higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1d abstract we study the generalized finite element methods gfems for the second order elliptic eigenvalue problem with an interface in 1d the linear stable generalized finite element methods sgfem were recently developed for the elliptic source problem with interfaces we first generalize sgfem to arbitrary order elements and establish the optimal error convergence of the approximate solutions for the elliptic source problem with an interface we then apply the abstract theory of spectral approximation of compact operators to establish the error estimation for the eigenvalue problem with an interface the error estimations on eigenpairs strongly depend on the estimation of the discrete solution operator for the source problem we verify our theoretical findings in various numerical examples including both source and eigenvalue problems

the globally weak attraction and global semi stability of lienard system this paper obtains some necessary and sufficient conditions for the globally weak attraction and global semi stability of the zero solution for lienard equation x h y f x y g x

spectral approximation properties of isogeometric analysis with variable continuity abstract we study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh which allows for large computational savings during the solution of the resulting linear system the continuity reduction results in extra degrees of freedom that modify the approximation properties of the method the convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis we show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra such as stopping bands and outliers and present a unified description of these effects in finite element method isogeometric analysis and refined isogeometric analysis accuracy of the refined isogeometric analysis approximations can be improved by using non standard quadrature rules in particular optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence as it occurs in standard isogeometric analysis

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

differential calculus on the quantum superplane wess and zumino 1990 succeeded in formulating a perfectly consistent differential calculus on the quantum hyperplane the author gives the natural extension of their scheme to superspace and discusses the various consistency checks that have been performed

z3 graded differential geometry of the quantum plane in this work the z3 graded differential geometry of the quantum plane is constructed the corresponding quantum lie algebra and its hopf algebra structure are obtained the dual algebra i e the universal enveloping algebra of the quantum plane is explicitly constructed

on the divergence constraint in mixed finite element methods for incompressible flows the divergence constraint of the incompressible navier stokes equations is revisited in the mixed finite element framework while many stable and convergent mixed elements have been developed throughout the past four decades most classical methods relax the divergence constraint and only enforce the condition discretely as a result these methods introduce a pressure dependent consistency error which can potentially pollute the computed velocity these methods are not robust in the sense that a contribution from the right hand side which influences only the pressure in the continuous equations impacts both velocity and pressure in the discrete equations this article reviews the theory and practical implications of relaxing the divergence constraint several approaches for improving the discrete mass balance or even for computing divergence free solutions will be discussed grad div stabilization higher order mixed methods derived on the basis of an exact de rham complex h div conforming finite

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

hypersymmetry a z3 graded generalization of supersymmetry we propose a generalization of non commutative geometry and gauge theories based on ternary z3 graded structures in the new algebraic structures we define all products of two entities are left free the only constraining relations being imposed on ternary products these relations reflect the action of the z3 group which may be either trivial i e abc bca cab generalizing the usual commutativity or non trivial i e abc jbca with j e 2 i 3 the usual z2 graded structures such as grassmann lie and clifford algebras are generalized to the z3 graded case certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed

conservation properties for the galerkin and stabilised forms of the advection diffusion and incompressible navier stokes equations abstract a common criticism of continuous galerkin finite element methods is their perceived lack of conservation this may in fact be true for incompressible flows when advective rather than conservative weak forms are employed however advective forms are often preferred on grounds of accuracy despite violation of conservation it is shown here that this deficiency can be easily remedied and conservative procedures for advective forms can be developed from multiscale concepts as a result conservative stabilised finite element procedures are presented for the advection diffusion and incompressible navier stokes equations

q integrals on the quantum complex plane q integration is defined for the complex quantum plane the existence of classical and grassmanian limits of q integration is proved quantum versions of cauchy s and stokes theorems are formulated

modified integration rules for reducing dispersion error in finite element methods abstract this paper describes a simple but effective technique for reducing dispersion errors in finite element solutions of time harmonic wave propagation problems the method involves a simple shift of the integration points to locations away from conventional gauss or gauss lobatto integration points for bilinear rectangular elements such a shift results in fourth order accuracy with respect to dispersion error error in wavelength as opposed to the second order accuracy resulting from conventional integration numerical experiments involving distorted meshes indicate that the method has superior performance in comparison with other dispersion reducing finite elements

