least squares mixed finite element methods for non selfadjoint elliptic problems i error estimates a least squares mixed finite element method for general second order non selfadjoint elliptic problems in two and three dimensional domains is formulated and analyzed the finite element spaces for the primary solution approximation u h and the flux approximation sigma h consist of piecewise polynomials of degree k and r respectively the method is mildly nonconforming on the boundary the cases k r and k 1 r are studied it is proved that the method is not subject to the lbb condition optimal l 2 and h 1 error estimates are derived for regular finite element partitions numerical experiments confirming the theoretical rates of convergence are presented

on the group classification of variable coefficient nonlinear diffusion convection equations we consider the variable coefficient diffusion convection equation of the form f x ut g x d u ux x h x k u ux which has considerable interest in mathematical physics biology and chemistry we present a complete group classification for this class of equations also we derive equivalence transformations between equations that admit lie symmetries furthermore we obtain mappings that connect variable and constant coefficient equations exact solutions of special forms of this equations are constructed using lie symmetries and equivalence transformations

classical point symmetries of a porous medium equation the lie group formalism is applied to deduce symmetries of the porous medium equation we study those spatial forms that admit the classical symmetry group the reduction obtained from the optimal system of subalgebras are derived some new exact solutions can be obtained

pointwise error estimates for first order div least squares finite element methods and applications to superconvergence and a posteriori error estimators first order div least squares finite element methods for second order elliptic partial differential equations are considered while there has been significant progress in error estimates for the methods to the best of our knowledge these estimates are based on the global norm of the error in this paper we provide highly localized pointwise error estimates for the primary variable u by establishing that the least squares solutions are locally higher order perturbations of the standard galerkin solutions as elementary consequences we identify the superconvergent points the set of superconvergent points for the primary function of the least squares solution is the same as that of the standard galerkin solution also we present a class of average type a posteriori error estimators for the method and conditions are given under which they are asymptotically exact or equivalent estimators in the maximum norm on each single element of the underlying mesh

performance of ds cdma systems with optimal hard decision parallel interference cancellation we study a multiuser detection system for code division multiple access cdma we show that applying multistage hard decision parallel interference cancellation hd pic significantly improves performance compared to the matched filter system in multistage hd pic estimates of the interfering signals are used iteratively to improve knowledge of the desired signal we use large deviation theory to show that the bit error probability bep is exponentially small when the number of users is fixed and the processing gain increases we investigate the exponential rate of the bep after several stages of hd pic we propose to use the exponential rate of the bep as a measure of performance rather than the signal to noise ratio snr which is often not reliable in multiuser detection models when the system is lightly loaded we show that the exponential rate of the bep remains fixed after a finite number of stages resulting in an optimal hard decision system when the number of users becomes large the exponential rate of the bep converges to log 2 2 1 4 we provide guidelines for the number of stages necessary to obtain this asymptotic exponential rate we also give chernoff bounds on the beps these estimates show that the beps are quite small as long as k o n log n when the number of stages of hd pic is fixed and even exponentially small when k o n for the optimal hd pic system and where k is the number of users in the system and n is the processing gain finally we extend the results to the case where the number of stages depends on k in a certain manner the above results are proved for a noiseless channel and we argue that we expect similar results in a noisy channel as long as the two sided spectrum of the noise decreases proportionally to n

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

enhanced group analysis and conservation laws of variable coefficient reaction diffusion equations with power nonlinearities a class of variable coefficient 1 1 dimensional nonlinear reaction diffusion equations of the general form f x ut g x unux x h x um is investigated different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements for the class under consideration the complete group classification is performed with respect to convenient equivalence groups generalized extended and conditional ones and with respect to the set of all local transformations usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results the corresponding set of admissible transformations is described exhaustively then using the most direct method we classify local conservation laws some exact solutions are constructed by the classical lie method

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

weak coupling of solutions of first order least squares method a theoretical analysis of a first order least squares finite element method for second order self adjoint elliptic problems is presented we investigate the coupling effect of the approximate solutions u h for the primary function u and h for the flux a u we prove that the accuracy of the approximate solution u h for the primary function u is weakly affected by the flux a u that is the bound for u u h 1 is dependent on but only through the best approximation for multiplied by a factor of meshsize h similarly we provide that the bound for h h div is dependent on u but only through the best approximation for u multiplied by a factor of the meshsize h this weak coupling is not true for the non selfadjoint case we provide the numerical experiment supporting the theorems in this paper

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

group classification of a class of equations arising in financial mathematics we provide a group classification of a class of nonlinearisable evolution partial differential equations which arise in financial mathematics sixteen different cases are identified for the general problem and another seven for a restricted version in the cases for which the algebra is suitable we determine the solution to the problem u 0 x u where u is a constant in addition we provide a number of solutions based upon reduction using inequivalent subalgebras

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

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

local error estimation and adaptive remeshing scheme for least squares mixed finite elements following the theme of our previous work on least squares finite elements 10 28 we describe an adaptive remeshing scheme using local residuals as the error indicator this choice of indicator is natural and exact at the element level in the norm associated with the corresponding least squares statement the remeshing strategy applied here involves mesh enrichment by point insertion in a delaunay scheme several refined grids and error plots are included for a representative model elliptic boundary value problem

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

analysis of least squares mixed finite element methods for nonlinear nonstationary convection diffusion problems some least squares mixed finite element methods for convection diffusion problems steady or nonstationary are formulated and convergence of these schemes is analyzed the main results are that a new optimal a priori l 2 error estimate of a least squares mixed finite element method for a steady convection diffusion problem is developed and that four fully discrete least squares mixed finite element schemes for an initial boundary value problem of a nonlinear nonstationary convection diffusion equation are formulated also some systematic theories on convergence of these schemes are established

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

on the accuracy of least squares methods in the presence of corner singularities abstract this paper treats elliptic problems with corner singularities finite element approximations based on variational principles of the least squares type tend to display poor convergence properties in such contexts moreover mesh refinement or the use of special singular elements do not appreciably improve matters here we show that if the least squares formulation is done in appropriately weighted spaces then optimal convergence results in unweighted spaces like l2

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

new results on group classification of nonlinear diffusion convection equations using a new method and additional conditional and partial equivalence transformations we performed group classification in a class of variable coefficient 1 1 dimensional nonlinear diffusion convection equations of the general form f x ut d u ux x k u ux we obtain new interesting cases of such equations with the density f localized in space which have non trivial invariance algebra exact solutions of these equations are constructed we also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration i e of describing all the possible partial equivalence transformations in this class

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives