a least squares method for second order noncoercive elliptic partial differential equations in this paper we consider a least squares method proposed by bramble lazarov and pasciak 1998 which can be thought of as a stabilized galerkin method for noncoercive problems with unique solutions we modify their method by weakening the strength of the stabilization terms and present various new error estimates the modified method has all the desirable properties of the original method indeed we shall show some theoretical properties that are not known for the original method at the same time our numerical experiments show an improvement of the method due to the modification

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

some least squares galerkin procedures for first order time dependent convection diffusion system abstract four least squares galerkin finite element schemes are formulated to solve first order time dependent convection diffusion systems and the convergence of these schemes is analyzed

domain decomposition for least squares finite element methods for the stokes equations in this paper we consider the solutions of the discrete linear system resulting from least squares finite element discretization applied to the stokes equations additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

a least squares based weak galerkin finite element method for second order elliptic equations in this article we introduce a least squares based weak galerkin finite element method for the second order elliptic equation this new method is shown to provide very accurate numerical approximations for both the primal and the flux variables in contrast to other existing least squares finite element methods this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon polyhedron shapes we also develop a schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element optimal order error estimates for both the primal and the flux variables are established an extensive set of numerical experiments are conducted to demonstrate the robustness reliability flexibility and accuracy of the least squares based weak galerkin finite element method the numerical examples cover a wide range of applied problems including singularly

a family of h div finite element approximations on polygonal meshes in this paper we present a family of h div compatible finite element spaces on strictly convex n gons whose construction makes use of generalized barycentric coordinates in particular for integers 0 leq k leq2 we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order k and whose vector fields have piecewise k th order polynomial normal traces along the element boundary these spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and as such their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence we present a general remedy for restoration of optimal convergence that involves polynomial corrections of vector fields and their divergence at the element level these corrections are consistent with one another and require computation of suitable polynomial projection maps in addition to the theo

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

the discrete first order system least squares the second order elliptic boundary value problem in z cai t manteuffel and s f mccormick siam j numer anal 34 1997 pp 425 454 an l2 norm version of first order system least squares fosls was developed for scalar second order elliptic partial differential equations a limitation of this approach is the requirement of sufficient smoothness of the original problem which is used for the equivalence of spaces between h1 d and h div cap h rm curl type where d 2 or 3 is the dimension by directly approximating h div cap h rm curl type space based on the helmholtz decomposition this paper develops a discrete fosls approach in two dimensions under general assumptions we establish error estimates in the l2 and h1 norms for the vector and scalar variables respectively such error estimates are optimal with respect to the required regularity of the solution a preconditioner for the algebraic system arising from this approach is also considered

mimo interaction measure and controller structure selection in this paper a mimo interaction measure is described and its use in structure selection of multivariable controllers is discussed the proposal is built upon a gramian based measure of interaction in multivariable plants and the results in this paper apply to stable processes this measure provides support for decentralized input output pairing as well as for a richer controller architecture selection in continuous and discrete time time frameworks including triangular block diagonal and sparse structures

error analysis for constrained first order system least squares finite element methods in this paper a general error analysis is provided for finite element discretizations of partial differential equations in a saddle point form with divergence constraint in particular this extends upon the work of j h adler and p s vassilevski springer proc math statist 45 springer new york 2013 pp 1 19 giving a general error estimate for finite element problems augmented with a divergence constraint and showing that these estimates are obtained for problems such as diffusion and stokes using the first order system least squares fosls finite element method the main result is that by enforcing the constraint on a mathbf h 1 equivalent fosls formulation one maintains optimal convergence of the fosls functional i e the energy norm of the error while guaranteeing the conservation of the divergence constraint i e mass conservation in some examples the error estimates and results depend on using finite elements for the constraint space that are inf sup stable when paired wi

first order least squares method with weakly imposed boundary condition for convection dominated diffusion problems we present and analyze a first order least squares method for convection dominated diffusion problems which provides robust l 2 a priori error estimate for the scalar variable even if the given data f l 2 the novel theoretical approach is to rewrite the method in the framework of discontinuous petrov galerkin dpg method and then show numerical stability by using a key equation discovered by gopalakrishnan and qiu 2014 this new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations we also show that the condition number of the global matrix is independent of the diffusion coefficient a key feature of the method is that there is no stabilization parameter chosen empirically in addition dirichlet boundary condition is weakly imposed numerical experiments verify our theoretical results and in particular show our way of weakly imposing dirichlet boundary condition is essential to the design of least squares methods numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes which are unstructured quasi uniform

