the hamiltonian structure for dynamic free boundary problems hamiltonian structures for 2 or 3 dimensional incompressible flows with a free boundary are determined which generalize a previous structure of zakharov for irrotational flow our poisson bracket is determined using the method of arnold namely reduction from canonical variables in the lagrangian material description using this bracket the hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated the structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained in the case of two dimensional flows the vorticity bracket is determined and the generalized enstrophy is shown to be a casimir function this investigation also clears up some confusion in the literature concerning the vorticity bracket even for fixed boundary flows

toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

multitarget error estimation and adaptivity in aerodynamic flow simulations important quantities in aerodynamic flow simulations are the aerodynamic force coefficients including the pressure induced and the viscous stress induced drag lift and moment coefficients in addition to the exact approximation of these quantities it is of increasing importance in particular in the field of uncertainty quantification to estimate the error in the computed quantities in recent years a posteriori error estimation and goal oriented refinement approaches have been developed for the accurate and efficient computation of single target quantities the current approaches are based on computing an adjoint solution related to each of the specific target quantities under consideration in this paper we extend this approach to the accurate and efficient computation of multiple target quantities instead of computing multiple adjoint solutions one for each target functional the new approach is based on the solution to one discrete adjoint problem and one discrete error problem this way only two auxiliary problems are required irrespective of the number of target functionals the practical performance of this approach is demonstrated for a laminar compressible flow in particular the proposed approach is compared to the standard approach of error estimation and goal oriented refinement as well as to residual based refinement the performance of the algorithms is measured in terms of computing resources required for meeting industrial as well as academic accuracy requirements on the computed force coefficients

error analysis for optimal control problem governed by convection diffusion equations dg method in the current paper we study the convergence properties of the dgfe approximation of optimal control problem governed by convection diffusion equations we derive a posteriori error estimates and a priori error estimates for both the states ad joint and the control variable approximation for the optimal control problem these estimates are apparently not available in the literature

error driven dynamical hp meshes with the discontinuous galerkin method for three dimensional wave propagation problems an hp adaptive discontinuous galerkin method for electromagnetic wave propagation phenomena in the time domain is proposed the method is highly efficient and allows for the first time the adaptive full wave simulation of large time dependent problems in three dimensional space refinement is performed anisotropically in the approximation order p and the mesh step size h regardless of the resulting level of hanging nodes for guiding the adaptation process a variant of the concept of reference solutions with largely reduced computational costs is proposed the computational mesh is adapted such that a given error tolerance is respected throughout the entire time domain simulation

canonical description of ideal magnetohydrodynamic flows and integrals of motion in the framework of the variational principle the canonical variables describing magnetohydrodynamic mhd flows of general type i e with spatially varying entropy and nonzero values of all topological invariants are introduced it is shown that the velocity representation of the clebsch type following from the variational principle with constraints is equivalent to that resulting from the generalization of the weber transformation performed in the paper for the case of arbitrary mhd flows using such complete velocity representation enables us not only to describe the general type flows in terms of single valued functions but also to solve the intriguing problem of the missing mhd integrals of motion the set of hitherto known mhd local invariants and integrals of motion appears to be incomplete for the vanishing magnetic field it does not reduce to the set of the conventional hydrodynamic invariants and if the analogs of the vorticity and helicity were discussed earlier for the particular cases the analog of ertel invariant has been so far unknown it is shown that all missing invariants are expressed in terms of the decomposition of the velocity representation into the hydrodynamic and magnetic parts in spite of the nonunique character of such representation it is shown that there exists a natural restriction of the gauge transformations set allowing one to make the invariants gauge independent it is found that on the basis of the new invariants introduced a wide set of high order invariants can be constructed the new invariants are relevant both for the deeper insight into the problem of the topological structure of the mhd flows as a whole and for the examination of the stability problems the additional advantage of the proposed approach is that it enables one to deal with discontinuous flows including all types of possible breaks

optimal order a posteriori error estimates for a class of runge kutta and galerkin methods we derive a posteriori error estimates which exhibit optimal global order for a class of time stepping methods of any order that include runge kutta collocation rk c methods and the continuous galerkin cg method for linear and nonlinear stiff odes and parabolic pdes the key ingredients in deriving these bounds are appropriate one degree higher continuous reconstructions of the approximate solutions and pointwise error representations the reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time step they do not hinge on asymptotics

computational modelling of multiscale multiphase fluid mixtures with application to tumour growth in this work we consider the discretization of a recently formulated collis et al 22 multiscale model for drug and nutrient limited tumour growth the key contribution of this work is the proposal of a discontinuous galerkin finite element scheme which incorporates a non standard coupling across a singular surface and the presentation of full details of a suitable discretization for the coupled flow and transport systems such as that arising in collis et al 22 and other similar works we demonstrate the application of the proposed discretizations via representative numerical experiments furthermore we present a short numerical study of convergence for the proposed microscale scheme in which we observe optimal rates of convergence for sufficiently smooth data

