discontinuous galerkin methods for friedrichs systems i general theory this paper presents a unified analysis of discontinuous galerkin methods to approximate friedrichs systems an abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator then a general discontinuous galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed all the design constraints of the method are fully stated and an abstract error analysis in the spirit of strang s second lemma is presented finally the method is formulated locally using element fluxes and links with other formulations are discussed details are given for three examples namely advection reaction equations advection diffusion reaction equations and the maxwell equations in the so called elliptic regime

local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

a reordering for the pagerank problem we describe a reordering particularly suited to the pagerank problem which reduces the computation of the pagerank vector to that of solving a much smaller system and then using forward substitution to get the full solution vector we compare the theoretical rates of convergence of the original pagerank algorithm to that of the new reordered pagerank algorithm showing that the new algorithm can do no worse than the original algorithm we present results of an experimental comparison on five datasets which demonstrate that the reordered pagerank algorithm can provide a speedup of as much as a factor of 6 we also note potential additional benefits that result from the proposed reordering

exceptional points of a hamiltonian of von neumann wigner type in this paper we present and discuss an example of exceptional points in the continuous spectrum of a hamiltonian of von neumann wigner type associated with a double pole in the normalized jost eigenfunctions as functions of the wave number the two unnormalized jost energy eigenfunctions are analytic functions of the wave number k for all k complex but at the exceptional points k q they coalesce to give rise to jordan cycles of two generalized bound state quadratically integrable eigenfunctions the green function has simple poles at the exceptional points but the regular scattering eigenfunction vanishes at those points the scattering matrix is a regular function of the wave number at the exceptional points that is s k does not have a pole at the exceptional points

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

geometric coupling thresholds in a two dimensional strip we consider the laplacian in a strip r 0 d with the boundary condition which is dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to neumann this operator is known to have a non empty and simple discrete spectrum for any a 0 there is a sequence 0 a1 a2 of critical values at which new eigenvalues emerge from the continuum when the neumann window expands we find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to a an 2 with an explicit coefficient expressed in terms of the corresponding threshold energy resonance eigenfunction

an efficient elimination strategy for solving pagerank problems in web link structures similar link distributions often occur especially for pages from same hosts for pagerank problems similar in link distributions of pages result in similar row patterns of the transition matrix we demonstrate that common row patterns in the transition matrix determine identical sub rows thus the identical sub rows with a large proportion of nonzeros can be eliminated to decrease the density of pagerank problems we propose an elimination strategy that exploits such identical sub rows and generates an elimination operator for transferring the problem to an equivalent but more sparse problem numerical experiments are reported to illustrate the effectiveness of this strategy for decreasing the computational cost of solving pagerank problems

a lower bound for the smallest singular value abstract a lower bound for the smallest singular value of a n x n is given in terms of the determinant and the 2 norm of the columns and the rows of a examples are given to show that this bound performs well for a large class of matrices

on the spectrum of two layer approach and multiplex pagerank abstract in this paper we present some results about the spectrum of the matrix associated with the computation of the multiplex pagerank defined by the authors in a previous paper these results can be considered as a natural extension of the known results about the spectrum of the google matrix in particular we show that the eigenvalues of the transition matrix associated with the multiplex network can be deduced from the eigenvalues of a block matrix containing the stochastic matrices defined for each layer we also show that as occurs in the classic pagerank the spectrum is not affected by the personalization vectors defined on each layer but depends on the parameter that controls the teleportation we also give some analytical relations between the eigenvalues and we include some small examples illustrating the main results

first order least squares method with weakly imposed boundary condition for convection dominated diffusion problems we present and analyze a first order least squares method for convection dominated diffusion problems which provides robust l 2 a priori error estimate for the scalar variable even if the given data f l 2 the novel theoretical approach is to rewrite the method in the framework of discontinuous petrov galerkin dpg method and then show numerical stability by using a key equation discovered by gopalakrishnan and qiu 2014 this new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations we also show that the condition number of the global matrix is independent of the diffusion coefficient a key feature of the method is that there is no stabilization parameter chosen empirically in addition dirichlet boundary condition is weakly imposed numerical experiments verify our theoretical results and in particular show our way of weakly imposing dirichlet boundary condition is essential to the design of least squares methods numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes which are unstructured quasi uniform