computing derivatives in interval type 2 fuzzy logic systems this paper makes type 2 fuzzy logic systems much more accessible to fuzzy logic system designers because it provides mathematical formulas and computational flowcharts for computing the derivatives that are needed to implement steepest descent parameter tuning algorithms for such systems it explains why computing such derivatives is much more challenging than it is for a type 1 fuzzy logic system it provides derivative calculations that are applicable to any kind of type 2 membership functions since the calculations are performed without prespecifying the nature of those membership functions some calculations are then illustrated for specific type 2 membership functions

application of basic hypergeometric series to stable analytic continuation abstract stable sampling formula using basic hypergeometric series for reconstructing analytic functions from exponentially spaced samples is considered criterion for selecting regularizing parameter and error estimates are obtained

convergence analysis for least squares finite element approximations of second order two point boundary value problems in this paper a c 0 least squares finite element method for second order two point boundary value problems is considered the problem is recast as a first order system standard and improved optimal error estimates in maximum norms are established superconvergence estimates at interelement lobatto and gauss points are developed numerical experiments are given to illustrate theoretical results

least squares for second order elliptic problems abstract in this paper we introduce and analyze two least squares methods for second order elliptic differential equations with mixed boundary conditions these methods extend to problems which involve oblique derivative boundary conditions as well as nonsymmetric and indefinite problems as long as the original problem has a unique solution with the methods to be developed neumann and oblique boundary conditions are imposed weakly and thus avoid compatibility conditions on the finite element subspaces the resulting least squares approximations are unconditionally stable no conditions on the step size h and will be shown to converge at an optimal rate the first least squares method involves a discrete computable h 1 norm of the residual and stabilization terms consisting of the jumps at the interelement boundaries and a weighted elementwise l 2 norm of the residual over the finite elements this method is developed without the introduction of additional problem variables the second method involves the use of the flux as an additional unknown although this method is similar to the least squares method for first order systems introduced in 7 it differs in that discontinuous finite elements are allowed it is also more general in that it extends to the oblique boundary problem

div first order system ll fosll for second order elliptic partial differential equations the first order system ll fosll approach for general second order elliptic partial differential equations was proposed and analyzed in z cai et al siam j numer anal 39 2001 pp 1418 1445 in order to retain the full efficiency of the l 2 norm first order system least squares fosls approach while exhibiting the generality of the inverse norm fosls approach the fosll approach of cai et al was applied to the div curl system with added slack variables and hence it is quite complicated in this paper we apply the fosll approach to the div system and establish its well posedness for the corresponding finite element approximation we obtain a quasi optimal a priori error bound under the same regularity assumption as the standard galerkin method but without the restriction to sufficiently small mesh size unlike the fosls approach the fosll approach does not have a free a posteriori error estimator we then propose an explicit residual error estimator and establish its reliability a

on km algorithms for solving type 2 fuzzy set problems computing the centroid and performing type reduction for type 2 fuzzy sets and systems are operations that must be taken into consideration karnik mendel km algorithms are the standard ways to do these operations however because these algorithms are iterative much research has been conducted during the past decade about centroid and type reduction computations this tutorial paper focuses on the research that has been conducted to 1 improve the km algorithms 2 understand the km algorithms leading to further improved algorithms 3 eliminate the need for km algorithms 4 use the km algorithms to solve other nonfuzzy logic system problems and 5 use or not use km algorithms for general type 2 fuzzy sets and fuzzy logic systems

stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems how in a discretized model to utilize the duality and complementarity of two saddle point variational principles is considered in the paper a homology family of optimality conditions different from the conventional saddle point conditions of the domain decomposed hellinger reissner principle is derived to enhance stability of hybrid finite element schemes based on this a stabilized hybrid method is presented by associating element interior displacement with an element boundary one in a nonconforming manner in addition energy compatibility of strain enriched displacements with respect to stress terms is introduced to circumvent poisson locking