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

isospectral problem interplay between liouville equations darboux transforms and mckean trubowitz flows systems of liouville equations are generated by isospectral transforms of the schrodinger operator and describe the isospectral evolution of both a potential and eigenfunctions relations of these equations with the mckean trubowitz isospectral flows are studied the role of these flows among other isospectral flows is discussed

error control for a class of runge kutta discontinuous galerkin methods for nonlinear conservation laws we propose an a posteriori error estimate for the runge kutta discontinuous galerkin method rk dg of arbitrary order in arbitrary space dimensions for stabilization of the scheme a general framework of projections is introduced finally it is demonstrated numerically how the a posteriori error estimate is used to design both an efficient grid adaption and gradient limiting strategy numerical experiments show the stability of the scheme and the gain in efficiency in comparison with computations on uniform grids

a goal oriented hp adaptive discontinuous galerkin approach for biharmonic problems summary a goal oriented algorithm is developed and applied for hp adaptive approximations given by the discontinuous galerkin finite element method for the biharmonic equation the methodology is based on the dual problem associated with the target functional we consider three error estimators and analyse their properties as basic tools for the design of the hp adaptive algorithm to improve adaptation the combination of two different error estimators is used each one at its best efficiency to guide the tasks of where and how to adapt the approximation spaces the performance of the resulting hp adaptive schemes is illustrated by numerical experiments for two benchmark problems copyright 2013 john wiley sons ltd

two new potentials for the free particle model in this work is proposed a very simple method for obtaining the generalized potential associated with a known standard potential the procedure is straightforward because it only uses two ricatti type relationships as enough condition to find a generalized potential one particular equation is needed to identify the specific potential under study and one general ricatti relationship is used to find the corresponding generalized potential moreover the method is completely general due to the fact that an arbitrary potential has been considered in its development for which the procedure can also be used to find new potentials which could be needed in the modeling of important quantum interactions the usefulness of the proposed approach is shown with the treatment of the three and one dimensional potential for the free particle model this work example leads to two new potentials whose hamiltonians are isospectral when they are compared with the former hamiltonian

on the nonlinear generalization of the fock method the generalized fock method is presented in the case of nonlinear dependence of the number of quanta operator on the hamiltonian operator that generalization is used for an analysis of the inverse problem in the case of strictly discrete spectra in r1 the relationship between the quantum description on a basis of the fock approach and the classical limit of this description is analysed

semi implicit discontinuous galerkin methods for the incompressible navier stokes equations on adaptive staggered cartesian grids in this paper a new high order semi implicit discontinuous galerkin method si dg is presented for the solution of the incompressible navier stokes equations on staggered space time adaptive cartesian grids amr in two and three space dimensions the pressure is written in the form of piecewise polynomials on the main grid which is dynamically adapted within a cell by cell amr framework according to the time dependent main grid different face based spatially staggered dual grids are defined for the piece wise polynomials of the respective velocity components although the resulting adaptive staggered grids are more complex than classical uniform cartesian meshes the numerical scheme can still be written in a rather compact form thanks to the use of a tensor product formulation for the definition of the nodal basis in the d dimensional space d 2 3 all the discrete operators can be efficiently written as a combination of linear one dimensional operators acting in the d space directions separately arbitrary high order of accuracy is achieved in space while a very simple semi implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation the real advantages of the staggered grid arise in the solution of the schur complement associated with the saddle point problem of the discretized incompressible navier stokes equations i e after substituting the discrete momentum equations into the discrete continuity equation this leads to a linear system for only one unknown the scalar pressure indeed the resulting linear pressure system is shown to be symmetric and positive definite in order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor an implicit discretization is also used for the diffusive terms in the momentum equation a particular feature of our staggered dg approach is that the viscous stress tensor is discretized on the dual mesh this corresponds to the use of a lifting operator but on the staggered grid both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix free conjugate gradient method for which fast convergence is observed moreover it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner due to the explicit discretization of the nonlinear convective terms the final algorithm is stable within the classical cfl type time step restriction for explicit dg methods based on the fluid velocity moreover by using a suitable local time stepping technique based on the ader dg framework each element can use its optimal local time step also within the amr context the new space time adaptive staggered dg scheme has been thoroughly verified for polynomial degrees up to n 9 for a large set of non trivial test problems in two and three space dimensions for which analytical numerical or experimental reference solutions exist the high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution to the knowledge of the authors this is the first staggered semi implicit dg scheme for the incompressible navier stokes equations on space time adaptive meshes in two and three space dimensions

