a subspace of the dssy nonconforming quadrilateral finite element space for the stokes equations in this paper we propose a subspace of the dssy nonconforming quadrilateral finite element space the product of this space together with the piecewise constant space can be used for approximating the velocity and pressure variables respectively in solving stokes problems more precisely this space consists of the p 1 nonconforming quadrilateral finite element space augmented by macro bubble functions based on the dssy nonconforming quadrilateral space under a hood taylor type assumption on meshes it is shown that the pair satisfies the discrete inf sup condition using a boundedness estimate of an interpolation operator based on edge integrals numerical results are presented

legendre multi galerkin methods for fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem abstract in this paper we consider legendre multi galerkin methods to solve fredholm integral equations of the second kind with weakly singular kernel and the corresponding eigenvalue problem we obtain the convergence rates for the approximated solution and iterated solution in weakly singular fredholm integral equations of the second kind in both l 2 and infinity norm we also establish error bounds of approximated eigenelements with exact eigenelements in the eigenvalue problem of a compact integral operator with weakly singular kernel in both infinity and l 2 norm numerical examples are presented to prove the theoretical estimates

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

a numerical study of elastic bodies that are described by constitutive equations that exhibit limited strains abstract recently a very general and novel class of implicit bodies has been developed to describe the elastic response of solids it contains as a special subclass the classical cauchy and green elastic bodies within the class of such bodies one can obtain through a rigorous approximation constitutive relations for the linearized strain as a nonlinear function of the stress such an approximation is not possible within classical theories of cauchy and green elasticity where the process of linearization will only lead to the classical linearized elastic body in this paper we study numerically the states of stress and strain in a finite rectangular plate with an elliptic hole and a stepped flat tension bar with shoulder fillets within the context of the new class of models for elastic bodies that guarantees that the linearized strain would stay bounded and limited below a value that can be fixed a priori thereby guaranteeing the validity of the use of the model this is in contrast to the classical linearized elastic model wherein the strains can become large enough in the body leading to an obvious inconsistency

discrete legendre spectral projection methods for fredholm hammerstein integral equations in this paper we discuss the discrete legendre galerkin and discrete legendre collocation methods for fredholm hammerstein integral equations with smooth kernel using sufficiently accurate numerical quadrature rule we obtain optimal convergence rates for both discrete legendre galerkin and discrete legendre collocation solutions in both infinity and l 2 norm numerical examples are given to illustrate the theoretical results

on the response of non dissipative solids in addition to being incapable of dissipation in any process that it is subject to there are other tacit requirements that a classical elastic body has to meet the class of solids that are incapable of dissipation is far richer than the class of bodies that is usually understood as being elastic we show that unlike the case of a classical elastic body the stress in non dissipative bodies is not necessarily derivable from a stored energy that depends only on the deformation gradient

numerical analysis for a new non conforming linear finite element on quadrilaterals starting with a short introduction of a new non conforming linear quadrilateral p1 finite element which has been recently proposed by park a study on locking phenomena in finite element methods ph d thesis seoul national university february 2002 and park and sheen p1 nonconforming quadrilateral finite element methods for second order elliptic problems siam j numer anal 41 2 2003 624 640 we examine in detail the numerical behaviour of this element with special emphasis on the treatment of dirichlet boundary conditions efficient matrix assembly and solver aspects furthermore we compare the numerical characteristics of p1 with other low order finite elements also regarding its use for the incompressible navier stokes equations several test examples show the efficiency and reliability of the proposed methods for elliptic second order problems

locally stabilized p1 nonconforming quadrilateral and hexahedral finite element methods for the stokes equations in this paper we consider locally stabilized pairs p 1 p 1 nonconforming quadrilateral and hexahedral finite element methods for the two and three dimensional stokes equations the stabilization is obtained by adding to the bilinear form the difference between an exact gaussian quadrature rule for quadratic polynomials and an exact gaussian quadrature rule for linear polynomials optimal error estimates are derived in the energy norm and the l 2 norm for the velocity and in the l 2 norm for the pressure in addition numerical experiments to confirm the theoretical results are presented from our numerical results we observe that the proposed stabilized p 1 p 1 nonconforming finite element method shows better performance than the standard method