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

local error estimates for least squares finite element methods for first order system we present local energy type error estimates for first order system div least squares ls finite element methods the estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution the estimate is similar to the one for the standard galerkin methods however our estimate needs to consider the effect of error of dual flux variables since ls methods approximate the primary and dual variables simultaneously the effect of error of the dual variables is shown to be of higher order moreover our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest as an elementary consequence of the estimate asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions

discontinuous least squares finite element method for the div curl problem in this paper we consider the div curl problem posed on nonconvex polyhedral domains we propose a least squares method based on discontinuous elements with normal and tangential continuity across interior faces as well as boundary conditions weakly enforced through a properly designed least squares functional discontinuous elements make it possible to take advantage of regularity of given data divergence and curl of the solution and obtain convergence also on nonconvex domains in general this is not possible in the least squares method with standard continuous elements we show that our method is stable derive a priori error estimates and present numerical examples illustrating the method

sampling theory for not necessarily band limited functions a historical overview shannon s sampling theorem is one of the most powerful results in signal analysis the aim of this overview is to show that one of its roots is a basic paper of de la vallee poussin of 1908 the historical development of sampling theory from 1908 to the present especially the matter dealing with not necessarily band limited functions which includes the duration limited case actually studied in 1908 is sketched emphasis is put on the study of error estimates as well as on the delicate point wise behavior of sampling sums at discontinuity points of the signal to be reconstructed

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

goal oriented local a posteriori error estimators for h div least squares finite element methods we propose a goal oriented local a posteriori error estimator for h div least squares ls finite element methods our main interest is to develop an a posteriori error estimator for the flux approximation in a preassigned region of interest d subset omega the estimator is obtained from the ls functional by scaling residuals with proper weight coefficients the weight coefficients are given in terms of local mesh size h t and a function omega d depending on the distance to d this new error estimator measures the pollution effect from the outside region of d and provides a basis for local refinement in order to efficiently approximate the solution in d numerical experiments show superior performances of our goal oriented a posteriori estimators over the standard ls functional and global error estimators

analysis of first order system least squares fosls for elliptic problems with discontinuous coefficients part i first order system least squares fosls is a recently developed methodology for solving partial differential equations among its advantages are that the finite element spaces are not restricted by the inf sup condition imposed for example on mixed methods and that the least squares functional itself serves as an appropriate error measure this paper studies the fosls approach for scalar second order elliptic boundary value problems with discontinuous coefficients irregular boundaries and mixed boundary conditions a least squares functional is defined and ellipticity is established in a natural norm of an appropriately scaled least squares bilinear form for some geometries this ellipticity is independent of the size of the jumps in the coefficients the occurrence of singularities at interface corners cross points reentrant corners and irregular boundary points is discussed and a basis of singular functions with local support around singular points is established a companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques also independent of the size of coefficient jumps for some geometries

on mixed finite element methods for first order elliptic systems a physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the galerkin finite element solution of these systems using this theory in conjunction with a certain hypothesis concerning approximation spaces optimal error estimates for galerkin type approximations are demonstrated an example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided

nonconforming elements in least squares mixed finite element methods in this paper we analyze the finite element discretization for the first order system least squares mixed model for the second order elliptic problem by means of using nonconforming and conforming elements to approximate displacement and stress respectively moreover on arbitrary regular quadrilaterals we propose new variants of both the rotated q1 nonconforming element and the lowest order raviart thomas element

least squares mixed method for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems having non symmetric matrix of coefficients is presented it is proved that the method is not subject to the ladyzhenskaya babuska brezzi lbb condition and that the finite element approximation yields a symmetric positive definite linear system with condition number o h 2 optimal error estimates are developed

a framework for block ilu factorizations using block size reduction the authors propose a block ilu factorization technique for block tridiagonal matrices that need not necessarily be m matrices the technique explores reduction by a coarse vector restriction of the block size of the approximate schur complements computed throughout the factorization process then on the basis of the sherman morrison woodbury formula these are easily inverted they prove the existence of the proposed factorization techniques in the case of nonsymmetric in general m matrices for block tridiagonal matrices with positive definite symmetric part they show the existence of a limit version of the factorization exact inverses of the reduced matrices are needed the theory is illustrated with numerical tests 24 refs 9 tabs

on different finite element methods for approximating the gradient of the solution to the helmholtz equation abstract we consider the numerical solution of the helmholtz equation by different finite element methods in particular we are interested in finding an efficient method for approximating the gradient of the solution we first approximate the gradient by the standard ritz galerkin method as a second method a two stage method due to aziz and werschulz 1980 is presented it is shown that this method gives the same accuracy in the computed gradient and in the computed solution also in the nonconforming case finally a direct method with asymptotic error estimates is given it turns out that the presented direct method is of lowest computational complexity test examples are presented to illustrate the accuracy of the methods