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

d3 0 d2 0 differential calculi on certain noncommutative super spaces in this paper we construct a covariant differential calculus on a quantum plane with two parametric quantum group as a symmetry group the two cases d2 0 and d3 0 are completely established we also construct differential calculi n 2 and n 3 nilpotent on super quantum spaces with one and two parametric symmetry quantum supergroup

a finite strain eulerian formulation for compressible and nearly incompressible hyperelasticity using high order b spline finite elements summary we present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions this finite element formulation is based on the eulerian description of motion and the transport of the deformation gradient when modeling a nearly incompressible solid the transport of the deformation gradient is decomposed into its isochoric part and the jacobian determinant as independent fields a homogeneous isotropic hyperelastic solid is assumed and b splines based finite elements are used for the spatial discretization a variational multiscale residual based approach is employed to stabilize the transport equations the performance of the scheme is explored for both compressible and nearly incompressible applications the numerical results are in good agreement with theory illustrating the viability of the computational scheme copyright 2011 john wiley sons ltd

dispersion minimized mass for isogeometric analysis abstract we introduce the dispersion minimized mass for isogeometric analysis to approximate the structural vibration which we model as a second order differential eigenvalue problem the dispersion minimized mass reduces the eigenvalue error significantly from the optimum order of 2 p to the superconvergence order of 2 p 2 for the p th order isogeometric elements with maximum continuity which in return leads to a more accurate method we first establish the dispersion error where the leading error term is explicitly written in terms of the stiffness and mass entries for arbitrary polynomial order isogeometric elements we derive the dispersion minimized mass in one dimension by solving a p dimensional local matrix problem for the p th order approximation and then extend it to multiple dimensions on tensor product grids we show that the dispersion minimized mass can also be obtained by approximating the mass matrix using optimally blended quadratures we generalize the lower order quadrature blending results to arbitrary polynomial order isogeometric approximations as well as to arbitrary quadrature rules various numerical examples validate the eigenvalue and eigenfunction error estimates

mixed isogeometric finite cell methods for the stokes problem we study the application of the isogeometric finite cell method iga fcm to mixed formulations in the context of the stokes problem we investigate the performance of the iga fcm when utilizing some isogeometric mixed finite elements namely taylor hood sub grid raviart thomas and nedelec elements these element families have been demonstrated to perform well in the case of conforming meshes but their applicability in the cut cell context is still unclear dirichlet boundary conditions are imposed by nitsche s method numerical test problems are performed with a detailed study of the discrete inf sup stability constants and of the convergence behavior under uniform mesh refinement

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

on the two parameter quantum supergroups and quantum superplanes starting from the assumption that the generators of quantum supergroup are not supercommutatives with the coordinates of quantum superplane instead of usual supercommutativity in the manin approach we construct in a natural way the two parameter quantum supergroup the virtue of this formulation is that it leads to many probable quantum superplanes associated with a given quantum supergroup some interesting examples are presented as a by product of this formulation a correspondence between the one and two parameter deformed differential structures on the quantum superplane is presented

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

analysis of locally stabilized mixed finite element methods for the stokes problem in this paper a locally stabilized finite element formulation of the stokes problem is analyzed a macroelement condition which is sufficient for the stability of locally stabilized mixed methods based on a piecewise constant pressure approximation is introduced by satisfying this condition the stability of the qi po quadrilateral and the pi po triangular element can be established

mixed stabilized finite element methods in nonlinear solid mechanics part ii strain localization this paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining a standard local isotropic rankine damage model with strain softening is used as exemplary constitutive model both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated from this analysis the issue of local discretization error in the pre localization regime is deemed as the main difficulty to be overcome in the discrete problem focus is placed on low order finite elements with continuous strain and displacement fields triangular p1p1 and quadrilateral q1q1 although the presented approach is very general numerical examples show that the resulting procedure is remarkably robust it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh bias dependence

homoclinic orbits in generalized lienard systems abstract this paper is devoted to the investigation on the existence of homoclinic orbits of the planar system of lienard type x h y f x y g x here h y is strictly increasing but is not imposed h sufficient conditions are given for a positive orbit of the system starting at a point on the curve h y f x to approach the origin without intersecting the x axis the obtained theorems include previous results as special cases our results are applied to a concrete system and their sharpness are improved

on curvature in noncommutative geometry a general definition of a bimodule connection in noncommutative geometry has been recently proposed for a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra the corresponding curvatures are also compared

