numerical wave propagation for the triangular p1dg p2 finite element pair the f plane and b plane wave propagation properties are examined for discretisations of the linearised rotating shallow water equations using the p1 d g p2 finite element pair on arbitrary triangulations in planar geometry a discrete helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations in the f plane planar geometry coriolis force independent of space case this decomposition is used to obtain decoupled equations for the geostrophic modes the inertia gravity modes and the inertial oscillations as has been noticed previously the geostrophic modes are steady the helmholtz decomposition is used to show that the resulting inertia gravity wave equation is third order accurate in space in general the p1 d g p2 finite element pair is second order accurate so this leads to very accurate wave propagation it is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency f and which do not propagate a restriction of the p1 d g velocity space is proposed in which these modes are not present leading to a finite element discretisation which is completely free of spurious modes the helmholtz decomposition also allows a simple derivation of the quasi geostrophic limit of the discretised p1 d g p2 equations in the b plane planar geometry coriolis force linear in space case resulting in a rossby wave equation which is also third order accurate this means that the dispersion relation for the wave propagation is very accurate an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles

geometric aspects of sturm liouville problems iii level surfaces of the nth eigenvalue for a given regular sturm liouville equation and any natural number n we explicitly describe the level surfaces of the nth eigenvalue of the equation regarded as a function on the space of self adjoint boundary conditions the description of each level surface uses only a particular value of the transfer matrix of the equation the shapes of these surfaces are also determined

matrix representations of sturm liouville problems with transmission conditions we identify a class of sturm liouville equations with transmission conditions such that any sturm liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem conversely given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self adjoint boundary condition and transmission condition we construct a class of sturm liouville problems with this specified boundary condition and transmission condition each of which is equivalent to the given matrix eigenvalue problem

dependence of eigenvalues on the diffusion operators with random jumps from the boundary abstract this paper deals with a non self adjoint eigenvalue problem a x y x b x y x y x y 0 0 1 y x d 0 x y 1 0 1 y x d 1 x which is associated with the generator of one dimensional diffusions with random jumps from the boundary we focus on the dependence of spectral gap eigenvalues and eigenfunctions on the coefficients a b and the probability distributions 0 1 to prove this we show that all the eigenvalues are confined to a parabolic neighborhood of the real axis moreover we also prove that zero is an algebraically simple eigenvalue of the problem

a semi implicit relaxation scheme for modeling two phase flow in a pipeline this work is devoted to the numerical approximation of the solutions to the system of conservation laws which arises in the modeling of two phase flows in pipelines the pdes are closed by two highly nonlinear algebraic relations namely a pressure law and a hydrodynamic law we have previously proposed an explicit relaxation scheme which allows us to cope with these nonlinearities but the system considered has eigenvalues which are of very different orders of magnitude which prevents the explicit scheme from being effective since the time step has to be very small in order to solve this effectiveness problem we now proceed to construct a scheme which is explicit with respect to the small eigenvalues and linearly implicit with respect to the large eigenvalues numerical evidences are provided

dissipative non self adjoint sturm liouville operators and completeness of their eigenfunctions abstract in this paper non self adjoint sturm liouville operators in weyl s limit circle case are studied we first determine all the non self adjoint boundary conditions yielding dissipative operators for each allowed sturm liouville differential expression then using the characteristic determinant the completeness of the system of eigenfunctions and associated functions for these dissipative operators is proved

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

global existence of weak solutions for a viscous two phase model abstract the purpose of this paper is to explore a viscous two phase liquid gas model relevant for well and pipe flow our approach relies on applying suitable modifications of techniques previously used for studying the single phase isothermal navier stokes equations a main issue is the introduction of a novel two phase variant of the potential energy function needed for obtaining fundamental a priori estimates we derive an existence result for weak solutions in a setting where transition to single phase flow is guaranteed not to occur when the initial state is a true mixture of both phases some numerical examples are also included in order to demonstrate characteristic behavior of solutions in particular we illustrate how two phase flow is genuinely different compared to single phase flow concerning the behavior of an initial mass discontinuity

dependence of eigenvalues of sturm liouville problems with interface conditions this paper investigates sturm liouville problems with interface conditions we obtain that the eigen values depend not only continuously but also smoothly on the coefficient functions boundary conditions and interface conditions moreover the derivative of the eigenvalues with respect to each of these parameters is given

left definite sturm liouville problems abstract left definite regular self adjoint sturm liouville problems with either separated or coupled boundary conditions are studied we give an elementary proof of the existence of eigenvalues for these problems for any fixed equation we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions some of our results here yield an algorithm for numerically computing the eigenvalues of a left definite problem with an arbitrary coupled boundary condition our inequalities imply that the well known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case moreover we study the continuous and differentiable dependence of the eigenvalues of the general left definite problem on all the parameters in its differential equation and boundary condition