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

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

a remark on least squares mixed element methods for reaction diffusion problems in this paper we propose a least squares mixed element procedure for a reaction diffusion problem based on the first order system by selecting the least squares functional properly the resulting procedure can be split into two independent symmetric positive definite schemes one of which is for the unknown variable and the other of which is for the unknown flux variable which lead to the optimal order h 1 w and l 2 w norm error estimates for the primal unknown and optimal h div w norm error estimate for the unknown flux finally we give some numerical examples

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

error estimates for least squares finite element methods abstract a least squares finite element scheme for a boundary value problem associated with a second order partial differential equation is considered previous work on this subject is generalized and improved by considering a larger class of equations by working in the natural context without additional smoothness conditions and by deriving error estimates not only in the h1 norm and the hdiv norm but also in the l2 norm some of these estimates are sharpened by using finite element spaces with the grid decomposition property the error estimates are supported by numerical results which extend previous numerical work

temporal spatial and thermal features of 3 d rayleigh benard convection by a least squares finite element method numerical solutions of 3 d time dependent rayleigh benard convection are presented in this work the temporal spatial and thermal features of convective patterns are studied for four different geometric aspect ratios 2 1 2 4 1 4 5 1 5 and 3 5 1 2 1 at supercritical rayleigh numbers ra 8 103 2 4 104 and prandtl numbers pr 0 71 2 5 several physical phenomena such as multicellular flow pattern oscillatory transient solution t shaped rolls at the ends of a rectangular box and roll alignment are observed in our simulations the numerical technique is based on an implicit fully coupled and time accurate method which consists of the crank nicolson scheme for time integration newton s method for the convective terms with extensive linearization steps and a least squares finite element method a matrix free algorithm of the jacobi conjugate gradient method is implemented to solve the symmetric positive definite linear system of equations

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

numerical solutions for nonlinear elliptic problems based on first order system we present a theoretical analysis of numerical solutions for nonlinear elliptic problems our analysis is based on the abstract approximation theory for branches of nonsingular solutions developed by brezzi rappaz and raviart brr in most cases of finite element analysis the same spaces are used for the domain and range of the nonlinear operator concerning the application of brr theory this results in a loss of accuracy in the error estimates while numerical experiments show optimal convergence rates the main contribution of this paper is a theoretical analysis for the optimal convergence rates this is achieved by choosing different spaces for the domain and range of the nonlinear operator used in the application of brr theory previously this idea was used for an analysis of petrov galerkin formulation for the brr theory our analysis is developed for the first order least squares finite element methods and the mixed galerkin methods based on first order systems of equations

a novel least squares finite element method enriched with residual free bubbles for solving convection dominated problems in this paper we devise a novel least squares finite element method lsfem for solving scalar convection dominated convection diffusion problems first we convert a second order convection diffusion problem into a first order system formulation by introducing the gradient mathbf p kappa nabla u as an additional variable the lsfem using continuous piecewise linear elements enriched with residual free bubbles for both variables u and mathbf p is applied to solve the first order mixed problem the residual free bubble functions are assumed to strongly satisfy the associated homogeneous second order convection diffusion equations in the interior of each element up to the contribution of the linear part and vanish on the element boundary to implement this two level least squares approach a stabilized method of galerkin least squares type is used to approximate the residual free bubble functions this enriched lsfem not only inherits the advantages of the primitive lsfem such as the resulting linear system being symmetric and positive definite but also exhibits the characteristics of the residual free bubble method without involving stability parameters several numerical experiments are given to demonstrate the effectiveness of the proposed enriched lsfem the accuracy and computational cost of this enriched lsfem are also compared with those of the primitive lsfem we find that for a small diffusivity kappa the enriched lsfem is much better able to capture the nature of layer structure in the solution than the primitive lsfem even if the primitive lsfem uses a very fine mesh or higher order elements in other words the enriched lsfem provides a significant improvement with a lower computational cost over the primitive lsfem for solving convection dominated problems

least squares finite element methods for the elasticity problem abstract a new first order system formulation for the linear elasticity problem in displacement stress form is proposed the formulation is derived by introducing additional variables of derivatives of the displacements whose combinations represent the usual stresses standard and weighted least squares finite element methods are then applied to this extended system these methods offer certain advantages such as that they need not satisfy the inf sup condition which is required in the mixed finite element formulation that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns that the resulting algebraic systems are symmetric and positive definite and that accurate approximations of the displacements and the stresses can be obtained simultaneously with displacement boundary conditions it is shown that both methods achieve optimal rates of convergence in the h 1 norm and in the l 2 norm for all the unknowns numerical experiments with various poisson ratios are given to demonstrate the theoretical error estimates

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

least squares mixed finite element methods for non selfadjoint elliptic problems ii performance of block ilu factorization methods the least squares mixed finite element technique developed in part i is applied to non selfadjoint second order elliptic problems this approach leads to a symmetric positive definite bilinear form which is coercive uniformly in the discretization parameter in this paper we consider an approximate block factorization technique recently proposed by chan and vassilevski in a framework for block ilu factorization using block size reduction math comp 64 1995 pp 129 156 and which is well defined for positive definite block tridiagonal matrices the method is analyzed and supported with extensive numerical experiments

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

on efficient least squares finite element methods for convection dominated problems abstract this paper focuses on the least squares finite element method and its three variants for obtaining efficient numerical solutions to convection dominated convection diffusion problems the coercivity estimates for the corresponding homogeneous least squares energy functionals are derived and based on which error estimates are established one of the common advantages of these least squares methods is that the resulting linear system is symmetric and positive definite numerical experiments that demonstrate the theoretical analysis of the developed methods are presented it was observed that the primitive least squares method performs poorly for convection dominated problems while the stabilized streamline diffusion and negatively stabilized streamline diffusion least squares methods perform considerably better for interior layer problems and the negatively stabilized streamline diffusion least squares method is able to better capture the boundary layer behavior when compared with other least squares methods but all the least squares methods do not give reasonable results for problem possessing both interior and boundary layer structures in the solution

dispersion induced by the pollution for the wave equation in this paper we give a computational method to measure the dispersion which is directly related to the pollution effect this pollution effect is observed by various researchers when the helmholtz equation is solved for high wave numbers the main effect of the pollution is that the wave number of the solution obtained via a finite element method is different from the wave number of the exact solution and this is what is called dispersion using fourier analysis we present a numerical method to measure the dispersion corresponding to a finite element method

interaction measures for control configuration selection based on interval type 2 takagi sugeno fuzzy model interaction measure determines decentralized and sparse control configurations for a multivariable process control this paper investigates interval type 2 takagi sugeno fuzzy it2tsf model based interaction measures using two different criteria one is controllability and observability gramians the other is relative normalized gain array rnga the main contributions are first a data driven it2tsf modeling method is introduced second explicit formulas to execute the two measures based on it2tsf models are given third two interaction indexes are defined from rnga to select sparse control configuration fourth the calculations to derive sensitivities of the two measures with respect to parametric variations in the it2tsf models are developed and fifth the discussion to compare the two measures is presented three multivariable processes are used as examples to show that the results calculated from it2tsf models are more accurate than that from their type 1 counterparts and compared to gramian based measure rnga selects more reasonable control configurations and is more robust to the parametric uncertainties

a posteriori error estimators for the first order least squares finite element method in this paper we propose a posteriori error estimators for certain quantities of interest for a first order least squares finite element method in particular we propose an a posteriori error estimator for when one is interested in email protected s email protected 0 where email protected u our a posteriori error estimators are obtained by assigning proper weight in terms of local mesh size h t to the terms of the least squares functional an a posteriori error analysis yields reliable and efficient estimates based on residuals numerical examples are presented to show the effectivity of our error estimators

