a locally divergence free interior penalty method for two dimensional curl curl problems an interior penalty method for certain two dimensional curl curl problems is investigated in this paper this method computes the divergence free part of the solution using locally divergence free discontinuous p 1 vector fields on graded meshes it has optimal order convergence up to an arbitrarily small epsilon for the source problem and the eigenproblem results of numerical experiments that corroborate the theoretical results are also presented

a material frame approach for evaluating continuum variables in atomistic simulations we present a material frame formulation analogous to the spatial frame formulation developed by hardy whereby expressions for continuum mechanical variables such as stress and heat flux are derived from atomic scale quantities intrinsic to molecular simulation this formulation is ideally suited for developing an atomistic to continuum correspondence for solid mechanics problems we derive expressions for the first piola kirchhoff p k stress tensor and the material frame heat flux vector directly from the momentum and energy balances using localization functions in a reference configuration the resulting p k stress tensor unlike the cauchy expression has no explicit kinetic contribution the referential heat flux vector likewise lacks the kinetic contribution appearing in its spatial frame counterpart using a proof for a special case and molecular dynamics simulations we show that our p k stress expression nonetheless represents a full measure of stress that is consistent with both the system virial and the cauchy stress expression developed by hardy we also present an expanded formulation to define continuum variables from micromorphic continuum theory which is suitable for the analysis of materials represented by directional bonding at the atomic scale

c 0 elements for generalized indefinite maxwell equations in this paper we develop the c 0 finite element method for a generalized curlcurl grad div indefinite maxwell problem in a lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the h r regularity for some r 1 the ingredients of our method are that two mass lumping l 2 projectors are applied to curl and div operators in the problem and that c 0 linear element or isoparametric bilinear element enriched with one element bubble on every triangle element or with two element bubbles on every quadrilateral element respectively is employed for each component of the nonsmooth solution due to the fact that the element bubbles can be statically eliminated at element levels our method is essentially three nodes or four nodes c 0 lagrange element method with two fortin type interpolations established a very technical duality argument is elaborated to estimate the error for the indefinite problem for the nonsmooth solution having the h r regularity where r may vary in the interval 0 1 we obtain the error bound mathcal o h r in an energy norm some numerical experiments are performed to confirm the theoretical error bounds

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

on finite generation of symbolic rees rings of space monomial curves and existence of negative curves abstract in this paper we shall study finite generation of symbolic rees rings of the defining ideal of the space monomial curves t a t b t c for pairwise coprime integers a b c such that a b c 1 1 1 if such a ring is not finitely generated over a base field then it is a counterexample to the hilbert s fourteenth problem finite generation of such rings is deeply related to existence of negative curves on certain normal projective surfaces we study a sufficient condition definition 3 6 for existence of a negative curve using it we prove that in the case of a b c 2 a b c a negative curve exists using a computer we shall show that there exist examples in which this sufficient condition is not satisfied

the separating variety for the basic representations of the additive group abstract for a group g acting on an affine variety x the separating variety is the closed subvariety of x x encoding which points of x are separated by invariants we concentrate on the indecomposable rational linear representations v n of dimension n 1 of the additive group of a field of characteristic zero and decompose the separating variety into the union of irreducible components we show that if n is odd divisible by four or equal to two the closure of the graph of the action which has dimension n 2 is the only component of the separating variety in the remaining cases there is a second irreducible component of dimension n 1 we conclude that in these cases there are no polynomial separating algebras

solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d polyhedral geometries with the help of a continuous approximation of the electromagnetic field in order to remove spurious eigenmodes their method required a parameterization of the variational formulation in order to avoid this difficulty we use a mixed variational setting instead of the parameterization which allows us to handle the divergence free constraint on the field in a straightforward manner the numerical analysis of the method is carried out and numerical examples are provided to show the efficiency of our approach