adjoint sensitivity formulation for discontinuous galerkin discretizations in unsteady inviscid flow problems this paper presents an unsteady discrete adjoint algorithm for high order implicit discontinuous galerkin discretizations in time dependent inviscid flow problems the major function of the adjoint approach is to obtain the sensitivity information in a time dependent functional output which in turn is used to drive an unsteady shape optimization process to deliver a minimum of the objective functional a gradient based optimization strategy is investigated in which the sensitivity derivatives of the objective functional with respect to input variables are formulated in the context of high order discontinuous galerkin discretizations while special emphasis is given to the variations and linearizations of curvilinear boundary elements implicit temporal discretizations consisting of a second order backward euler scheme and a fourth order implicit runge kutta scheme are considered exclusively in this work where the corresponding adjoint problem is required to be solved in a backward time integration manner due to the associated transpose operation two numerical examples for the unsteady shape design techniques are presented to verify the derived sensitivity formulations and to demonstrate the performance of the adjoint approach the first involves an inverse shape optimization case by matching a time dependent target pressure profile for a two dimensional periodic vortical gust impinging on a rae 2822 airfoil and the second considers minimization of the acoustic noise produced by subsonic flow over a naca0012 airfoil with a 0 03c thick blunt trailing edge

a posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems abstract in this paper we consider the a posteriori error analysis of discontinuous galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions we establish several residual based a posteriori error estimators which provide global upper bounds and a local lower bound on the error further for nonsteady problem we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space our a posteriori error analysis is based on the mesh dependent a priori estimates for the first order hyperbolic problems these a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods

generalization of the darboux transform this article presents a generalization of the standard darboux transform applied to sturm liouville differential equations this is achieved with the aid of an ansatz as a particular solution for the riccati relationship involved which in turn led us to obtain its generalized darboux solution that contains as a particular case the standard darboux transform the proposed generalized darboux transform gdt applied to the quantum mechanical field gives the opportunity to prove the existence of standard and generalized darboux potentials that match with the so called isospectral potentials this is exemplified by obtaining through the gdt a set of standard and generalized darboux potentials that form the partner of the one dimensional harmonic oscillator model for any quantum principal number the worked example indicates how the gdt can be used to obtain the isospectral potentials associated to any known specific potential we consider also the application of our method as proposed to the theory of

an h1 ph coercive discontinuous galerkin formulation for the poisson problem 1 d analysis coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite element discretizations by the classical lax milgram theorem any conforming discretization of a coercive variational problem is stable i e discrete approximations are well posed and possess unique solutions irrespective of the specifics of the underlying approximation space based on the prototypical one dimensional poisson problem we establish in this work that most concurrent discontinuous galerkin formulations for second order elliptic problems represent instances of a generic conventional formulation and that this generic formulation is noncoercive consequently all conventional discontinuous galerkin formulations are a fortiori noncoercive and typically their well posedness is contingent on approximation space dependent stabilization parameters moreover we present a new symmetric nonconventional discontinuous galerkin formulation based on element green s functions and the data local to the edges we show that the new discontinuous galerkin formulation is coercive on the broken sobolev space h1 ph viz the space of functions that are elementwise in the h1 sobolev space the coercivity of the new formulation is supported by calculations of discrete inf sup constants and numerical results are presented to illustrate the optimal convergence behavior in the energy norm and in the l2 norm

hp adaptive discontinuous galerkin finite element methods for first order hyperbolic problems we consider the a posteriori error analysis of hp discontinuous galerkin finite element approximations to first order hyperbolic problems in particular we discuss the question of error estimation for linear functionals such as the outflow flux and the local average of the solution based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance this involves exploiting both local polynomial degree variation and local mesh subdivision the theoretical results are illustrated by a series of numerical experiments

on the nonlinear fock description of quantum systems with quadratic spectra the nonlinear version of the fock approach developed previously for systems with extended equidistant spectra eleonsky v m and korolev v g 1995 j phys a math gen 28 4973 is also shown to describe systems with quadratic spectra some general properties of shift operators for such spectra are studied it is found in particular that families of shift operators related to the generators of the algebra su 1 1 as well as supersymmetry ladder operators arise naturally in that approach in a unified way