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

high order sobolev preconditioning abstract this paper compares the use of first and second order sobolev gradients to solve differential equations using the method of least squares steepest descent the use of high order sobolev gradients offers a very effective preconditioning strategy for the linear part of a nonlinear differential equation

a posteriori error estimates for the primary and dual variables for the div first order least squares finite element method abstract we propose a posteriori error estimators for first order div least squares ls finite element method for linear elasticity stokes equations and general second order scalar elliptic problems our main interest is obtaining a posteriori error estimators for the dual variables fluxes strains stress etc which are main quantity of interest in many applications we also provide a posteriori error estimators for the primary variable these estimators are obtained from the local least squares functional by assigning weight coefficients scaling the respective residuals the weight coefficients are given in terms of local meshsize h k we establish the global upper bounds and local lower bounds for the estimators the estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution the indicators and estimators are identical numerical experiments show a superior performance of our a posteriori estimators for user specific norm over the standard ls functional

numerical analysis of a least squares finite element method for the time dependent advection diffusion equation a mixed finite element scheme designed for solving the time dependent advection diffusion equations expressed in terms of both the primal unknown and its flux incorporating or not a reaction term is studied once a time discretization of the crank nicholson type is performed the resulting system of equations allows for a stable approximation of both fields by means of classical lagrange continuous piecewise polynomial functions of arbitrary degree in any space dimension convergence in the norm of h 1xh div in space and in appropriate senses in time applying to this pair of fields is demonstrated

first order system least squares and the energetic variational approach for two phase flow this paper develops a first order system least squares fosls formulation for equations of two phase flow the main goal is to show that this discretization along with numerical techniques such as nested iteration algebraic multigrid and adaptive local refinement can be used to solve these types of complex fluid flow problems in addition from an energetic variational approach it can be shown that an important quantity to preserve in a given simulation is the energy law we discuss the energy law and inherent structure for two phase flow using the allen cahn interface model and indicate how it is related to other complex fluid models such as magnetohydrodynamics finally we show that using the fosls framework one can still satisfy the appropriate energy law globally while using well known numerical techniques

a least squares virtual element method for second order elliptic problems abstract in this paper a least squares virtual element method is presented for approximating the vector and scalar variables of second order elliptic problems the h div conforming and scalar conforming virtual elements are used to approximate the vector and scalar variables respectively the method allows the use of very general polygonal meshes and leads to a symmetric positive definite system the optimal a priori error estimates are established for the vector variable in h div norm and the scalar variable in h 1 norm a simple a posteriori error estimator is also presented and proved to be reliable and efficient the virtual element method handles the hanging nodes naturally thus the local mesh post processing to remove hanging nodes is not required numerical experiments are conducted to verify the accuracy of the method and show the effectiveness and flexibility of the adaptive strategy driven by the proposed estimator and suitable mesh refinement strategy

on finite element approximation of the gradient for solution of poisson equation a nonconforming mixed finite element method is presented for approximation of w with w f w r 0 convergence of the order left nabla w u h right 0 omega mathcal o h 2 is proved when linear finite elements are used only the standard regularity assumption on triangulations is needed

recursive least squares identification algorithms with incomplete excitation convergence analysis and application to adaptive control the convergence properties of a fairly general class of adaptive recursive least squares algorithms are studied under the assumption that the data generation mechanism is deterministic and time invariant first the open loop identification case is considered by a suitable notion of excitation subspace the convergence analysis of the identification algorithm is carried out with no persistent excitation hypothesis i e it is proven that the projection of the parameter error on the excitation subspace tends to zero while the orthogonal component of the error remains bounded the convergence of an adaptive control scheme based on the minimum variance control law is then dealt with it is shown that under the standard minimum phase assumption the tracking error converges to zero whenever the reference signal is bounded furthermore the control variable turns out to be bounded

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

quasi least squares finite element method for steady flow and heat transfer with system rotation two quasi least squares finite element schemes based on l2 inner product are proposed to solve a steady navier stokes equations coupled to the energy equation by the boussinesq approximation and augmented by a coriolis forcing term to account for system rotation the resulting nonlinear systems are linearized around a characteristic state resulting in linearized least squares models that yield algebraic systems with symmetric positive definite coefficient matrices existence of solutions are investigated and a priori error estimates are obtained the performance of the formulation is illustrated by using a direct iteration procedure to treat the nonlinearities and shown theoretical convergent rate for general initial guess