fosll method for the eddy current problem with three dimensional edge singularities in the case that the domain has reentrant edges the standard finite element method loses its global accuracy because of singularities on the boundary to overcome this difficulty foslla is applied in this paper foslla is a methodology for solving pdes using the dual operator here a modified foslla method is developed that employs a partially weighted functional and allows the use of a standard finite element scheme without losing global accuracy

a least squares finite element method for the magnetostatic problem in a multiply connected lipschitz domain a new least squares finite element method is developed for the curl div magnetostatic problem in lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials in order to deal with possibly low regularity of the solution local l 2 projectors are introduced to standard least squares formulation and applied to both curl and div operators coercivity is established by adding suitable mesh dependent bilinear terms as a result any continuous finite elements lower order elements are enriched with suitable bubbles can be employed a desirable error bound is obtained bf u bf u h 0 le c bf u tilde bf u 0 where bf u h and tilde bf u are the finite element approximation and the finite element interpolant of the exact solution bf u respectively numerical tests confirm the theoretical results

weitzenbock derivations of free metabelian lie algebras abstract a nonzero locally nilpotent linear derivation of the polynomial algebra k x d k x 1 x d in several variables over a field k of characteristic 0 is called a weitzenbock derivation the classical theorem of weitzenbock states that the algebra of constants k x d which coincides with the algebra of invariants of a single unipotent transformation is finitely generated similarly one may consider the algebra of constants of a locally nilpotent linear derivation of a finitely generated not necessarily commutative or associative algebra which is relatively free in a variety of algebras over k now the algebra of constants is usually not finitely generated except for some trivial cases this holds for the algebra of constants l d l d of the free metabelian lie algebra l d l d with d generators we show that the vector space of the constants l d l d in the commutator ideal l d l d is a finitely generated k x d module for small d we calculate the hilbert series of l d l d and find the generators of the k x d module l d l d this gives also an infinite set of generators of the algebra l d l d

connecting molecular dynamics to micromorphic theory ii balance laws abstract in this paper the time evolution laws for the averaged conserved dynamical variables are derived based on a molecular dynamic model it is found that they follow exactly the balance laws in micromorphic continuum derived by eringen in a top down approach since all the mechanical variables in micromorphic theory can be obtained from an atomic model through statistical ensemble averaging and those averaged quantities satisfy the balance laws that govern the behavior of the continuum it is thus concluded that the atomic model is consistent with the continuum model and correspondence between these two models is analytically achieved

numerical solution to the time dependent maxwell equations in axisymmetric singular domains the singular complement method in this paper we present a method to solve numerically the axisymmetric time dependent maxwell equations in a singular domain in math methods appl sci 25 2002 49 math methods appl sci 26 2003 861 the mathematical tools and an in depth study of the problems posed in the meridian half plane were exposed the numerical method and experiments based on this theory are now described here it is also the generalization to axisymmetric problems of the singular complement method that we developed to solve maxwell equations in 2d singular domains see c r acad sci paris t 330 2000 391 it is based on a splitting of the space of solutions in a regular subspace and a singular one derived from the singular solutions of the laplace problem numerical examples are finally given to illustrate our purpose in particular they show how the singular complement method captures the singular part of the solution

singularities of electromagnetic fields in polyhedral domains in this paper we investigate the singular solutions of time harmonic maxwell equations in a domain which has edges and polyhedral corners it is now well known that in the presence of non convex edges the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the laplace operator 4 5 we show how this type of result can be derived from the classical mellin analysis and how this analysis leads to sharper results concerning the singular parts which belong to h 1 for the singular functions we exhibit simple and explicit formulas based on generalized dirichlet and neumann singularities for the laplace operator these formulas are more explicit than the results announced in our note 10

approximation of the eigenvalue problem for the time harmonic maxwell system by continuous lagrange finite elements we propose and analyze an approximation technique for the maxwell eigenvalue problem using h1 conforming finite elements the key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional sobolev space h with 1 2 1 the method is shown to be convergent and spectrally correct