bound states in coupled guides i two dimensions we compute bound state energies in two three dimensional coupled waveguides each obtained from the two dimensional configuration considered in paper i j math phys 45 1359 1379 2004 by rotating the geometry about a different axis the first geometry consists of two concentric circular cylindrical waveguides coupled by a finite length gap along the axis of the inner cylinder and the second is a pair of planar layers coupled laterally by a circular hole we have also extended the theory for this latter case to include the possibility of multiple circular windows both problems are formulated using a mode matching technique and in the cylindrical guide case the same residue calculus theory as used in paper i is employed to find the bound state energies for the coupled planar layers we proceed differently computing the zeros of a matrix derived from the matching analysis directly

asymptotic properties of bound states in coupled quantum wave guides we investigate the motion of bound state poles in two quantum wave guides laterally coupled through a window the imaginary momentum ik at the bound state poles is studied as a function of the size a of the window both bound and virtual states appear as a spans the whole range from 0 up to we are able to find simple scaling laws relating the critical value of the window size at which the nth bound state appears to the number n of bound states in the limit of large n a similar relation is also provided for the slope and the second derivative of the pole trajectories in the k a plane these relations are characterized by an extremely high numerical accuracy we also evaluate the exact value of the first two derivatives for a 0

discontinuous galerkin methods for friedrichs systems part iii multifield theories with partial coercivity this paper is the third and last part of a work attempting to give a unified analysis of discontinuous galerkin methods the purpose of this paper is to extend the framework that has been developed in part ii for two field friedrichs systems associated with second order pdes we now consider two field friedrichs systems with partial l 2 coercivity and three field friedrichs systems with an even weaker l 2 coercivity hypothesis in particular this work generalizes the discontinuous galerkin methods of part ii to compressible and incompressible linear continuum mechanics we also show how the stabilizing parameters of the method must be set when the two field friedrichs system is composed of terms that may be of different magnitude thus accounting for instance for advection diffusion equations at high peclet numbers

asymptotic behavior of jost functions near resonance points for wigner von neumann type potentials in this work are considered radial schrodinger operators v r e where v r a sin br r w r with w r bounded w r o r 2 at infinity a b real the asymptotic behavior of the jost function and the scattering matrix near the resonance point e0 b2 4 are studied if a b then this point may be an eigenvalue embedded in the continuous spectrum the leading behavior of the jost function for all values of a and b were determined somewhat surprisingly situations were found where the jost function becomes singular as e e0 even if e0 is an embedded eigenvalue moreover it was found that the scattering matrix is always discontinuous at e0 except in a few special cases it is also shown that the asymptotics for the jost function and the scattering matrix hold under weaker assumptions on w r in particular an angular momentum term l l 1 r 2 may be incorporated into w r the results were also applied to a whole line problem with a potential v x such that v x 0 for x 0 and v x of wigner von neumann

consistent discontinuous finite elements in elastodynamics finite element discontinuities with respect to time have recently been extremely used in elastodynamic problems due to their natural utilization in combination with adaptive methods and their efficiency in discontinuity capturing techniques for non smooth problems in this work we present some theoretical aspects and numerical results concerning the use of spatial discontinuities in a consistent finite element method for the same class of problems we first review some formulations for the elastostatic problem and prove two korn like inequalities which are very useful for the derivation of convergence rates in sobolev norms next we present formulations for the dynamic case along with comments on their properties and estimates of convergence rates for smooth solutions followed by numerical investigations of a typically non smooth problem involving classical and emerging variational formulations we also show some numerical experiments with finite element spaces enriched by discontinuous functions other than piecewise lagrangian polynomials

mixed hp finite element methods for stokes and non newtonian flow abstract we analyze the stability of hp finite elements for viscous incompressible flow for the classical velocity pressure formulation we give new estimates for the discrete inf sup constants on geometric meshes which are explicit in the polynomial degree k of the elements in particular we obtain new bounds for p elements on triangles for the three field stokes problem describing linearized non newtonian flow we estimate discrete inf sup constants explicit in both h and k for various subspace choices continuous and discontinuous for the extra stress we also give a stability analysis of the hp version of an elastic viscous split stress evss method and present elements that are stable and optimal in h and k finally we present numerical results that show the exponential convergence of the hp version for stokes flow over unsmooth domains