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

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

a least squares finite element analysis for diffraction problems the diffraction of a time harmonic wave incident upon a grating or periodic structure is treated by a least squares finite element method that incorporates the jump conditions at interfaces into the objective functional two fundamental polarizations are considered coercivity of the quadratic functional is established and optimal discretization error estimates are obtained in both cases the theoretical results indicate that for sufficiently smooth interfaces the error estimates are superior to those of standard finite element methods

least squares galerkin procedures for pseudohyperbolic equations abstract two least squares galerkin finite element schemes are formulated to solve pseudohyperbolic equations the advantage of this method is that it is not subject to the lbb condition the convergence analysis shows that the methods both yield the approximate solutions with optimal accuracy in l 2 2 l 2 moreover the two methods get the approximate solutions with first order and second order accuracy in time increment respectively numerical example shows that the two schemes are effective

preconditioners for regularized saddle point problems with an application for heterogeneous darcy flow problems saddle point problems arise in the modeling of many important practical situations preconditioners for the corresponding matrices of block triangular form based on coupled inner outer iteration methods are analyzed and applied to a darcy flow problem possibly with strong heterogeneity and non symmetric saddle point systems using proper regularized forms of the given matrix and its preconditioner it is shown that the eigenvalues cluster about one or two points on the real axis for large values of the regularization parameters and that eigenvalue bounds do not depend on this variation therefore just two outer iterations can suffice on the other hand several iterations may be needed to solve the inner iteration systems

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

an h 1 galerkin mixed finite element method for parabolic partial differential equations in this paper an h1 galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts compared to the standard h1 galerkin procedure c1 continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method moreover it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method but without lbb consistency condition and quasiuniformity requirement on the finite element mesh finally a better rate of convergence for the flux in l2 norm is derived using a modified h1 galerkin mixed method in two and three space dimensions which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution

an h 1 galerkin mixed finite element method for an evolution equation with a positive type memory term an h1 galerkin mixed finite element method is analyzed for a class of evolution equations with memory when a classical mixed method is applied to such problems it has not been possible to obtain any estimate for the flux however the proposed approach yields optimal order convergence without the lbb consistency condition and quasi uniformity of the finite element mesh compared to the results proved for one space variable the l2 estimate of the flux is not optimal for problems in two and three space dimensions therefore a modification of the method is proposed and analyzed a maximum norm estimate is also derived in one and two space variables a backward euler approximation of the modified method is also analyzed

a bubble stabilized least squares finite element method for steady mhd duct flow problems at high hartmann numbers in this paper we devise a stabilized least squares finite element method using the residual free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow we convert the original system of second order partial differential equations into a first order system formulation by introducing two additional variables then the least squares finite element method using c 0 linear elements enriched with the residual free bubble functions for all unknowns is applied to obtain approximations to the first order system the most advantageous features of this approach are that the resulting linear system is symmetric and positive definite and it is capable of resolving high gradients near the layer regions without refining the mesh thus this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the hartmann number before incoorperating the bubble functions into the global problem we apply the galerkin least squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements therefore we indeed introduce a two level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual free bubble functions numerical results confirming theoretical findings are presented for several examples including the shercliff problem

first order system cl cl fosll scalar elliptic partial differential equations the l2 norm version of first order system least squares fosls attempts to reformulate a given system of partial differential equations so that applying a least squares principle yields a functional whose bilinear part is h1 elliptic this ellipticity means that the minimization process amounts to solving a weakly coupled system of poisson like scalar equations an unfortunate limitation of the l2 norm fosls approach is that this product h1 equivalence generally requires sufficient smoothness of the original problem inverse norm fosls overcomes this limitation but at a substantial loss of real efficiency the fosll approach introduced here is a promising alternative that is based on recasting the original problem as a minimization principle involving the adjoint equations this paper provides a theoretical foundation for the fosll methodology and illustrates its performance by applying it numerically to several examples results for the so called two stage approach applied to discontinuous coefficient problems show promising robustness and optimality indeed fosll appears to exhibit the generality of the inverse norm fosls approach while retaining the full efficiency of the l2 norm approach