the spectral class of the quantum mechanical harmonic oscillator the purpose of this paper is to study the so calledspectral classq of anharmonic oscillatorsq d2 q having the same spectrum n 2n n 0 as the harmonic oscillatorq0 d2 x2 1 thenorming constants t n mathop lim limits x uparrow infty ell g 1 n e n x mathord left vphantom e n x e n right kern nulldelimiterspace e n x of the eigenfunctions ofq form a complete set of coordinates inq in terms of which the potential may be expressed asq x2 1 2d2 lg with theta det left delta ij e ti 1 int limits x infty e i 0 e j 0 0 leqq i j infty right e n 0 being thenth eigenfunctionq0 the spectrum and norming constants are canonically conjugate relative to the bracket f g fd gdx to wit i j 0 ti 2 j 1 or 0 according to whetheri j or not and ti tj 0 this prompts an investigation of the symplectic geometry ofq the function is related to the theta function of a singular algebraic curve numerical results are also presented

numerical study of transport properties in continuum percolation the authors present numerical simulations of ac conductance for a random resistor capacitor network the conductance obeys a probability density function p g varies as g alpha 0 alpha 1 they use a highly efficient propagation algorithm to calculate the effective conductance of a long strip of a lattice at low frequencies they find that for the concentration p of conducting bonds less than the percolation threshold pc the imaginary part of conductance is proportional to frequency im geff approximately omega and the real part of conductance shows an anomalous frequency dependence re geff approximately omega 2 alpha the results of simulations in such a continuum system are in agreement with the predictions of the effective medium and the maxwell garnett approximation they also calculate the non universal dc conductivity exponents in continuum percolation the results are consistent with earlier theoretical predictions and numerical calculations

variational principle and canonical variables in hydrodynamics with discontinuities the paper discusses variational principles describing ideal fluid motion with discontinuities including shocks in canonical variables the three types of variational principles are discussed the general one that describes all types of discontinuities and the two simplified ones that describe separately the case of slide and contact discontinuities and the shock case

goal oriented error estimation and adaptivity for free boundary problems the domain map linearization approach in free boundary problems the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation we develop duality based a posteriori error estimates for functional outputs of solutions of free boundary problems that include both sources of error the derivation of an appropriate dual problem linearized adjoint is however nonobvious for free boundary problems to derive an appropriate dual problem we present the domain map linearization approach in this approach the free boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map we show for a bernoulli type free boundary problem that this dual problem corresponds to a poisson problem with a nonlocal robin type boundary condition furthermore we present numerical experiments that demonstrate the effectivity of the dual based error estimate and its usefulness in goal oriented adaptive mesh refinement

adaptive discontinuous galerkin finite element methods for the compressible euler equations in this paper a recently developed approach for the design of adaptive discontinuous galerkin finite element methods is applied to physically relevant problems arising in inviscid compressible fluid flows governed by the euler equations of gas dynamics in particular we employ weighted type i a posteriori bounds to drive adaptive finite element algorithms for the estimation of the error measured in terms of general target functionals of the solution these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated this general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest as well as providing reliable and efficient error estimation the superiority of the proposed approach over mesh refinement algorithms which employ standard unweighted type ii error indicators which do not require the solution of an auxiliary problem are illustrated by a series of numerical experiments here we consider transonic flow through a nozzle as well as subsonic and supersonic flows around different airfoil geometries

adjoint based h p adaptive discontinuous galerkin methods for the 2d compressible euler equations in this paper we investigate and present an adaptive discontinuous galerkin algorithm driven by an adjoint based error estimation technique for the inviscid compressible euler equations this approach requires the numerical approximations for the flow i e primal problem and the adjoint i e dual problem which corresponds to a particular simulation objective output of interest the convergence of these two problems is accelerated by an hp multigrid solver which makes use of an element gauss seidel smoother on each level of the multigrid sequence the error estimation of the output functional results in a spatial error distribution which is used to drive an adaptive refinement strategy which may include local mesh subdivision h refinement local modification of discretization orders p enrichment and the combination of both approaches known as hp refinement the selection between h and p refinement in the hp adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values numerical results for the inviscid compressible flow over an idealized four element airfoil geometry demonstrate that both pure h refinement and pure p enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach but requiring fewer degrees of freedom the proposed hp adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom and results in significant savings in computational cost a high speed flow test case is used to demonstrate the ability of the hp refinement approach for capturing strong shocks or discontinuities while improving functional accuracy

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related