the study of jacobi and cyclic jacobi matrix eigenvalue problems using sturm liouville theory abstract we study the eigenvalues of matrix problems involving jacobi and cyclic jacobi matrices as functions of certain entries of particular interest are the limits of the eigenvalues as these entries approach infinity our approach is to use the recently discovered equivalence between these problems and a class of sturm liouville problems and then to apply the sturm liouville theory

entropy satisfying relaxation method with large time steps for euler ibvps this paper could have been given the title how to positively and implicitly solve euler equations using only linear scalar advections the new relaxation method we propose is able to solve euler like systems as well as initial and boundary value problems with real state laws at very low cost using a hybrid explicit implicit time integration associated with the arbitrary lagrangian eulerian formalism furthermore it possesses many attractive properties such as i the preservation of positivity for densities ii the guarantee of min max principle for mass fractions iii the satisfaction of entropy inequality under an expressible bound on the cfl ratio the main feature that will be emphasized is the design of this optimal time step which takes into account data not only from the inner domain but also from the boundary conditions

analysis of numerically induced oscillations in 2d finite element shallow water models part i inertia gravity waves the numerical approximation of shallow water models is a delicate problem for most of the discretization schemes the coupling between the momentum and the continuity equations usually leads to anomalous dispersion in the representation of fast waves a dispersion relation analysis is employed here to ascertain the presence and determine the form of spurious modes as well as the dispersive nature of the finite element galerkin mixed formulation of the two dimensional linearized shallow water equations nine popular finite element pairs are considered using a variety of mixed interpolation schemes for each pair the frequency or dispersion relation is obtained and analyzed and the dispersion properties are compared analytically and graphically with the continuous case it is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of fast waves the p nc 1 p 1 and rt0 pairs are identified as a promising compromise provided the grid resolution is high relative to the rossby radius of deformation for the rt0 element the numerical solutions of two test problems to simulate fast waves are in good agreement with the analytical results

analyticity of the spectrum and dirichlet to neumann operator technique for quantum graphs in some previous works the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph were established however a specific local coordinate chart on the grassmannian of all possible vertex conditions was used thus creating an erroneous impression that something wrong can happen at the boundaries of the chart here we show that the analyticity of the corresponding dispersion relation holds over the whole grassmannian as well as over other parameter spaces we also address the dirichlet to neumann technique of relating quantum and discrete graph operators which allows one to transfer some results from the discrete to the quantum graph case but which has issues at the dirichlet spectrum we conclude that this difficulty as in the first part of the paper stems from the use of specific coordinates in a grassmannian and show how to avoid it to extend some of the consequent results to the general situation in some previous works the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph were established however a specific local coordinate chart on the grassmannian of all possible vertex conditions was used thus creating an erroneous impression that something wrong can happen at the boundaries of the chart here we show that the analyticity of the corresponding dispersion relation holds over the whole grassmannian as well as over other parameter spaces we also address the dirichlet to neumann technique of relating quantum and discrete graph operators which allows one to transfer some results from the discrete to the quantum graph case but which has issues at the dirichlet spectrum we conclude that this difficulty as in the first part of the paper stems from the use of specific coordinates in a grassmannian and show how to avoid it to extend some of the consequent results to the general situation

dependence of the nth sturm liouville eigenvalue on the problem abstract we consider the nth eigenvalue as a function on the space of self adjoint regular sturm liouville problems with positive leading coefficient and weight functions the discontinuity of the nth eigenvalue is completely characterized

mixed finite elements for numerical weather prediction we show how mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo uniform grids this family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal c grid finite difference methods there are a few major advantages the mixed finite element methods do not require an orthogonal grid and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation these methods preserve several properties of the c grid method when applied to linear barotropic wave propagation namely a energy conservation b mass conservation c no spurious pressure modes and d steady geostrophic modes on the f plane we explain how these properties are preserved and describe two examples that can be used on pseudo uniform grids the recently developed modified rtk q k 1 element pairs on quadrilaterals and the bdfm1 p 1 dg element pair on triangles all of these mixed finite element methods have an exact 2 1 ratio of velocity degrees of freedom to pressure degrees of freedom finally we illustrate the properties with some numerical examples

a note on eigenvalues of a class of singular continuous and discrete linear hamiltonian systems abstract in this paper we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear hamiltonian systems are equal where both endpoints are in the limit circle cases the proof is fundamental and is given for both continuous and discrete hamiltonian systems the method used in this paper also works for both endpoints are regular or one endpoint is regular and the other is in the limit circle case