off diagonal low rank preconditioner for difficult pagerank problems abstract pagerank problem is the cornerstone of google search engine and is usually stated as solving a huge linear system moreover when the damping factor approaches 1 the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve in this paper we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets in particular the off diagonal part of the block coefficient matrix can be compressed by a simple low rank factorization which can be beneficial for solving the pagerank problem hence a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization then a preconditioner based on the low rank factorization is proposed for solving difficult pagerank problems numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods

theory of a curved planar waveguide with robin boundary conditions a model of a thin straight strip with a uniformly curved section and with boundary requirements zeroing at the edges a linear superposition of the wave function and its normal derivative robin boundary condition is analyzed theoretically within the framework of the linear schrodinger equation and is applied to the study of the processes in the bent magnetic multilayers superconducting films and metallic ferrite filled waveguides in particular subband thresholds of the straight and curved parts of the film are calculated and analyzed as a function of the robin parameter 1 lambda with lambda being an extrapolation length entering robin boundary condition for the arbitrary robin coefficients which are equal on the opposite interfaces of the strip and for all bend parameters the lowest mode energy of the continuously curved duct is always smaller than its straight counterpart accordingly the bound state below the fundamental propagation threshold of the straight arms always exists as a result of the bend in terms of the superconductivity language it means an increased critical temperature of the curved film compared to its straight counterpart localized level dependence on the parameters of the curve is investigated with its energy decreasing with increasing bend angle and decreasing bend radius conditions of the bound state existence for the different robin parameters on the opposite edges are analyzed too in particular it is shown that the bound state below the first transverse threshold of the straight arm always exists if the inner extrapolation length is not larger than the outer one in the opposite case there is a range of the bend parameters where the curved film cannot trap the wave and form the localized mode for example for the fixed bend radius the bound state emerges from the continuum at some nonzero bend angle that depends on the difference of the two lengths lambda at the opposite interfaces various transport properties of the film such as interference blockade of the current flow at some special energies is also discussed with the special attention being paid to the transformation from the dirichlet to the neumann case as the extrapolation length lambda sweeps the positive axis

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

discrete functional analysis tools for discontinuous galerkin methods with application to the incompressible navier stokes equations two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes i a discrete counterpart of the continuous sobolev embeddings in both hilbertian and non hilbertian settings ii a compactness result for bounded sequences in a suitable discontinuous galerkin norm together with a weak convergence property for some discrete gradients the proofs rely on techniques inspired by the finite volume literature which differ from those commonly used in finite element analysis the discrete functional analysis tools are used to prove the convergence of discontinuous galerkin approximations of the steady incompressible navier stokes equations two discrete convective trilinear forms are proposed a non conservative one relying on temam s device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure

generalized hardy inequality for the magnetic dirichlet forms we obtain lower bounds for the magnetic dirichlet form in dimensions d 2 for d 2 the results generalize a well known lower bound by the magnetic field strength we replace the actual magnetic field bby an non vanishing effective field which decays outside the support of bas dist x suppb 2 in the case d 3 we establish that the magnetic form is bounded from below by the magnetic field strength if one assumes that the field does not vanish and its direction is slowly varying

a simple separable hamiltonian having bound states in the continuum the author discusses a simple separable two freedom hamiltonian h describing the motion of a point particle in a plane region v 0 between two infinite parallel hard walls v infinity and with a rectangular finite potential well v v0 the quantal spectrum of h is discrete below the continuum limit ionisation energy different from the classical escape energy e 0 by the zero point energy of the transverse motion and has infinitely many bound states embedded in the continuum the author shows that this spectrum is fragile against perturbations of h if the hard walls are made diverging the bound states in the continuum disappear while if they are converging the continuum disappears and the spectrum is pure point