linear counterexamples to the fourteenth problem of hilbert abstract following ideas of a campo neuen indag math n s 5 1994 253 257 we construct linear counterexamples to the fourteenth problem of hilbert from non linear counterexamples d daigle g freudenburg j algebra 221 1999 528 535 g freudenburg transform groups 5 2000 61 71

a nodal based finite element approximation of the maxwell problem suitable for singular solutions a new mixed finite element approximation of maxwell s problem is proposed its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term which yields a very weak control on the divergence of the unknown the method is shown to be stable and convergent in the natural h rm mathbf curl 0 omega norm for this unknown in particular convergence also applies to singular solutions for which classical nodal based interpolations are known to suffer from spurious convergence upon mesh refinement

analysis of a continuous finite element method for h mathrm curl mathrm div elliptic interface problem in this paper we develop a continuous finite element method for the curlcurl grad div vector second order elliptic problem in a three dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials in spite of the fact that the curlcurl grad div interface problem is closely related to the elliptic interface problem of vector laplace operator type the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges to discretize the curlcurl grad div interface problem by the continuous finite element method we apply an element local l 2 projector to the curl operator and a pseudo local l 2 projector to the div operator where the continuous lagrange linear element enriched by suitable element and face bubbles may be employed it is shown that the finite element problem retains the same coercivity property as the continuous problem an error estimate mathcal o h r in an energy norm is obtained between the analytical solution and the continuous finite element solution where the analytical solution is in prod l 1 l h r omega l 3 for some r in 1 2 1 due to the domain boundary reentrant corners and edges e g nonconvex polyhedron and due to the interfaces between the different material domains in omega bigcup l 1 l omega l

error estimates for a vectorial second order elliptic eigenproblem by the local l 2 projected c 0 finite element method in this paper a theoretical framework with a completely new theory is presented for the general curlcurl graddiv second order elliptic eigenproblem when discretized by the recently developed local l 2 projected c 0 finite element method the theoretical framework consists of two fortin type interpolations and an inf sup inequality associated with a trilinear curl div form from this framework error estimates are readily derived for the source problem as well as the eigenproblem the new theory is verified for the local l 2 projected c 0 finite element method with elementwise linear element l 2 projections applied to div and curl operators and the c 0 linear elements enriched with some face bubbles and element bubbles for the singular solution the challenging question is whether c 0 elements can give spectrally correct approximations when eigenfunctions are singular not in h 1 omega d space it is shown that optimal error bounds can be obtained from this theoretical framework with

a singular field method for maxwell s equations numerical aspects for 2d magnetostatics the present paper deals with the solution of maxwell type problems by means of nodal h1 conforming finite elements in a nonconvex piecewise regular domain surrounded by a perfect conductor such a discretization cannot in general approximate the singular behavior of the electromagnetic field near reentrant corners or edges the singular field method consists of adding to the finite element discretization space some particular fields which take into account the singular behavior the latter are deduced from the singular functions associated with the scalar laplace operator the theoretical justification of this approach as well as the analysis of the convergence of the approximation are presented for a very simple model problem arising from magnetostatics in a translation invariant setting but the study can be easily extended to numerous maxwell type problems the numerical implementation of both variants is studied for a domain containing a single reentrant corner

factorial algebraic group actions and categorical quotients abstract given an action of an affine algebraic group with only trivial characters on a factorial variety we ask for categorical quotients we characterize existence in the category of algebraic varieties moreover allowing constructible sets as quotients we obtain a more general existence result which for example settles the case of a finitely generated algebra of invariants as an application we provide a combinatorial git type construction of categorical quotients for actions of not necessarily reductive groups on e g complete varieties with finitely generated cox ring via lifting to the characteristic space

atomistic viewpoint of the applicability of microcontinuum theories abstract microcontinuum field theories including micromorphic theory microstructure theory micropolar theory cosserat theory nonlocal theory and couple stress theory are the extensions of the classical field theories for the applications in microscopic space and time scales they have been expected to overlap atomic model at microscale and encompass classical continuum mechanics at macroscale this work provides an atomic viewpoint to examine the physical foundations of those well established microcontinuum theories and to justify their applicability through lattice dynamics and molecular dynamics