dispersion relation analysis of the p nc_1 p _1 finite element pair in shallow water models shallow water models typically employ gridpoint finite and spectral element techniques for most of these methods the coupling between the momentum and continuity equations is a delicate problem and usually leads to spurious solutions in the representation of inertia gravity waves the spurious modes have a wide range of characteristics and may take the form of pressure surface elevation velocity and or coriolis modes the modes usually cause aliasing and an accumulation of energy in the smallest resolvable scale leading to noisy solutions the triangular finite element pair p nc 1 p 1 is shown to properly model the dispersion of the inertia gravity waves and this is achieved at a reasonable computational cost compared to traditional finite difference schemes two tests are proposed the first test examines the propagation and dispersion of fast surface gravity waves in a circular basin and their reflection at the lateral boundary in the second test results of the stommel problem are presented using the proposed finite element pair and are compared with those obtained with the c and cd finite difference grids they illustrate the promise of the proposed approach

a relaxation method for two phase flow models with hydrodynamic closure law this paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two phase flows in pipelines the pdes are closed by two highly nonlinear algebraic relations namely a pressure law and a hydrodynamic one the severe nonlinearities encoded in these laws make the classical approximate riemann solvers virtually intractable at a reasonable cost of evaluation we propose a strategy for relaxing solely these two nonlinearities the relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate such a property not only makes it trivial to solve the riemann problem but also enables us to enforce some further stability requirements in addition to those coming from a chapman enskog analysis the new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions extensive numerical evidences are provided

a robust and accurate riemann solver for a compressible two phase flow model in this paper we analyze the riemann problem for the widely used drift flux two phase flow model this analysis introduces the complete information that is attained in the representation of solutions to the riemann problem it turns out that the riemann waves have rarefactions a contact discontinuity and shocks within this respect an exact riemann solver is developed to accurately resolve and represent the complete wave structure of the gas liquid two phase flows to verify the solver a series of test problems selected from the literature are presented including validation against independent numerical simulations where the solution of the riemann problem is fully numerical in this framework the governing equations are discretized by finite volume techniques facilitating the application godunov methods of centred type it is shown that both analytical and numerical results demonstrate the broad applicability and robustness of the new exact riemann solver

eigenvalues of regular sturm liouville problems abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem an expression for the derivative of an eigenvalue with respect to a given parameter an endpoint a boundary condition a coefficient or the weight function is found

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

floquet theory for left definite sturm liouville problems abstract we determine the essential spectrum of left definite sturm liouville problems with periodic coefficients and a weight function which changes sign

a priori error estimates for mixed finite element approximations of the acoustic wave equation in this paper we derive optimal a priori l infty l 2 error estimates for mixed finite element displacement formulations of the acoustic wave equation the computational complexity of this approach is equivalent to the traditional mixed finite element formulations of the second order hyperbolic equations in which the primary unknowns are pressure and the gradient of pressure however the displacement formulations with the physical variables of interest displacement and pressure requires less regularity on the displacement

a finite element exterior calculus framework for the rotating shallow water equations we describe discretisations of the shallow water equations on the sphere using the framework of finite element exterior calculus which are extensions of the mimetic finite difference framework presented in ringler 2010 11 the exterior calculus notation provides a guide to which finite element spaces should be used for which physical variables and unifies a number of desirable properties we present two formulations a primal formulation in which the finite element spaces are defined on a single mesh and a primal dual formulation in which finite element spaces on a dual mesh are also used both formulations have velocity and layer depth as prognostic variables but the exterior calculus framework leads to a conserved diagnostic potential vorticity in both formulations we show how to construct discretisations that have mass consistent constant potential vorticity stays constant stable and oscillation free potential vorticity advection

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

symplectic mixed finite element approximation of linear acoustic wave equations we apply mixed finite element approximations to the first order form of the acoustic wave equation the semidiscrete method exactly conserves the system energy a fully discrete method employing the symplectic euler time method in time exactly conserves a positive definite pertubed energy functional that is equivalent to the actual energy under a cfl condition in addition to proving optimal order l infty l 2 l l2 estimates we also develop a bootstrap technique that allows us to derive stability and error bounds for the time derivatives and divergence of the vector variable beyond the standard under some additional regularity assumptions

an isoperimetric type inequality for the principal eigenvalue of schrodinger operators depending on the curvature of a loop abstract let be the curvature of a smooth closed curve of length 2 the 2 periodic eigenvalue problems u g 2 u u 0 where g is a real parameter are considered and upper and lower bounds for their principal eigenvalues are obtained the bounds are of isoperimetric type and characterize uniquely the curve in the case where equality occurs

mixed finite elements for global tide models with nonlinear damping we study mixed finite element methods for the rotating shallow water equations with linearized momentum terms but nonlinear drag by means of an equivalent second order formulation we prove long time stability of the system without energy accumulation we also give rates of damping in unforced systems and various continuous dependence results on initial conditions and forcing terms a priori error estimates for the momentum and free surface elevation are given in l 2 as well as for the time derivative and divergence of the momentum numerical results confirm the theoretical results regarding both energy damping and convergence rates