analysis of an upwind mixed finite element method for nonlinear contaminant transport equations in this paper the numerical approximation of a nonlinear diffusion equation arising in contaminant transport is studied the equation is characterized by advection diffusion and adsorption assuming the adsorption term is modeled by a freundlich isotherm it can be nonlinear in concentration and nondifferentiable as the concentration approaches zero we consider the approximation of this equation by a method which upwinds the advection and incorporates diffusion using a mixed finite element method error estimates for a semidiscrete formulation are derived and numerical results for a fully discrete formulation are given

friedrichs systems in a hilbert space framework solvability and multiplicity abstract the friedrichs 1958 theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics irrespective of their type this theory was recast in an abstract hilbert space setting by ern guermond and caplain 2007 and by antonic and burazin 2010 in this work we make a further step presenting a purely operator theoretic description of abstract friedrichs systems and proving that any pair of abstract friedrichs operators admits bijective extensions with a signed boundary map moreover we provide sufficient and necessary conditions for existence of infinitely many such pairs of spaces and by the universal operator extension theory grubb 1968 we get a complete identification of all such pairs which we illustrate on two concrete one dimensional examples

the dpg method for the stokes problem we discuss well posedness and convergence theory for the dpg method applied to a general system of linear partial differential equations pdes and specialize the results to the classical stokes problem the stokes problem is an iconic troublemaker for standard bubnov galerkin methods if discretizations are not carefully designed they may exhibit non convergence or locking by contrast dpg does not require us to treat the stokes problem in any special manner we illustrate and confirm our theoretical convergence estimates with numerical experiments

godunov mixed methods for advection diffusion equations in multidimensions time split methods for multidimensional advection diffusion equations are considered in these methods advection is approximated by a godunov type procedure and diffusion is approximated by a low

a straight quantum wave guide with mixed dirichlet and neumann boundary conditions in uniform magnetic fields a straight quasi one dimensional dirichlet wave guide with a neumann window of length l on one or two confining surfaces is considered theoretically with and without perpendicular homogeneous magnetic field it is shown that for the field free case a bound state in the continuum bic for one neumann window exists for some critical lengths only while for the two neumann segments symmetrically located on the opposite walls due to the restored transverse symmetry of the system bics exist for the arbitrary l bound states lying below the fundamental propagation threshold of the dirichlet strip survive any strength of the uniform magnetic field and do not depend on its direction moreover an increasing field induces new bound states regularly arranged with the levels present at b 0 for two neumann windows strong magnetic fields lead to the degeneracy of the adjacent odd and even bound states with their energies almost equal to each other and to their corresponding counterpart for one neumann segment which is explained by mapping the problem onto the field free one or two purely attractive one dimensional quantum wells with field dependent depth miscellaneous magnetotransport characteristics of the structures are also considered in particular it is demonstrated that small fields applied to the channel with two neumann windows destroy bics by coupling them to the continuum states this is manifested in the conductance fermi energy dependence by fano resonances currents flowing in the wave guide are investigated too and it is shown that current density patterns near the resonances form vortices which change their chirality as energy sweeps through the resonant region generalizations to any other arbitrary combination of the boundary conditions are provided comparison with other structures such as window coupled dirichlet wave guides a bent strip or straight dirichlet channel with electrostatic impurity inside is performed

parabolic h measures abstract classical h measures introduced by tartar 1990 and independently by gerard 1991 are not well suited for the study of parabolic equations recently several parabolic variants have been proposed together with a number of applications we introduce a new parabolic variant and call it the parabolic h measure which is suitable for these known applications moreover for this variant we prove the localisation and propagation principle establishing a basis for more demanding applications of parabolic h measures similarly as it was the case with classical h measures in particular the propagation principle enables us to write down a transport equation satisfied by the parabolic h measure associated to a sequence of solutions of a schrodinger type equation some applications to specific equations are presented illustrating the possible use of this new tool a comparison to similar results for classical h measures has been made as well

boundary operator from matrix field formulation of boundary conditions for friedrichs systems abstract following the recent progress in understanding the abstract setting for friedrichs symmetric positive systems by ern guermond and caplain 2007 8 as well as antonic and burazin 2010 3 an attempt is made to relate these results to the classical friedrichs theory a comparison of two approaches via the trace operator and the boundary operator has been made favouring the latter finally a particular set of sufficient conditions for a boundary matrix field to define a boundary operator in that case is given and the applicability of this procedure in realistic situations is shown by examples

