stochastic and mixed flower graphs stochasticity is introduced to a well studied class of recursively grown graphs u v flower nets which have power law degree distributions as well as small world properties when u 1 the stochastic variant interpolates between different deterministic flower graphs thus adding flexibility to the model the random multiplicative growth process involved however leads to a spread ensemble of networks with finite variance for the number of links nodes and loops nevertheless the degree exponent and loopiness exponent attain unique values in the thermodynamic limit of infinitely large graphs we also study a class of mixed flower networks closely related to the stochastic flowers but which are grown recursively in a deterministic way the deterministic growth of mixed flower nets eliminates ensemble spreads and their recursive growth allows for exact analysis of their uniquely defined mixed properties

on monotonicity preserving stabilized finite element approximations of transport problems the aim of this work is to design monotonicity preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear shock capturing stabilization as linear stabilization we consider and analyze a novel symmetric projection stabilization technique based on a local scott zhang projector next we design a weighting of the aforementioned linear stabilization such that when combined with a finite element discretization enjoying a discrete maximum principle usually attained via nonlinear stabilization it does not spoil these monotonicity properties then we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes for the nodal scheme we prove a discrete maximum principle for time dependent multidimensional transport problems numerical experiments support the numerical analysis and we show that the resulting methods prov

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

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

exact solution for mean first passage time on a pseudofractal scale free web the explicit determinations of the mean first passage time mfpt for trapping problem are limited to some simple structure e g regular lattices and regular geometrical fractals and determining mfpt for random walks on other media especially complex real networks is a theoretical challenge in this paper we investigate a simple random walk on the the pseudofractal scale free web psfw with a perfect trap located at a node with the highest degree which simultaneously exhibits the remarkable scale free and small world properties observed in real networks we obtain the exact solution for the mfpt that is calculated through the recurrence relations derived from the structure of psfw the rigorous solution exhibits that the mfpt approximately increases as a power law function of the number of nodes with the exponent less than 1 we confirm the closed form solution by direct numerical calculations we show that the structure of psfw can improve the efficiency of transport by diffusion compared with some other structure such as regular lattices sierpinski fractals and t graph the analytical method can be applied to other deterministic networks making the accurate computation of mfpt possible

a stabilized finite element method for the incompressible magnetohydrodynamic equations we propose and analyze a stabilized finite element method for the incompressible magnetohydrodynamic equations the numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint we explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hydrodynamic diffusion is small and even if the magnetic diffusion is large this observation is confirmed by our numerical tests

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

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

mixed finite element methods for stationary incompressible magneto hydrodynamics a new mixed variational formulation of the equations of stationary incompressible magneto hydrodynamics is introduced and analyzed the formulation is based on curl conforming sobolev spaces for the magnetic variables and is shown to be well posed in possibly non convex lipschitz polyhedra a finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf sup stable velocity pressure space pairs and the magnetic ones by a mixed approach using nedelec s elements of the first kind an error analysis is carried out that shows that the proposed finite element approximation leads to quasi optimal error bounds in the mesh size

domination number and minimum dominating sets in pseudofractal scale free web and sierpinski graph abstract the minimum dominating set mds problem is a fundamental subject of theoretical computer science and has found vast applications in different areas including sensor networks protein interaction networks and structural controllability however the determination of the size of a mds and the number of all mdss in a general network is np hard and it thus makes sense to seek particular graphs for which the mds problem can be solved analytically in this paper we study the mds problem in the pseudofractal scale free web and the sierpinski graph which have the same number of vertices and edges for both networks we determine explicitly the domination number as well as the number of distinct mdss we show that the pseudofractal scale free web has a unique mds and its domination number is only half of that for the sierpinski graph which has many mdss we argue that the scale free topology is responsible for the difference of the size and number of mdss between the two studied graphs which in turn indicates that power law degree distribution plays an important role in the mds problem and its applications in scale free networks

maximum matchings and minimum dominating sets in apollonian networks and extended tower of hanoi graphs abstract the apollonian networks display the remarkable power law and small world properties as observed in most realistic networked systems their dual graphs are extended tower of hanoi graphs which are obtained from the tower of hanoi graphs by adding a special vertex linked to all its three extreme vertices in this paper we study analytically maximum matchings and minimum dominating sets in apollonian networks and their dual graphs both of which have found vast applications in various fields e g structural controllability of complex networks for both networks we determine their matching number domination number the number of maximum matchings as well as the number of minimum dominating sets

