convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction diffusion problems the numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems the quasi optimal order error estimates are proved in the e weighted h 1 norm valid uniformly up to a logarithmic factor in the singular perturbation parameter by using the interpolation postprocessing technique the global superconvergent error estimates in e weighted h 1 norm are obtained numerical experiments are given to demonstrate validity of our theoretical analysis

quasioptimal uniformly convergent finite element methods for the elliptic boundary layer problem we consider the bilinear finite element method for the singularly perturbed elliptic boundary value problem in two dimensions we construct our method on a shishkin mesh through the boundary layer correction technique and elaborate barrier functions the method is shown to be convergent uniformly in the perturbation parameter of order n 2 ln2 n in the l2 norm numerical results confirm our theoretical analysis

a globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions we analyze a new galerkin finite element method for numerically solving a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of order h1 2 in a global energy norm which is stronger than the l2 norm this order is optimal in this norm for our choice of trial functions

a posteriori error estimation based on conservative flux reconstruction for nonconforming finite element approximations to a singularly perturbed reaction diffusion problem on anisotropic meshes abstract based on conservative flux reconstruction we derive a posteriori error estimates for the nonconforming finite element approximations to a singularly perturbed reaction diffusion problem on anisotropic meshes since the solution exhibits boundary or interior layers and this anisotropy is reflected in a discretization by using anisotropic meshes without the assumption that the meshes are shape regular our estimates give the upper bounds on the error only containing the alignment measure factor and therefore could provide actual numerical bounds if the alignment measure was approximated very well two resulting error estimators are presented to be equivalent to each other up to a data oscillation one of which can be directly constructed without solving local neumann problems and provide computable error bound numerical experiments confirm that our estimates are reliable and efficient as long as the singularly perturbed problem is discretized by a suitable mesh which leads to a small alignment measure

layer adapted meshes for convection diffusion problems abstract in the present paper a classification and a survey are given of layer adapted meshes for stationary convection dominated convection diffusion problems for a number of standard numerical schemes theoretical results are presented that demonstrate that the use of properly layer adapted meshes yields robust methods i e methods that perform equally well no matter how dominant the convection we review a number of techniques used in the convergence analysis of these methods

the integral closure of subrings associated to graphs letgbe a graph on the vertex setv andr k v a polynomial ring over a fieldk we give a very explicit description of the integral closure of the monomial subringk g r generated by the squarefree monomials of degree two defining the edges of the graphg

uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems i reaction diffusion type abstract we consider the bilinear finite element method on a shishkin mesh for the singularly perturbed elliptic boundary value problem 2 2 u x 2 2 u y 2 a x y u f x y in two space dimensions by using a very sophisticated asymptotic expansion of han et al 1 and the technique we used in 2 we prove that our method achieves almost second order uniform convergence rate in l2 norm numerical results confirm our theoretical analysis

nonconforming streamline diffusion finite element methods for convection diffusion problems we analyze nonconforming finite element approximations of streamline diffusion type for solving convection diffusion problems both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same o h k 1 2 order of convergence in l 2 as in the conforming case for convection dominated problems a rigorous error analysis supported by numerical experiments is given

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction diffusion problems the bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems in each case the quasi optimal order error estimates are proved in the e weighted h 1 norm uniformly in singular perturbation parameter e up to a logarithmic factor by using the interpolation postprocessing technique the global superconvergent error estimates in e weighted h 1 norm are obtained numerical experiments are given to demonstrate validity of our theoretical analysis

uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids the lowest order raviart thomas rectangular element is considered for solving the singular perturbation problem mbox div a nabla p bp f where the diagonal tensor a varepsilon 2 1 or a varepsilon 2 varepsilon 2 global uniform convergence rates of o n 1 for both p and a1 2 nabla p in the l2 norm are obtained in both cases where n is the number of intervals in either direction the pointwise interior away from the boundary layers convergence rates of o n 1 for p are also proved superconvergence i e o n 2 at special points and o n 2 global l2 estimate for both p and a 1 2 nabla p are obtained by a local postprocessing numerical results support our theoretical analysis moreover numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh

positivity preserving non markovian master equations a general class of integrodifferential non markovian master equations is developed which is representative of the dynamics of small subsystems interacting with open reservoirs with memory conditions which guarantee positivity of the subsystem reduced density are established