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

exponential splitting of bound states in a waveguide with a pair of distant windows we consider the laplacian in a straight planar strip with dirichlet boundary which has two neumann windows of the same length the centres of which are 2l apart and study the asymptotic behaviour of the discrete spectrum as l it is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single window strip and their distances vanish exponentially in the limit l we derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue

the general inner outer iteration method based on regular splittings for the pagerank problem abstract in this paper combined the regular splittings of the coefficient matrix i p with the inner outer iteration framework 9 a general inner outer gio iteration method is presented for solving the pagerank problem firstly the aor and modified aor maor methods for solving the pagerank problem are constructed and several comparison results are also given next the gio iteration scheme is developed and its overall convergence is analyzed in detail furthermore the preconditioner derived from the gio iteration can be used to accelerate the krylov subspace methods such as gmres method finally some numerical experiments on several pagerank problems are provided to illustrate the efficiency of the proposed algorithm

a unified discontinuous petrov galerkin method and its analysis for friedrichs systems we propose a unified discontinuous petrov galerkin dpg framework with optimal test functions for friedrichs like systems which embrace a large class of elliptic parabolic and hyperbolic partial differential equations pdes the well posedness i e existence uniqueness and stability of the dpg solution is established on a single abstract dpg formulation and two abstract dpg methods corresponding to two different but equivalent norms are devised we then apply the single dpg framework to several linear ized pdes including but not limited to scalar transport laplace diffusion convection diffusion convection diffusion reaction linear ized continuum mechanics e g linear ized elasticity a version of linearized navier stokes equations etc time domain acoustics and a version of the maxwell s equations the results show that we not only recover several existing dpg methods but also discover new dpg methods for both pdes currently considered in the dpg community and new ones as

photonic crystals with bound states in continuum and their realization by an advanced digital grading method the helmholtz equation can be reshaped into a form analogous to the schrodinger equation with the term labeled the optical potential by following this analogy we conclude that there exist certain profiles of optical potentials which possess bound states of electric field in the continuous part of the spectrum one of the methods for generating these specific optical potentials is the application of supersymmetric formalism which transforms a real initial potential into a family of complex potentials which all have one bound state in the continuum we present general steps of this procedure and illustrate its use through the example of flat initial optical potential in this particular case conditions are found for the existence of the bound field in continuum as well as the expression for the field and the corresponding complex optical potential in an analytic form in addition the approximation of digital grading is applied to the generated complex supersymmetric optical potential and the bound state is calculated the complex nature and the sharp variations of the supersymmetric optical potential impose the development of an original and sophisticated method of digital grading

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

h measures and variants applied to parabolic equations since their introduction h measures have been mostly used in problems related to propagation effects for hyperbolic equations and systems in this study we give an attempt to apply the h measure theory to other types of equations through a number of examples we present how do the differences between parabolic and hyperbolic equations reflect in the properties of h measures corresponding to the solutions secondly we apply the h measures to the schrodinger equation where we succeed in proving a propagation property however our conclusion is that a variant of h measures should be sought which would be better suited to parabolic problems we propose such a variant show some fundamental properties and illustrate its applicability by some examples in particular we show that the variant provides new information in a number of situations where the original h measures did not finally we describe how the new variant can be used in small amplitude homogenisation of parabolic equations

normalizable states through deformation of lame and the associated lame potentials we construct new aperiodic potentials with positive energy bound states starting from the lame and the associated lame potentials by confining them to the half line the method used is the isospectral deformation discussed in supersymmetric quantum mechanics the new potentials have normalizable eigenfunctions with energies the same as certain levels of the original potentials on the half line these states are similar to the von neumann wigner states as they have an infinite number of zeros the new potentials and their eigenfunctions are obtained numerically and their plots are given

a tent pitching scheme motivated by friedrichs theory certain friedrichs systems can be posed on hilbert spaces normed with a graph norm functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet motivated by this continuity property an explicit space time finite element scheme of the tent pitching type with spaces that conform to the continuity property is designed numerical results for a model one dimensional wave propagation problem are presented