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

efficient sub 5 approximations for minimum dominating sets in unit disk graphs abstract a unit disk graph is the intersection graph of n congruent disks in the plane dominating sets in unit disk graphs are widely studied due to their applicability in wireless ad hoc networks because the minimum dominating set problem for unit disk graphs is np hard numerous approximation algorithms have been proposed in the literature including some ptass however since the proposal of a linear time 5 approximation algorithm in 1995 the lack of efficient algorithms attaining better approximation factors has aroused attention we introduce an o n m algorithm that takes the usual adjacency representation of the graph as input and outputs a 44 9 approximation this approximation factor is also attained by a second algorithm which takes the geometric representation of the graph as input and runs in o n log n time regardless of the number of edges additionally we propose a 43 9 approximation which can be obtained in o n 2 m time given only the graph s adjacency representation it is noteworthy that the dominating sets obtained by our algorithms are also independent sets

time dependent subscales in the stabilized finite element approximation of incompressible flow problems abstract in this paper we analyze a stabilized finite element approximation for the incompressible navier stokes equations based on the subgrid scale concept the essential point is that we explore the properties of the discrete formulation that results allowing the subgrid scales to depend on time this apparently natural idea avoids several inconsistencies of previous formulations and also opens the door to generalizations

a finite element method for magnetohydrodynamics abstract this paper presents a finite element method for the solution of 3d incompressible magnetohydrodynamic mhd flows two important issues are thoroughly addressed first appropriate formulations for the magnetic governing equations and the corresponding weak variational forms are discussed the selected b q formulation is conservative in the sense that the local divergence free condition of the magnetic field is accounted for in the variational sense a galerkin least squares variational formulation is used allowing equal order approximations for all unknowns in the second issue a solution algorithm is developed for the solution of the coupled problem which is valid for both high and low magnetic reynolds numbers several numerical benchmark tests are carried out to assess the stability and accuracy of the finite element method and to test the behavior of the solution algorithm

a combined nodal continuous discontinuous finite element formulation for the maxwell problem abstract continuous galerkin formulations are appealing due to their low computational cost whereas discontinuous galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters since many electromagnetic problems involve materials with different physical properties this last point is very important for this reason in this article we have developed a combined cg dg formulation for maxwell s problem that allows arbitrary finite element spaces with functions continuous in patches of finite elements and discontinuous on the interfaces of these patches in particular the second formulation we propose comes from a novel continuous galerkin formulation that reduces the amount of stabilization introduced in the numerical system in all cases we have performed stability and convergence analyses of the methods the outcome of this work is a new approach that keeps the low cpu cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones all these methods have been tested using a problem with singular solution and another one with different materials in order to prove that in fact the resulting formulations can properly deal with these problems

inapproximability of dominating set on power law graphs we prove the first logarithmic lower bounds for the approximability of the minimum dominating set problem for the case of connected power law graphs for being a size parameter and the power law exponent we give also a best up to now upper approximation bound for this problem in the case of the parameters 2 we develop also a new functional method for proving lower approximation bounds and display a sharp approximation phase transition area between approximability and inapproximability of the underlying problems our results depend on a method which could be also of independent interest

approximation of the thermally coupled mhd problem using a stabilized finite element method a numerical formulation to solve the mhd problem with thermal coupling is presented in full detail the distinctive feature of the method is the design of the stabilization terms which serve several purposes first convective dominated flows in the navier stokes and the heat equation can be dealt with second there is no restriction in the choice of the interpolation spaces of all the variables and finally flows highly coupled with the magnetic field can be accounted for different aspects related to the design of the final fully discrete and linearized algorithm are also discussed

on the existence uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics the authors consider the equations of stationary incompressible magneto hydrodynamics posed in a bounded domain in three dimensions and treat the full coupled system of equations with inhomogeneous boundary conditions under certain conditions on the data they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed they discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution

on the number of minimal dominating sets on some graph classes a dominating set in a graph is a subset of vertices such that each vertex is either in the dominating set or adjacent to some vertex in the dominating set it is known that graphs have at most o 1 7159 n minimal dominating sets in this paper we establish upper bounds on this maximum number of minimal dominating sets for split graphs cobipartite graphs and interval graphs for each of these graph classes we provide an algorithm to enumerate them for split and interval graphs we show that the number of minimal dominating sets is at most 3 n 3 1 4423 n which is the best possible bound this settles a conjecture by couturier et al sofsem 2012 1 for cobipartite graphs we lower the o 1 5875 n upper bound from couturier et al to o 1 4511 n