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

galerkin methods for second kind integral equations with singularities this paper discusses the numerical solution of fredholm integral equations of the second kind which have weakly singular kernels and inhomogeneous terms global convergence estimates are derived for the galerkin and iterated galerkin methods using splines on arbitrary quasiuniform meshes as approximating subspaces it is observed that due to the singularities present in the solution being approximated the resulting convergence may be slow it is then shown that convergence will be improved greatly by the use of splines based on a mesh which has been suitably graded to accommodate these singularities in fact it is shown that under suitable conditions the galerkin method converges optimally and the iterated galerkin method is superconvergent numerical ilustrations are given

a hybrid collocation method for volterra integral equations with weakly singular kernels the commonly used graded piecewise polynomial collocation method for weakly singular volterra integral equations may cause serious round off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time dependent equations to round off errors the singularity preserving nonpolynomial collocation method is known to have only local convergence to overcome the shortcoming of these well known methods we introduce a hybrid collocation method for solving volterra integral equations of the second kind with weakly singular kernels in this hybrid method we combine a singularity preserving nonpolynomial collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain we prove the optimal order of global convergence for this method the convergence analysis of this method is based on a singularity expansion of the exact solution of the equations we prove that the solutions of such equations can be decomposed into two parts with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function a numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method

a 2 d coupled bem fem simulation of electro elastostatics at large strain the numerical simulation of nonlinear electro elastostatics is considered in this work using the coupled boundary and finite element method the objective of the work is to properly simulate the deformation of an electroelastic body undergoing large deformation subjected to electric stimulations in the case where the surrounding space has significant influence on the electric field inside the body finite elements are used to model the nonlinear electroelastic body in which both mechanical nonlinearity and electrical nonlinearity are taken into account boundary elements are used to model the surrounding space and account for the large deformation of the boundary of the body

on a new class of electro elastic bodies ii boundary value problems in part i of this two part paper a new theoretical framework was presented to describe the response of electro elastic bodies the constitutive theory that was developed consists of two implicit c

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

superconvergence results of legendre spectral projection methods for weakly singular fredholm hammerstein integral equations abstract in this paper we consider the galerkin method to approximate the solution of fredholm hammerstein integral equations of second kind with weakly singular kernels using legendre polynomial bases we prove that for both the algebraic and logarithmic kernels the legendre galerkin method has order of convergence o n r whereas the iterated legendre galerkin method converges with the order o n r 1 2 for the algebraic kernel and order o log n n r 1 2 for logarithmic kernel in both l 2 norm and infinity norm where n is the highest degree of the legendre polynomial employed in the approximation and r is the smoothness of the solution we also propose the legendre multi galerkin and iterated legendre multi galerkin methods we prove that iterated legendre multi galerkin method has order of convergence o 1 c l o g n n r 2 1 2 for the algebraic kernel and order of convergence o log n 2 1 c l o g n n r 3 2 for logarithmic kernel in both l 2 norm and infinity norm numerical examples are given to illustrate the theoretical results

numerical solution of finite geometry boundary value problems in nonlinear magnetoelasticity this paper provides examples of the numerical solution of boundary value problems in nonlinear magnetoelasticity involving finite geometry based on the theoretical framework developed by dorfmann and co workers specifically using a prototype constitutive model for isotropic magnetoelasticity we consider two two dimensional problems for a block with rectangular cross section and of infinite extent in the third direction in the first problem the deformation induced in the block by the application of a uniform magnetic field far from the block and normal to its larger faces without mechanical load is examined while in the second problem the same magnetic field is applied in conjunction with a shearing deformation produced by in plane shear stresses on its larger faces for each problem the distribution of the magnetic field throughout the block and the surrounding space is illustrated graphically along with the corresponding deformation of the block the rapidly in space changing magnitude of the magnetic field in the neighbourhood of the faces of the block is highlighted