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

multidomain spectral method for the helically reduced wave equation we consider the 2 1 and 3 1 scalar wave equations reduced via a helical killing field respectively referred to as the 2 dimensional and 3 dimensional helically reduced wave equation hrwe the hrwe serves as the fundamental model for the mixed type pde arising in the periodic standing wave psw approximation to binary inspiral we present a method for solving the equation based on domain decomposition and spectral approximation beyond describing such a numerical method for solving strictly linear hrwe we also present results for a nonlinear scalar model of binary inspiral the psw approximation has already been theoretically and numerically studied in the context of the post minkowskian gravitational field with numerical simulations carried out via the eigenspectral method despite its name the eigenspectral technique does feature a finite difference component and is lower order accurate we intend to apply the numerical method described here to the theoretically well developed post minkowski psw formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation

a preconditioned and shifted gmres algorithm for the pagerank problem with multiple damping factors google has become one of the most popular and successful search engines in recent years google s success can be attributed to its simple and elegant algorithm pagerank in practice one often needs to solve the pagerank problem with multiple damping factors or with multiple damping factors and multiple personalization vectors the conventional pagerank algorithm has to solve these problems one by one the shifted gmres m algorithm can be used to solve them in the same search subspace however there are two disadvantages to this algorithm the first is near singularity and the second is stagnation in this paper we first present a modified and shifted gmres m algorithm to deal with the problem of near singularity in order to overcome the drawback of stagnation and to improve convergence we propose a polynomial preconditioner for the modified algorithm we show that the resulting algorithm can circumvent the drawbacks of near singularity and stagnation that occur in its original counterpart

discontinuous galerkin methods for advection diffusion reaction problems we apply the weighted residual approach recently introduced in f brezzi et al comput methods appl mech engrg 195 2006 pp 3293 3310 to derive discontinuous galerkin formulations for advection diffusion reaction problems we devise the basic ingredients to ensure stability and optimal error estimates in suitable norms and propose two new methods

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

discontinuous galerkin methods for anisotropic semidefinite diffusion with advection we construct and analyze a discontinuous galerkin method to solve advection diffusion reaction pdes with anisotropic and semidefinite diffusion the method is designed to automatically detect the so called elliptic hyperbolic interface on fitted meshes the key idea is to use consistent weighted average and jump operators optimal estimates in the broken graph norm are proven these are consistent with well known results when the problem is either hyperbolic or uniformly elliptic the theoretical results are supported by numerical evidence

factorization method and singular hamiltonians the oscillator singular supersymmetric partners are studied in detail and their relationship with the equivalent problem of an oscillator plus an infinite barrier is clearly established interesting potentials with possible applications to different physical models appear as a bonus

a new c0 discontinuous galerkin method for kirchhoff plates abstract a general framework of constructing c 0 discontinuous galerkin cdg methods is developed for solving the kirchhoff plate bending problem following some ideas in castillo et al 2000 10 and cockburn 2003 12 the numerical traces are determined based on a discrete stability identity which lead to a class of stable cdg methods a stable cdg method called the lcdg method is particularly interesting in our study it can be viewed as an extension to fourth order problems of the ldg method studied in castillo et al 2000 10 and cockburn 2003 12 for this method optimal order error estimates in certain broken energy norm and h 1 norm are established some numerical results are reported confirming the theoretical convergence orders

heat equation as a friedrichs system abstract inspired by recent advances in the theory of friedrichs symmetric positive systems we apply newly developed results to the heat equation by showing how the intrinsic theory of ern guermond and caplain 2007 can be used in order to get a well posedness result for the dirichlet initial boundary value problem we also demonstrate the application of the two field theory with partial coercivity of ern and guermond 2008 originally developed for elliptic problems and also discuss different possibilities for the construction of the appropriate boundary operator