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

unified stabilized finite element formulations for the stokes and the darcy problems in this paper we propose stabilized finite element methods for both stokes and darcy s problems that accommodate any interpolation of velocities and pressures apart from the interest of this fact the important issue is that we are able to deal with both problems at the same time in a completely unified manner in spite of the fact that the functional setting is different concerning the stabilization formulation we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters both in what refers to stability and to accuracy this choice is shown to be crucial in the case of darcy s problem as an additional feature of this work we treat two types of stabilized formulations showing that they have a very similar behavior

enumerating minimal connected dominating sets in graphs of bounded chordality enumerating objects of specified type is one of the principal tasks in algorithmics in graph algorithms one often enumerates vertex subsets satisfying a certain property we study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality we establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs in particular we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time o 1 7159 n of split graphs in time o 1 3803 n and of at free strongly chordal and distance hereditary graphs in time o 3 n 3 where n is the number of vertices of the input graph our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes

modularity of certain potentially barsotti tate galois representations where is the reduction of p these results were subject to hypotheses on the local behavior of p at x i e the restriction of p to a decomposition group at x and to irreducibility hypotheses on 1p in this paper we build on the methods of 461 451 and 121 and relax the hypotheses on local behavior in particular we treat certain adic representations which are not semistable at x but potentially semistable we do this using results of 61 generalizing a theorem of ramakrishna 321 see fontaine mazur 22 131 for a slightly different point of view the results in 61 show that certain potentially barsotti tate deformation problems are smooth allowing us to define certain universal deformations for with the necessary galoistheoretic properties to apply wiles method to carry out the proof that these deformations are indeed realized in the cohomology of modular curves i e that the universal deformation rings are hecke algebras we need to identify the corresponding cohomology groups and prove they have the modular theoretic properties needed to apply wiles method as in 151 and 121 the identification is made by matching local behavior of automorphic representations and galois representations via the local langlands correspondence together with fontaine s theory at the prime we work directly with cohomology of modular curves instead of

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

decoupled schemes for unsteady mhd equations ii finite element spatial discretization and numerical implementation in this article a decoupled fully discrete scheme for solving 2d magnetohydrodynamics mhd equations is proposed the decoupled scheme is used for time discretization and the finite element method is used for spatial discretization firstly the almost unconditional stability t c of this scheme is established then optimal l2 and h1 error estimates of numerical solution are provided finally a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme

analysis of a semi implicit structure preserving finite element method for the nonstationary incompressible magnetohydrodynamics equations abstract we revise the structure preserving finite element method in k hu y ma and j xu 2017 stable finite element methods preserving b 0 exactly for mhd models numer math 135 371 396 the revised method is semi implicit in time discretization we prove the linearized scheme preserves the divergence free property for the magnetic field exactly at each time step further we showed the linearized scheme is unconditionally stable and we obtain optimal convergence in the energy norm of the revised method even for solutions with low regularity

approximation of the inductionless mhd problem using a stabilized finite element method in this work we present a stabilized formulation to solve the inductionless magnetohydrodynamic mhd problem using the finite element fe method the mhd problem couples the navier stokes equations and a darcy type system for the electric potential via lorentz s force in the momentum equation of the navier stokes equations and the currents generated by the moving fluid in ohm s law the key feature of the fe formulation resides in the design of the stabilization terms which serve several purposes first the formulation is suitable for convection dominated flows second there is no need to use interpolation spaces constrained to a compatibility condition in both sub problems and therefore equal order interpolation spaces can be used for all the unknowns finally this formulation leads to a coupled linear system this monolithic approach is effective since the coupling can be dealt by effective preconditioning and iterative solvers that allows to deal with high hartmann numbers

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

on an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics in this work we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations the novelty of this formulation is the fact that it always converges to the physical solution even for singular ones a detailed set of numerical experiments have been performed in order to validate our approach

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

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

a mixed finite element method with exactly divergence free velocities for incompressible magnetohydrodynamics abstract we introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem in two and three dimensions the velocity field is discretized using divergence conforming brezzi douglas marini bdm elements and the magnetic field is approximated by curl conforming nedelec elements the h 1 continuity of the velocity field is enforced by a dg approach a central feature of the method is that it produces exactly divergence free velocity approximations and captures the strongest magnetic singularities we prove that the energy norm error is convergent in the mesh size in general lipschitz polyhedra under minimal regularity assumptions and derive nearly optimal a priori error estimates for the two dimensional case we present a comprehensive set of numerical experiments which indicate optimal convergence of the proposed method for two dimensional as well as three dimensional problems

nodal based finite element methods with local projection stabilization for linearized incompressible magnetohydrodynamics abstract we consider the linearized resistive magnetohydrodynamics mhd model in incompressible media and its numerical solution conforming nodal based finite element approximations of the mhd model both for inf sup stable and equal order finite elements with respect to velocity and pressure are considered as opposed to a residual based stabilization method by badia et al 2013 we consider a local projection stabilization for the numerical solution a detailed stability and error analysis for the arising discrete problem is given some numerical experiments like hartmann s mhd problem and singular solutions are examined

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100