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

optimal error estimate for the div least squares method with data f inl 2 and application to nonlinear problems the div least squares methods have been studied by many researchers for the second order elliptic equations elasticity and the stokes equations and optimal error estimates have been obtained in the h mathrm div times h 1 norm however there is no known convergence rate when the given data f belongs only to l 2 space in this paper we will establish an optimal error estimate in the l 2 times h 1 norm with the given data f in l 2 and hence fill a theoretical gap of least squares methods as a consequence of this estimate we will provide a convergence analysis for the linearization process on solving navier stokes equations which uses the div least squares method for solving the corresponding stokes equations

on least squares finite element methods for the poisson equation and their connection to the dirichlet and kelvin principles least squares finite element methods for first order formulations of the poisson equation are not subject to the inf sup condition and lead to stable solutions even when all variables are approximated by equal order continuous finite element spaces for such elements one can also prove optimal convergence in the energy norm equivalent to a norm on h 1 omega times h omega mbox rm div for all variables and optimal l2 convergence for the scalar variable however showing optimal l2 convergence for the flux has proven to be impossible without adding the redundant curl equation to the first order system of partial differential equations in fact numerical evidence strongly suggests that nodal continuous flux approximations do not posses optimal l2 accuracy in this paper we show that optimal l2 error rates for the flux can be achieved without the curl constraint provided that one uses the div conforming family of brezzi douglas marini or brezzi douglas duran fortin elements then we proceed to reveal an interesting connection between a least squares finite element method involving h omega mbox rm div conforming flux approximations and mixed finite element methods based on the classical dirichlet and kelvin principles we show that such least squares finite element methods can be obtained by approximating through an l2 projection the hodge operator that connects the kelvin and dirichlet principles our principal conclusion is that when implemented in this way a least squares finite element method combines the best computational properties of finite element methods based on each of the classical principles

multilevel boundary functionals for least squares mixed finite element methods for least squares mixed finite element methods for the first order system formulation of second order elliptic problems a technique for the weak enforcement of boundary conditions is presented this approach is based on least squares boundary functionals which are equivalent to the h 1 2 and h1 2 norms on the trace spaces of lowest order raviart thomas elements for the flux and standard continuous piecewise linear elements for the pressure respectively continuity and coercivity of the resulting bilinear form is proved implying optimal order convergence of the resulting galerkin approximation the boundary least squares functional is implemented using multilevel principles and the technique is tested numerically for a model problem

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

domain decomposition for a least square finite element method for second order elliptic problem in this paper we consider the solutions of the discrete linear system resulting from least square mixed finite element discretization applied to a second order elliptic problems additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

a stabilized mixed finite element method for steady and unsteady reaction diffusion equations abstract in this paper we propose a new mixed finite element method called stabilized mixed finite element method for the approximation of steady reaction diffusion partial differential equations pdes the method is obtained by translating the primal second order pdes into a first order mixed system and then adding some suitable elementwise residual terms multiplied by a stabilization parameter to the weak formulation the new method is compatible i e the added terms equal to zero in the continuous case furthermore it is mesh independent i e the stabilization parameter is independent of the mesh size we prove both coercive and continuous properties in a weighted norm for the corresponding new mixed bilinear formulation these assure that the finite element function spaces do not require to satisfy the classical ladyzhenkaya babuska brezzi lbb consistency condition therefore the widely used lagrange finite element can be adopted a simple proof of a priori error estimate with lower order regularity requirement is discussed and numerical experiments confirm the efficiency and reliability of the new stabilized mixed method finally the method is applied to solving unsteady reaction diffusion equations error estimates are also given and numerical examples still support the theoretical analysis very well

wavelet least squares methods for boundary value problems this paper is concerned with least squares methods for the numerical solution of operator equations our primary focus is the discussion of the following conceptual issues the selection of appropriate least squares functionals their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations we describe first a general format of variational problems that are well posed in a certain natural topology in order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems their formulation as a first order system transmission problems the system of stokes equations or more general saddle point problems particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions we propose a unified treatment based on wavelet expansions in particular we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context this provides access to difficult norms efficient preconditioners and in the case of first order systems optimal l2 error estimates