the weak galerkin finite element method for the symmetric hyperbolic systems abstract in this paper we present and analyze a weak galerkin finite element wg method for solving the symmetric hyperbolic systems this method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other by introducing special weak derivative we construct a stable weak galerkin scheme and derive the optimal l 2 error estimate of o h k 1 2 order for the discrete solution when the k order polynomials are used for k 0 as application we discuss this wg method for solving the singularly perturbed convection diffusion reaction equation and derive an e uniform error estimate of order k 1 2 numerical examples are provided to show the effectiveness of the proposed wg method

pagerank beyond the web google s pagerank method was developed to evaluate the importance of web pages via their link structure the mathematics of pagerank however are entirely general and apply to any graph or network in any domain thus pagerank is now regularly used in bibliometrics social and information network analysis and for link prediction and recommendation it s even used for systems analysis of road networks as well as biology chemistry neuroscience and physics we ll see the mathematics and ideas that unite these diverse applications

von neumann wigner type potentials and the wavefunctions asymptotics for discrete levels in continuum one to one correspondence between the decay law of von neumann wigner type potentials and the asymptotic behaviour of the wavefunctions representing bound states in the continuum is established

localized mode evolution in a curved planar waveguide with combined dirichlet and neumann boundary conditions we present a theoretical study of a planar waveguide with a uniformly curved section opposite sides of the channel satisfy different boundary conditions it is shown that if the dirichlet condition is applied to the inner side of the strip and the neumann one to the outer wall then properties of such a system in many respects resemble those with the dirichlet requirements on both surfaces namely in both cases a propagation threshold for the curved section is smaller than its counterpart for the straight channel as a consequence a localized mode exists with its energy below the propagation threshold of the straight waveguide analysis of such states is presented as a function of the bend parameters for the transport in the fundamental mode an interaction of a quasibound level split off from the higher lying threshold with its degenerate continuum counterpart causes a dip in the transmission such a resonance is characterized by a location of its zero minimum e min and the half width gamma changing the bend angle and radius one varies e min and gamma in particular for some critical parameters of the bend it is possible to turn the half width to zero i e to eliminate the dip in the transmission this corresponds to the absence of the interaction between the split off level and the continuum and consequently to the formation of the true bound state in the continuum vortex structure of the currents flowing in the waveguide near the resonance is also shown to strongly resemble the analogous results for the dirichlet case it is pointed out that the properties of the waveguide with the neumann inner condition and the dirichlet outer one mimic the duct with the neumann requirements on the two sides since for both these cases the propagation threshold in the curved section is greater than in the straight channel

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

fom accelerated by an extrapolation method for solving pagerank problems this paper formulates the pagerank problem a x x into a consistent singular linear system i a x 0 and applies the full orthogonalization method fom to solve it this singular system is characterized by index one namely i n d e x i a 1 we analyze the breakdown performance of fom on a general singular linear system and conclude that fom can determine a solution if it converges without any unfortunate breakdowns for our target problem then we propose to use a vector extrapolation method to speed up the convergence performance of fom this extrapolation procedure is based on ritz values which directly stems from the arnoldi extrapolation algorithm wu and wei 2010 eventually numerical experiments are presented to illustrate the effectiveness of our approaches

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

comparison of krylov subspace methods on the pagerank problem pagerank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions the problem incorporates a parameter a that determines the difficulty of the problem in this paper the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of a we see that stationary methods are very reliable and more competitive when the problem is well conditioned that is for small values of a however for large values of the parameter a the problem becomes more difficult and methods such as preconditioned bicgstab or restarted preconditioned gmres become competitive with stationary methods in terms of mflops count as well as in number of iterations necessary to reach convergence

complex friedrichs systems and applications recently there has been a significant development of the abstract theory of friedrichs systems in hilbert spaces ern a guermond j l and caplain g commun partial differ equations 32 317 341 2007 and antonic n and burazin k commun partial differ equations 35 1690 1715 2010 and its applications to specific problems in mathematical physics however these applications were essentially restricted to real systems we check that the already developed theory of abstract friedrichs systems can be adjusted to the complex setting with some necessary modifications which allows for applications to partial differential equations with complex coefficients we also provide examples where the involved hilbert space is not the space of square integrable functions as it was the case in previous studies but rather its closed subspace or the space hs rd cr for real s this setting appears to be suitable for particular systems of partial differential equations such as the dirac system the di

