non radial solutions abstract in this work we study the following henon type equation div u p 2 u x a p x f u in b u 0 in b u 0 on b where b x r n x 1 is a ball centered at the origin the parameters verify the inequalities 0 a n p p n 4 0 2 p n p p n p a 1 and the nonlinearity f is nonhomogeneous by minimization on the nehari manifold we prove that for large values of the parameter there is a symmetry breaking and non radial solutions appear

a note on fuzzy linear systems in this note we show by a counterexample that the so called weak solution of a fuzzy linear system defined by friedman et al fuzzy linear systems fuzzy sets and systems 96 1998 201 209 is not always a fuzzy number vector

p version least squares finite element formulation of burgers equation a p version least squares finite element formulation for non linear problems is presented and applied to the steady state one dimensional burgers equation the second order equation is recast as a set of first order equations which permit the use of c0 elements the primary and auxiliary variables are approximated using equal order p version hierarchical approximation functions the system of non linear simultaneous algebraic equations resulting from the least squares process is solved using newton s method with a line search the use of exact and reduced quadrature rules is investigated and the results are compared the formulation is found to produce excellent results when the exact integration rule is used the combination of least squares finite element formulation and p version works extremely well for burgers equation and appears to have great potential in fluid dynamics problems

a mixed variable continuously deforming finite element method for parabolic evolution problems part iii numerical implementation and computational results in part ii a system of ordinary differential equations with respect to time was derived for solving a system of three coupled evolution equations by the deforming grid mixed variable least squares rate variational finite element method in this paper we present the method we have adopted to integrate with respect to time the resulting spatially discretized system of non linear differential equations we present computational results obtained using the code in which this deforming mixed finite element method was implemented the problems were selected to exercise these features with the objective of demonstrating that the formulation is correct and that the numerical procedures adopted converge to the correct solutions

continuation techniques for a penalty approximation of the navier stokes equations continuation techniques are used to solve a penalty finite element approximation of the navier stokes equations sufficient conditions are given for convergence of the euler newton continuation method in a reynolds parameter to an isolated solution of the finite element problem numerical results are presented for the driven cavity problem the approach is extended to arc length continuation to treat problems with singular points on the continuation path

interval linear systems as a necessary step in fuzzy linear systems this article clarifies what it means to solve a system of fuzzy linear equations relying on the fact that they are a direct extension of interval linear systems of equations already studied in a specific interval mathematics literature we highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals each of which based on a generalization of scalar equality to intervals each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals we also consider the smallest multidimensional intervals containing the solution sets we propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals this unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

non radial least energy solutions of the generalized henon equation in this paper we study the generalized henon equation with a radial coefficient function in the unit ball and show the existence of a positive non radial solution our result is applicable to a wide class of coefficient functions our theorem ensures that if the ratio of the density of the coefficient function in x a to that in a x 1 is small enough and a is sufficiently close to 1 then a least energy solution is not radially symmetric

existence of positive solutions for the henon equation involving critical sobolev terms abstract let n 3 2 2 n n 2 and r n be a bounded domain with a smooth boundary and 0 our purpose in this paper is to consider the existence of solutions of henon equation u x x u x 2 1 in u 0 in u 0 on where 0

existence of infinitely many spike solutions for a critical henon type biharmonic equation abstract following the idea of wei and yan 32 with new ingredients to take care of the critical dimension n 5 we construct infinitely many solutions of the biharmonic equation 2 v x v n 4 n 4 in the unit ball of r n n 5 0 with the navier conditions v v 0 on the boundary

adaptive refinement for least squares finite elements with element by element conjugate gradient solution an error analysis and an adaptive refinement scheme are developed for use with a class of least squares finite element methods the error indicators for the refinement procedure are based on element residuals which are calculated as part of the least squares method the solution scheme used in the supporting numerical studies employs an elment by element conjugate gradient algorithm consequently storage and computation are not influenced by nodal numbering

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100