analysis of a stabilized finite element method for stokes equations of velocity boundary condition and of pressure boundary condition abstract in this paper a new pressure stabilized finite element method is analyzed for solving the stokes equations the key feature of the method is using the curl integral instead of the standard dirichlet integral and using the pressure stabilization instead of the inf sup condition thus the method is very flexibly applicable to two problems of stokes equations in terms of velocity and pressure with either pressure dirichlet boundary condition or velocity dirichlet boundary condition for both stokes problems the finite element scheme and the finite element space of velocity are the same and one where only the finite element spaces of pressure have small differences a general analysis of stability and error estimates is developed and applications to both stokes problems are further analyzed the method covers the low regularity solution of very weak non h 1 space solution of stokes problem of pressure dirichlet boundary condition as well as higher regularity solutions of stokes problems of either pressure dirichlet boundary condition or velocity dirichlet boundary condition numerical results are presented to illustrate the performance of the method and to confirm the theoretical results

bubble functions prompt unusual stabilized finite element methods a second order linear scalar differential equation including a zero th order term is approximated using first the standard galerkin method enriched with bubble functions static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations the method is generalized to allow for a convection operator in the equation this work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena flows with chemical reactions and other important problems

computation of maxwell singular solution by nodal continuous elements abstract in this paper we propose and analyze a nodal continuous and h 1 conforming finite element method for the numerical computation of maxwell s equations with singular solution in a fractional order sobolev space h r where r may take any value in the most interesting interval 0 1 the key feature of the method is that mass lumping linear finite element l 2 projections act on the curl and divergence partial differential operators so that the singular solution can be sought in a setting of l 2 space we shall use the nodal continuous linear finite elements enriched with one element bubble in each element to approximate the singular and non h 1 solution discontinuous and nonhomogeneous media are allowed in the method some error estimates are given and a number of numerical experiments for source problems as well as eigenvalue problems are presented to illustrate the superior performance of the proposed method

a mixed h 1 conforming finite element method for solving maxwell s equations with non h 1 solution in this paper we propose and analyze a mixed h 1 conforming finite element method for solving maxwell s equations in terms of electric field and lagrange multiplier where the multiplier is intr

a singular field method for the solution of maxwell s equations in polyhedral domains it is well known that in the case of a regular domain the solution of the time harmonic maxwell s equations allows a discretization by means of nodal finite elements this is achieved by solving a regularized problem similar to the vector helmholtz equation the present paper deals with the same problem in the case of a nonconvex polyhedron it is shown that a nodal finite element method does not approximate in general the solution to maxwell s equations but actually the solution to a neighboring variational problem involving a different function space indeed the solution to maxwell s equations presents singularities near the edges and corners of the domain that cannot be approximated by lagrange finite elements a new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly this singular field method is presented in various situations such as electri

the local l 2 projected c 0 finite element method for maxwell problem an element local l 2 projected c 0 finite element method is presented to approximate the nonsmooth solution being not in h 1 of the maxwell problem on a nonconvex lipschitz polyhedron with reentrant corners and edges the key idea lies in that element local l 2 projectors are applied to both curl and div operators the c 0 linear finite element enriched with certain higher degree bubble functions is employed to approximate the nonsmooth solution the coercivity in l 2 norm is established uniform in the mesh size and the condition number cal o h 2 of the resulting linear system is proven for the solution and its curl in h r with r 1 we obtain an error bound cal o h r in an energy norm numerical experiments confirm the theoretical error bound

existence of minimizers for a finite strain micromorphic elastic solid we investigate geometrically exact generalized continua of micromorphic type in the sense of eringen the two field problem for the macrodeformation and the affine microdeformation p gl 3 r in the quasistatic conservative load case is investigated in a variational form depending on material constants two existence theorems in sobolev spaces are given for the resulting nonlinear boundary value problems these results comprise existence results for the micro incompressible case p sl 3 r and the cosserat micropolar case p so 3 r in order to treat external loads a new condition called bounded external work has to be included which overcomes the conditional coercivity of the formulation the possible lack of coercivity is related to fracture of the micromorphic solid the mathematical analysis uses an extended korn first inequality the methods of choice are the direct methods of the calculus of variations