supercloseness on graded meshes for q1 finite element approximation of a reaction diffusion equation in this paper we analyze the standard piece wise bilinear finite element approximation of a model reaction diffusion problem we prove supercloseness results when appropriate graded meshes are used the meshes are those introduced in duran and lombardi 2005 8 but with a stronger restriction on the graduation parameter as a consequence we obtain almost optimal error estimates in the l 2 norm thus completing the error analysis given in duran and lombardi 2005 8

corrigendum corrigendum to uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems i reaction diffusion type comput math appl 35 3 1998 57 70

error estimates on anisotropic elements for functions in weighted sobolev spaces in this paper we prove error estimates for a piecewise q 1 average interpolation on anisotropic rectangular elements i e rectangles with sides of different orders in two and three dimensions our error estimates are valid under the condition that neighboring elements have comparable size this is a very mild assumption that includes more general meshes than those allowed in previous papers in particular strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses moreover we generalize the error estimates allowing on the right hand side some weighted sobolev norms this extension is of interest in singularly perturbed problems finally we consider the approximation of functions vanishing on the boundary by finite element functions with the same property a point that was not considered in previous papers on average interpolations for anisotropic elements as an application we consider the approximation of a singularly perturbed reaction diffusion equation and show that as a consequence of our results almost optimal order error estimates in the energy norm valid uniformly in the perturbation parameter can be obtained

resurgences for ideals of special point configurations in bf p n coming from hyperplane arrangements abstract symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years with a notable recent focus on containment relations between symbolic powers and ordinary powers see for example 3 7 13 16 18 19 20 to cite just a few several invariants have been introduced and studied in the latter context including the resurgence and asymptotic resurgence 3 15 there have been exciting new developments in this area recently it had been expected for several years that i n r n 1 i r should hold for the ideal i of any finite set of points in p n for all r 0 but in the last year various counterexamples have now been constructed see 11 17 8 all involving point sets coming from hyperplane arrangements in the present work we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

counterexamples to the i 3 i2 containment abstract the purpose of this short note is to show that there is in general no containment i 3 i 2 for an ideal i of points in p 2 this answers in the negative a question asked by huneke and generalized by harbourne the sets of points constituting counterexamples come from the dual of the hesse configuration and more generally from fermat arrangements

normal polytopes arising from finite graphs abstract let g be a finite connected graph on the vertex set 1 d allowing loops and having no multiple edge let k t 1 t d denote the polynomial ring in d indeterminates over a field k and let k g be the subalgebra of k t 1 t d generated by all quadratic monomials t i t j such that i j is an edge of g and by all quadratic monomials t 2 i such that g has a loop at i we describe the normalization of k g explicitly and we give a combinatorial criterion for k g to be normal

a streamline diffusion method for nonconforming finite element approximations applied to convection diffusion problems abstract we consider a nonconforming streamline diffusion finite element method for solving convection diffusion problems the theoretical and numerical investigation for triangular and tetrahedral meshes recently given by john maubach and tobiska has shown that the usual application of the sdfem gives not a sufficient stabilization additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case the error analysis has been essentially based on the existence of a conforming finite element subspace of the nonconforming space thus the analysis can be applied for example to the crouzeix raviart element but not to the nonconforming quadrilateral elements proposed by rannacher and turek in this paper parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case numerical experiments support the theoretical predictions

asymptotic resurgences for ideals of positive dimensional subschemes of projective space abstract recent work of ein lazarsfeld smith and hochster huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal bocci harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings with a focus on zero dimensional subschemes of projective space here we take the first steps toward extending this work to higher dimensional subschemes we introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals i of smooth subschemes generalizing what is done in bocci and harbourne 2010 5 we apply these bounds to ideals of unions of general lines in p n we also pose a nagata type conjecture for symbolic powers of ideals of lines in p 3

linear equivalence of ideal topologies it is proved that whenever p is a prime ideal in a commutative noetherian ring such that the p adic and thep symbolic topologies are equivalent then the two topologies are equivalent linearly several explicit examples are calculated in particular for all prime ideals corresponding to non torsion points on nonsingular elliptic cubic curves