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that 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 it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

on a class of non dissipative materials that are not hyperelastic the purpose of this brief note is to develop fully eulerian implicit constitutive equations for the mechanical response of a class of materials that do not dissipate mechanical work in any process we show that such materials can be modelled by obtaining a form for the helmholtz potential as a function of the current mass density the cauchy stress and certain other parameters that capture anisotropic response the resulting constitutive equations are of the form graphic 1 where graphic 2 and graphic 3 are functions of the state variables of the system the class of materials that can be obtained from such a constitutive relation is considerably more general than conventional green elastic hyperelastic materials such response functions may be suitable for the modelling of biological tissue where due to the constant remodelling that takes place there may be no physical meaning to a reference configuration 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif 3 embed inline graphic 3 gif

convergence analysis of discrete legendre spectral projection methods for hammerstein integral equations of mixed type in this paper we consider the discrete legendre spectral galerkin and discrete legendre spectral collocation methods to approximate the solution of mixed type hammerstein integral equation with smooth kernels the convergence of the discrete approximate solutions to the exact solution is discussed and the rates of convergence are obtained we have shown that when the quadrature rule is of certain degree of precision the rates of convergence for the legendre spectral galerkin and legendre spectral collocation methods are preserved we obtain superconvergence rates for the iterated discrete legendre galerkin solution by choosing the collocation nodes and quadrature points to be same we also obtain superconvergence rates for the iterated discrete legendre collocation solution

approximation methods for second kind weakly singular volterra integral equations abstract projection methods are applied to obtain the convergence rates for volterra integral equations with weakly singular kernels we consider galerkin and multi galerkin methods and their iterated versions to solve volterra integral equations with weakly singular kernels in the space of piecewise polynomials subspaces based on graded mesh we will show that the iterated multi galerkin method improves over iterated galerkin method in fact we show that iterated multi galerkin solution converges with the convergence rates o n 3 m and o n 3 m log n 2 for algebraic and logarithmic type kernels respectively we prove that iterated galerkin method for algebraic kernel converges with the convergence rate o n 2 m and for logarithmic type kernel converges with the convergence rate o n 2 m log n where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations theoretical results are justified by the numerical results

a cheapest nonconforming rectangular finite element for the stationary stokes problem abstract we introduce a least degrees of freedom nonconforming finite element pair to approximate the stokes equations based on quadrilateral meshes the finite element space for the velocity field is composed of the p 1 nonconforming quadrilateral element plus only one additional global dssy douglas santos sheen ye or rt rannacher turek type of bubble function the pressure field is approximated by the piecewise constant function we show that the discrete problem is nonsingular and the element pair satisfies a weak discrete inf sup condition several numerical examples are shown to confirm the efficiency and reliability of the proposed method

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

nonlinear electroelasticity material properties continuum theory and applications in the last few years it has been recognized that the large deformation capacity of elastomeric materials that are sensitive to electric fields can be harnessed for use in transducer devices such as actuators and sensors this has led to the reassessment of the mathematical theory that is needed for the description of the electromechanical in particular electroelastic interactions for purposes of material characterization and prediction after a review of the key experiments concerned with determining the nature of the electromechanical interactions and a discussion of the range of applications to devices we provide a short account of the history of developments in the nonlinear theory this is followed by a succinct modern treatment of electroelastic theory including the governing equations and constitutive laws needed for both material characterization and the analysis of general electroelastic coupling problems for illustration the theory is then applied to two simple representative boundary value problems that are relevant to the geometries of activation devices in particular a a rectangular plate and b a circular cylindrical tube in each case with compliant electrodes on the major surfaces and a potential difference between them in a an electric field is generated normal to the major surfaces and in b a radial electric field is present this is followed by a short section in which other problems addressed on the basis of the general theory are described briefly

polynomially based multi projection methods for fredholm integral equations of the second kind in this paper we propose a multi projection and iterated multi projection methods for fredholm integral equations of the second kind with a smooth kernel using polynomial bases we obtain super convergence rates for the approximate solutions more precisely we prove that in m galerkin and m collocation methods not only iterative solution u n approximates the exact solution u in the supremum norm with the order of convergence n 4 k but also the derivatives of u n approximate the corresponding derivatives of u in the supremum norm with the same order of convergence n being the degree of polynomial approximation and k being the smoothness of the kernel