discontinuous galerkin methods for friedrichs systems part ii second order elliptic pdes this paper is the second part of a work attempting to give a unified analysis of discontinuous galerkin methods the setting under scrutiny is that of friedrichs systems endowed with a particular 2 times 2 structure in which one unknown can be eliminated to yield a system of second order elliptic like pdes for the remaining unknown a general discontinuous galerkin method for approximating such systems is proposed and analyzed the key feature is that the unknown that can be eliminated at the continuous level can also be eliminated at the discrete level by solving local problems all the design constraints on the boundary operators that weakly enforce boundary conditions and on the interface operators that penalize interface jumps are fully stated examples are given for advection diffusion reaction linear continuum mechanics and a simplified version of the magnetohydrodynamics equations comparisons with well known discontinuous galerkin approximations for the poisson equation are presented

symmetric positive systems with boundary characteristic of constant multiplicity the theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity no such hypothesis is needed on a neighborhood of the boundary both regularity theorems and mixed initial boundary value problems are discussed many classical ideas are sharpened in the process

curved planar quantum wires with dirichlet and neumann boundary conditions we investigate the discrete spectrum of the hamiltonian describing a quantum particle living in the two dimensional curved strip we impose the dirichlet and neumann boundary conditions on opposite sides of the strip the existence of the discrete eigenvalue below the essential spectrum threshold depends on the sign of the total bending angle for the asymptotically straight strips

the local discontinuous galerkin method for the oseen equations we introduce and analyze the local discontinuous galerkin method for the oseen equations of incompressible fluid flow for a class of shape regular meshes with hanging nodes we derive optimal a priori estimates for the errors in the velocity and the pressure in l 2 and negative order norms numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of reynolds numbers

bound states in straight quantum waveguides with combined boundary conditions we investigate the discrete spectrum of the hamiltonian describing a quantum particle living in the two dimensional straight strip we impose the combined dirichlet and neumann boundary conditions on different parts of the boundary several statements on the existence or the absence of the discrete spectrum are compared for two models with combined boundary conditions examples of eigenfunctions and eigenvalues are computed numerically

a domain decomposition method for solving the three dimensional time harmonic maxwell equations discretized by discontinuous galerkin methods we present here a domain decomposition method for solving the three dimensional time harmonic maxwell equations discretized by a discontinuous galerkin method in order to allow the treatment of irregularly shaped geometries the discontinuous galerkin method is formulated on unstructured tetrahedral meshes the domain decomposition strategy takes the form of a schwarz type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains a multifrontal sparse direct solver is used at the subdomain level the resulting domain decomposition strategy can be viewed as a hybrid iterative direct solution method for the large sparse and complex coefficients algebraic system resulting from the discretization of the time harmonic maxwell equations by a discontinuous galerkin method

a remark on von neumann wigner type potentials a general formulation of the modulation function approach to von neumann wigner type potentials is given covering recent discussions of such potentials as special cases

solution of the time harmonic maxwell equations using discontinuous galerkin methods we present numerical results concerning the solution of the time harmonic maxwell equations discretized by discontinuous galerkin methods in particular a numerical study of the convergence which compares different strategies proposed in the literature for the elliptic maxwell equations is performed in the two dimensional case

numerical analysis of the modified evss method abstract in this paper we present a proof of the stability of a new mixed finite element method introduced recently by the authors 9 within the context of viscoelastic fluids the mixed formulation is related to the evss elastic viscous split stress method proposed by rajagopalan et al 14 and is based on the introduction of the rate of deformation tensor as an additional unknown the proof applies to the stokes flow and some linearized constitutive equations of slow viscoelastic flows existence and uniqueness of the continuous and the discrete problems are derived from a generalized brezzi babuska theory it is shown that no additional compatibility condition is required between the various variables except the usual one for the velocity and the pressure fields this result allows to choose low order finite element for the stress several numerical experiments on the 4 1 contraction stokes flow will be presented which will confirm the improved stability obtained with this new formulation