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

weighted regularization of maxwell equations in polyhedral domains we present a new method of regularizing time harmonic maxwell equations by a bf grad div term adapted to the geometry of the domain this method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions in the presence of reentrant corners or edges the usual regularization is known to produce wrong solutions due the non density of smooth fields in the variational space we get rid of this undesirable effect by the introduction of special weights inside the divergence integral standard finite elements can then be used for the approximation of the solution this method proves to be numerically efficient

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

new mixed elements for maxwell equations new inf sup stable mixed elements are proposed and analyzed for solving the maxwell equations in terms of electric field and lagrange multiplier nodal continuous lagrange elements of any order on simplexes in two and three dimensional spaces can be used for the electric field the multiplier is compatibly approximated always by the discontinuous piecewise constant elements a general theory of stability and error estimates is developed when applied to the eigenvalue problem we show that the proposed mixed elements provide spectral correct spurious free approximations essentially optimal error bounds only up to an arbitrarily small constant are obtained for eigenvalues and for both singular and smooth solutions numerical experiments are performed to illustrate the theoretical results

a condition for finite generation of the kernel of a derivation abstract we consider a homogeneous derivation on a finitely generated graded normal domain over a field and give a sufficient condition for finite generation of its kernel as its consequence we show some sufficient conditions for finite generation of the kernel of a derivation on a polynomial ring in particular we prove that the kernel of a locally nilpotent monomial derivation on a polynomial ring in four variables is finitely generated

a family of optimal lagrange elements for maxwell s equations abstract in this paper we propose and study a new lagrange finite element method for the two dimensional maxwell s equations its solution may be singular because of the nonsmooth domain with reentrant corners the proposed method allows the standard lagrange elements on barycentric refinements of any order greater than or equal to two we analyze the proposed method for the eigenvalue problem and the indefinite source problem obtaining the well posedness and optimal error estimates numerical results are presented for confirming the theoretical results

a new approximation technique for div curl systems in this paper we describe an approximation technique for div curl systems based in l where is a domain in 3 we formulate this problem as a general variational problem with different test and trial spaces the analysis requires the verification of an appropriate inf sup condition this results is a very weak formulation where the solution space is l and the data reside in various negative norm spaces subsequently we consider finite element approximations based on this weak formulation the main approach of this paper involves the development of stable pairs of discrete test and trial spaces with this approach we enlarge the test space so that the discrete inf sup condition holds and use a negative norm least squares formulation to reduce to a uniquely solvable linear system this leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low sobolev regularity e g h with 0 s 1 2 the resulting algebraic equations are symmetric positive definite and well conditioned a second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned some numerical results are also presented

a counterexample to hilbert s fourteenth problem in dimension 5 abstract the kernel of a certain triangular derivation of the polynomial ring k x 1 x 2 x 3 x 4 x 5 is shown to be non finitely generated over k a field of characteristic zero thus giving a new counterexample to hilbert s fourteenth problem in the lowest dimension to date

numerical solution to the time dependent maxwell equations in two dimensional singular domains in this paper we present a method to solve numerically the time dependent maxwell equations in nonsmooth and nonconvex domains indeed the solution is not of regularity h1 in space in general moreover the space of h1 regular fields is not dense in the space of solutions thus an h1 conforming finite element method can fail even with mesh refinement the situation is different than in the case of the laplace problem or of the lame system for which mesh refinement or the addition of conforming singular functions work to cope with this difficulty the singular complement method is introduced this method consists of adding some well chosen test functions these functions are derived from the singular solutions of the laplace problem also the scm preserves the interesting features of the original method easiness of implementation low memory requirements small cost in terms of the cpu time to ascertain its validity some concrete problems are solved numerically