mixed finite elements for global tide models we study mixed finite element methods for the linearized rotating shallow water equations with linear drag and forcing terms by means of a strong energy estimate for an equivalent second order formulation for the linearized momentum we prove long time stability of the system without energy accumulation the geotryptic state a priori error estimates for the linearized momentum and free surface elevation are given in l 2 l2 as well as for the time derivative and divergence of the linearized momentum numerical results confirm the theoretical results regarding both energy damping and convergence rates

singular sturm liouville problems the friedrichs extension and comparison of eigenvalues a new characterization of singular self adjoint boundary conditions for sturm liouville problems is given these are an exact parallel of the regular case they are given explicitly in terms of principal and non principal solutions the special nature of the friedrichs extension is clearly apparent and highlighted inequalities among the eigenvalues of different boundary conditions separated and coupled are obtained most of all e want to stress the method of proof it is based on a very elementary transformation which transforms any singular non oscillatory limit circle endpoint into a regular one

dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary in this paper we consider the dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary we show that the eigenvalues depend not only continuously but smoothly on boundary points and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation in addition we prove that as the length of the interval shrinks to zero all higher fourth order dirichlet eigenvalues march off to plus infinity this is also true for the first i e lowest eigenvalue

regular third order boundary value problems abstract in this paper we consider some boundary value problems generated by third order formally symmetric self adjoint regular differential expression and separated real coupled and complex coupled boundary conditions it is shown that these problems generate self adjoint operators moreover the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced

existence of multiple positive solutions for sturm liouville like four point boundary value problem with p laplacian abstract in this paper we consider the sturm liouville like four point boundary value problem with p laplacian p u t f t u t 0 t 0 1 u 0 u 0 u 1 u 0 where p s s p 2 s p 1 by means of a fixed point theorem for operators on a cone the existence of multiple at least three positive solutions to the above boundary value problem is obtained

geometric aspects of sturm liouville problems ii space of boundary conditions for left definiteness for a given regular sturm liouville equation with an indefinite weight function we explicitly describe the space of left definite selfadjoint boundary conditions the description only uses one value of a fundamental solution of the matrix form of the equation as a consequence we show that this space has the shape of a solid consisting of two cones sharing a common base

analysis of numerically induced oscillations in two dimensional finite element shallow water models part ii free planetary waves the behavior of planetary rossby waves in finite element numerical models is investigated by using a quasi geostrophic approximation to the midlatitude beta plane two dimensional linear shallow water equations a dispersion relation analysis is employed here to determine the possible occurrence of spurious modes and to ascertain the dispersive dissipative nature of the finite element galerkin mixed formulation three finite element pairs are considered by using a variety of mixed interpolation schemes for each pair the frequency or dispersion relation is obtained and analyzed and the dispersion properties are compared analytically and graphically with the continuous case it is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of slow rossby waves due to the coupling between the momentum and continuity equations numerical solutions of two test problems to simulate slow rossby waves are in good agreement with the analytical results

error estimate of eigenvalues of perturbed second order discrete sturm liouville problems abstract this paper is concerned with eigenvalues of perturbed second order vector discrete sturm liouville problems by some variational properties of eigenvalues of discrete sturm liouville problems error estimates of eigenvalues of perturbed problems sufficiently close to a given sturm liouville problem are given under a certain non singularity condition perturbations of the coefficient functions of the difference equation the weight function and the coefficients of the boundary condition are all considered this together with higher dimension involved results in a certain complexity of the problem and difficulty of study as a direct consequence continuous dependence of eigenvalues on problems is obtained under the non singularity condition in addition an example is presented to illustrate the necessity of the non singularity condition

on the boundary conditions in self adjoint multi interval sturm liouville problems abstract in this paper we completely solve a problem whose two interval case has been studied recently by wang et al in a wang j sun a zettl two interval sturm liouville operators in modified hilbert spaces j math anal appl 328 2007 390 399 the problem is to give an explicit description of all boundary conditions defining the self adjoint differential operators associated with several sturm liouville equations together i e the so called multi interval sturm liouville problems the explicit description obtained in this paper only uses hermitian matrices as an application we then show that a generic self adjoint boundary condition can not come from any direct sum of self adjoint operators associated with the individual sturm liouville equations and it depends non trivially on the multiples used in the definition of the so called maximum operator

dependence of eigenvalues of sturm liouville problems on the boundary abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on boundary points the derivative of the n th eigenvalue as a function of an endpoint satisfies a first order differential equation this for arbitrary separated or coupled self adjoint regular boundary conditions in addition as the length of the interval shrinks to zero all higher eigenvalues march off to plus infinity this is also true for the first i e lowest dirichlet eigenvalue but not for the lowest neumann eigenvalue the latter has a finite limit