differential geometry of the q superplane hopf algebra structures on the extended q superplane and its differential algebra are defined an algebra of forms which are obtained from the generators of the extended q superplane is introduced and its hopf algebra structure is given

optimal spectral approximation of 2n order differential operators by mixed isogeometric analysis abstract we approximate the spectra of a class of 2 n order differential operators using isogeometric analysis in mixed formulations this class includes a wide range of differential operators such as those arising in elliptic biharmonic cahn hilliard swift hohenberg and phase field crystal equations the spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations which require auxiliary parameters the mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying b spline space we improve this order to be 2 p 2 by applying optimally blended quadrature rules developed in puzyrev et al 2017 caloet al 0000 and this order is an optimum in the view of dispersion error we also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations

n 3 differential calculus and gauge theory on a reduced quantum plane we discuss the algebra of n n matrices as a reduced quantum plane a n 3 nilpotent deformed differential calculus involving a complex parameter q is constructed the two cases q 3rd and nth root of unity are completely treated as an application we establish a gauge field theory for the particular cases n 2 and n 3

an energy stable time integrator for phase field models we introduce a provably energy stable time integration method for general classes of phase field models with polynomial potentials we demonstrate how taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy stability implicitly which are second order accurate in time the spatial discretization relies on a mixed finite element formulation and isogeometric analysis we also propose an adaptive time stepping discretization that relies on a first order backward approximation to give an error estimator this error estimator is accurate robust and does not require the computation of extra solutions to estimate the error this methodology can be applied to any second order accurate time integration scheme we present numerical examples in two and three spatial dimensions which confirm the stability and robustness of the method the implementation of the numerical schemes is done in petiga a high performance isogeometric analysis framework

optimal quadrature rules for odd degree spline spaces and their application to tensor product based isogeometric analysis abstract we introduce optimal quadrature rules for spline spaces that are frequently used in galerkin discretizations to build mass and stiffness matrices using the homotopy continuation concept barto and calo 2016 that transforms optimal quadrature rules from source spaces to target spaces we derive optimal rules for splines defined on finite domains starting with the classical gaussian quadrature for polynomials which is an optimal rule for a discontinuous odd degree space we derive rules for target spaces of higher continuity we further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points we demonstrate it by deriving optimal rules for various odd degree spline spaces particularly with non uniform knot sequences and non uniform multiplicities we also discuss convergence of our rules to their asymptotic counterparts that is the analogues of the midpoint rule of hughes et al 2010 that are exact and optimal for infinite domains for spaces of low continuities we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values

postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems a new procedure for accelerating the convergence of mixed finite element approximations of the eigenpairs and of the biharmonic operator is proposed it is based on a postprocessing technique that involves an additional solution of a source problem on an augmented finite element space this space could be obtained either by substantially refining the grid the two grid method or by using the same grid but increasing the order of polynomials by one the two space method the numerical results presented and discussed in the paper illustrate the efficiency of the postprocessing method

exterior differentials of higher order and their covariant generalization we investigate a particular realization of generalized q differential calculus of exterior forms on a smooth manifold based on the assumption that dn 0 while dk 0 for k n it implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order special attention is paid to the cases n 3 and n 4 a covariant basis of the algebra of such q grade forms is introduced and the analogs of torsion and curvature of higher order are considered we also study a zn graded exterior calculus on a generalized clifford algebra

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

optimally blended spectral finite element scheme for wave propagation and nonstandard reduced integration we study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent finite element mass matrix and lumped spectral element mass matrix for the small wave number limit we find and prove that for the optimum blending the resulting scheme a provides 2p 4 order accuracy for p th order method two orders more accurate compared with finite and spectral element schemes b has an absolute accuracy which is mathcal o p 3 and mathcal o p 2 times better than that of the pure finite and spectral element schemes respectively c tends to exhibit phase lag moreover we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived

gaussian quadrature for splines via homotopy continuation we introduce a new concept for generating optimal quadrature rules for splines to generate an optimal quadrature rule in a given target spline space we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one while preserving the number of quadrature points and therefore optimality the quadrature nodes and weights are considered as a higher dimensional point a zero of a particular system of polynomial equations as the space is continuously deformed by changing the source knot vector the quadrature rule gets updated using polynomial homotopy continuation for example starting with c 1 cubic splines with uniform knot sequences we demonstrate the methodology by deriving the optimal rules for uniform c 2 cubic spline spaces where the rule was only conjectured to date we validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen