a structure preserving split finite element discretization of the split wave equations abstract we introduce a new finite element fe discretization framework applicable for covariant split equations the introduction of additional differential forms df that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric dependent closure equations that apply the hodge star operator our discretization framework conserves this geometrical structure and provides for all dfs proper fe spaces such that the differential operators here gradient and divergence hold in strong form we introduce lowest possible order discretizations of the split 1d wave equations in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant fe spaces omitting partial integrations approximating the hodge star by nontrivial galerkin projections gp the two discrete metric equations follow by projections onto either the piecewise constant gp0 or piecewise linear gp1 space out of the four possible realizations our framework gives us three schemes with significantly different behavior the split scheme using twice gp1 is unstable and shares the dispersion relation with the p1 p1 fe scheme that approximates both variables by piecewise linear spaces p1 the split schemes that apply a mixture of gp1 and gp0 share the dispersion relation with the stable p1 p0 fe scheme that applies piecewise linear and piecewise constant p0 spaces however the split schemes exhibit second order convergence for both quantities of interest for the split scheme applying twice gp0 we are not aware of a corresponding standard formulation to compare with though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find fe schemes by new combinations of fe spaces

a constraint generation approach for two machine shop problems with jobs selection abstract we consider job selection problems in two stage flow shops and job shops the aim is to select the best job subset with a given cardinality to minimize the makespan these problems are known to be ordinary np hard and the current state of the art algorithms can solve flow shop problems with up to 3000 jobs we introduce a constraint generation approach to the integer linear programming ilp formulation of these problems according to which the constraints associated with nearly all potential critical paths are relaxed and then only the ones violated by the relaxed solution are sequentially reinstated the proposed approach is capable of solving problems with up to 100 000 jobs

finite volume transport on various cubed sphere grids the performance of a multidimensional finite volume transport scheme is evaluated on the cubed sphere geometry advection tests with prescribed winds are used to evaluate a variety of cubed sphere projections and grid modifications including the gnomonic and conformal mappings as well as two numerically generated grids by an elliptic solver and spring dynamics we explore the impact of grid non orthogonality on advection tests over the corner singularities of the cubed sphere grids using some variations of the transport scheme including the piecewise parabolic method with alternative monotonicity constraints the advection tests revealed comparable or better accuracy to those of the original latitudinal longitudinal grid implementation it is found that slight deviations from orthogonality on the modified cubed sphere quasi orthogonal grids do not negatively impact the accuracy in fact the more uniform version of the quasi orthogonal cubed sphere grids provided better overall accuracy than the most orthogonal and therefore much less uniform conformal grid it is also shown that a simple non orthogonal extension to the transport equation enables the use of the highly non orthogonal and computationally more efficient gnomonic grid with acceptable accuracy

oracle dependent properties of the lattice of np sets abstract we consider under the assumption p np questions concerning the structure of the lattice of np sets together with the sublattice p we show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize the two questions which we consider are whether every infinite np set contains an infinite p subset and whether there exists an np simple set we construct several oracles all of which make p np and which in addition make the above mentioned statements either true or false in particular we give a positive answer to the question raised by bennett and gill 1981 whether an oracle b exists making p b np b and such that every infinite set in np b has an infinite subset in p b the constructions of the oracles are finite injury priority arguments

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

optimal constrained interpolation in mesh adaptive finite element modeling mesh to mesh galerkin l 2 projection allows piecewise polynomial unstructured finite element data to be interpolated between two nonmatching unstructured meshes of the same domain the interpolation is by definition optimal in an l 2 sense and subject to fairly weak assumptions conserves the integral of an interpolated function however other properties such as the l 2 norm or the weak divergence of a vector valued function can still be adversely affected by the interpolation this is an important issue for calculations in which numerical dissipation should be minimized or for simulations of incompressible flow this paper considers extensions to mesh to mesh galerkin l 2 projection which are l 2 optimal and ensure exact conservation of key discrete properties including preservation of both the l 2 norm and the integral and preservation of both the l 2 norm and weak incompressibility the accuracy of the interpolants is studied the utility of the interpolants is studied via adaptive

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

a polyhedral approach to multicommodity survivable network design the design of cost efficient networks satisfying certain survivability constraints is of major concern to the telecommunications industry in this paper we study a problem of extending the capacity of a network by discrete steps as cheaply as possible such that the given traffic demand can be accommodated even when a single edge or node in the network fails we derive valid and nonredundant inequalities for the polyhedron of capacity design variables by exploiting its relationship to connectivity network design and knapsack like subproblems a cutting plane algorithm and heuristics for the problem are described and preliminary computational results are reported

weak form of stokes dirac structures and geometric discretization of port hamiltonian systems we present the mixed galerkin discretization of distributed parameter port hamiltonian systems due to the inherent definition of boundary interconnection ports this system representation is particularly useful for the modeling of interconnected multi physics systems and control at the prototypical example of a system of two conservation laws in arbitrary spatial dimension we derive the main contributions i a weak formulation of the underlying geometric stokes dirac structure ii its geometric approximation by a finite dimensional dirac structure using a mixed galerkin approach and power preserving maps on the space of discrete power variables and iii the approximation of the hamiltonian to obtain finite dimensional port hamiltonian state space models the power preserving maps on the discrete bond space offer design degrees of freedom for the discretization which is illustrated at the example whitney finite elements on a 2d simplicial triangulation the resulting schemes can be considered as trade offs between centered approximations and upwinding

a gradient robust well balanced scheme for the compressible isothermal stokes problem abstract a novel notion for constructing a well balanced scheme a gradient robust scheme is introduced and a showcase application for the steady compressible isothermal stokes equations in a nearly hydrostatic situation is presented gradient robustness means that gradient fields in the momentum balance are well balanced by the discrete pressure gradient which is possible on arbitrary unstructured grids the scheme is asymptotic preserving in the sense that it reduces for low mach numbers to a recent inf sup stable and pressure robust discretization for the incompressible stokes equations the convergence of the coupled fem fvm scheme for the nonlinear isothermal stokes equations is proved by compactness arguments numerical examples illustrate the numerical analysis and show that the novel approach can lead to a dramatically increased accuracy in nearly hydrostatic low mach number flows numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible

a proximal cutting plane method using chebychev center for nonsmooth convex optimization an algorithm is developed for minimizing nonsmooth convex functions this algorithm extends elzinga moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones thus removing compactness assumption our method is to elzinga moore s algorithm what a proximal bundle method is to kelley s algorithm instead of lower approximations used in proximal bundle methods the present approach is based on some objects regularizing translated functions of the objective function we propose some variants and using some academic test problems we conduct a numerical comparative study with elzinga moore algorithm and two other well known nonsmooth methods

numerical wave propagation on the hexagonal c grid inertio gravity mode and rossby mode dispersion properties are examined for discretizations of the linearized rotating shallow water equations on a regular hexagonal c grid in planar geometry it is shown that spurious non zero rossby mode frequencies found by previous authors in the f plane case can be avoided by an appropriate discretization of the coriolis terms three generalizations of this discretization that conserve energy even for non constant coriolis parameter are presented a quasigeostrophic b plane analysis is carried out to investigate the rossby mode dispersion properties of these three schemes the rossby mode dispersion relation is found to have two branches the primary branch modes are good approximations in terms of both structure and frequency to corresponding modes of the continuous governing equations and offer some improvements over a quadrilateral c grid scheme the secondary branch modes have vorticity structures approximating those of small scale modes of the continuous governing equations suggesting that the hexagonal c grid might have an advantage in terms of resolving extra rossby modes however the frequencies of the secondary branch rossby modes are much smaller than those of the corresponding continuous modes so this potential advantage is not fully realized

conservative interpolation of edge and face data on n dimensional structured grids using differential forms interpolation methods for edge and face centered data are described which preserve line and area integrals under regridding these interpolation methods complement the multilinear nodal and conservative interpolation methods which are widely used in climate data processing and other areas the presented interpolation schemes ensure that curl free and divergence free fields remain so after regridding these edge and face conservative interpolation methods are suitable for general curvilinear structured grids including those with singular points poles support for masked invalid regions is implicitly provided by attaching partial line surface integral field values to cell edges faces

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

discrete hodge operators many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms discrete differential forms provide a natural and canonical approach to their discretization however much freedom remains concerning the choice of discrete hodge operators that is discrete analogues of constitutive laws a generic discrete hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations we reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus

vertical slice modelling of nonlinear eady waves using a compatible finite element method a vertical slice model is developed for the euler boussinesq equations with a constant temperature gradient in the direction normal to the slice the eady boussinesq model the model is a solution of the full three dimensional equations with no variation normal to the slice which is an idealised problem used to study the formation and subsequent evolution of weather fronts a compatible finite element method is used to discretise the governing equations to extend the charney phillips grid staggering in the compatible finite element framework we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space for the time discretisation we solve the semi implicit equations together with an explicit strong stability preserving runge kutta scheme to all of the advection terms the model reproduces several quasi periodic lifecycles of fronts despite the presence of strong discontinuities an asymptotic limit analysis based on the semi geostrophic theory shows that the model solutions are converging to a solution in cross front geostrophic balance the results are consistent with the previous results using finite difference methods indicating that the compatible finite element method is performing as well as finite difference methods for this test problem we observe dissipation of kinetic energy of the cross front velocity in the model due to the lack of resolution at the fronts even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi geostrophic limit solution

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

almost every set in exponential time is p bi immune a set a is p bi immune if neither a nor its complement has an infinite subset in p we investigate here the abundance of p bi immune languages in linear exponential time e using resource bounded measure we prove that the class of p bi immune sets has measure 1 in e this implies that almost every language in e is p bi immune a bit further we show that every p random pseudorandom language is e bi immune regarding the existence of p bi immune sets in np we show that if np does not have measure 0 in e then np contains a p bi immune set another consequence is that the class of 1 tt p complete languages for e has measure 0 in e in contrast using the approach of resource bounded category it is shown that in e the class of p bi immune languages lacks the property of baire the baire category analogue of lebesgue measurability

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 framework for mimetic discretization of the rotating shallow water equations on arbitrary polygonal grids accurate simulation of atmospheric flow in weather and climate prediction models requires the discretization of the governing equations to have a number of desirable properties although these properties can be achieved relatively straightforwardly on a latitude longitude grid they are much more challenging on the various quasi uniform spherical grids that are now under consideration a recently developed scheme called trisk has these desirable properties on grids that have an orthogonal dual the present work extends the trisk scheme into a more general framework suitable for grids that have a nonorthogonal dual such as the equiangular cubed sphere we also show that this framework fits within the wider framework of mimetic discretizations and discrete exterior calculus one key ingredient is the definition of certain mapping operators that are discrete analogues of the hodge star operator enabling the definition of a compatible inner product discrete coriolis terms are also included within the mimet

numerical representation of geostrophic modes on arbitrarily structured c grids a c grid staggering in which the mass variable is stored at cell centers and the normal velocity component is stored at cell faces or edges in two dimensions is attractive for atmospheric modeling since it enables a relatively accurate representation of fast wave modes however the discretization of the coriolis terms is non trivial for constant coriolis parameter the linearized shallow water equations support geostrophic modes stationary solutions in geostrophic balance a naive discretization of the coriolis terms can cause geostrophic modes to become non stationary causing unphysical behaviour of numerical solutions recent work has shown how to discretize the coriolis terms on a planar regular hexagonal grid to ensure that geostrophic modes are stationary while the coriolis terms remain energy conserving in this paper this result is extended to arbitrarily structured c grids an explicit formula is given for constructing an appropriate discretization of the coriolis terms the general formula is illustrated by showing that it recovers previously known results for the planar regular hexagonal c grid and the spherical longitude latitude c grid numerical calculation confirms that the scheme does indeed give stationary geostrophic modes for the hexagonal pentagonal and triangular geodesic c grids on the sphere

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

a constraint generation algorithm for large scale linear programs using multiple points separation in order to solve linear programs with a large number of constraints constraint generation techniques are often used in these algorithms a relaxation of the formulation containing only a subset of the constraints is first solved then a separation procedure is called which adds to the relaxation any inequality of the formulation that is violated by the current solution the process is iterated until no violated inequality can be found in this paper we present a separation procedure that uses several points to generate violated constraints the complexity of this separation procedure and of some related problems is studied also preliminary computational results about the advantages of using multiple points separation procedures over traditional separation procedures are given for random linear programs and survivable network design they illustrate that for some specific families of linear programs multiple points separation can be computationally effective

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

a unified approach to energy conservation and potential vorticity dynamics for arbitrarily structured c grids a numerical scheme applicable to arbitrarily structured c grids is presented for the nonlinear shallow water equations by discretizing the vector invariant form of the momentum equation the relationship between the nonlinear coriolis force and the potential vorticity flux can be used to guarantee that mass velocity and potential vorticity evolve in a consistent and compatible manner underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field the numerical scheme also exhibits conservation of total energy to within time truncation error simulations of the standard shallow water test cases confirm the analysis and show convergence rates between 1st and 2nd order accuracy when discretizing the system with quasi uniform spherical voronoi diagrams the numerical method is applicable to a wide class of meshes including latitude longitude grids voronoi diagrams delaunay triangulations and conformally mapped cubed sphere meshes

higher order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere abstract we describe a compatible finite element discretisation for the shallow water equations on the rotating sphere concentrating on integrating consistent upwind stabilisation into the framework although the prognostic variables are velocity and layer depth the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a taylor galerkin scheme this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency we also use upwind discontinuous galerkin schemes for the transport of layer depth these transport schemes are incorporated into a semi implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed helmholtz equation we demonstrate that our discretisation achieves the expected second order convergence and provide results from some standard rotating sphere test problems

the analogue of grad div stabilization in dg methods for incompressible flows limiting behavior and extension to tensor product meshes abstract grad div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems for mitigating velocity errors that are sometimes called poor mass conservation such errors arise due to the relaxation of the divergence constraint in classical mixed methods and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures in this contribution an analogue of grad div stabilization for discontinuous galerkin methods is studied here the key is the penalization of the jumps of the normal velocities over facets of the triangulation which controls the measure valued part of the distributional divergence of the discrete velocity solution our contribution is twofold first we characterize the limit for arbitrarily large penalization parameters which shows that the stabilized nonconforming discontinuous galerkin methods remain robust and accurate in this limit second we extend these ideas to the case of non simplicial meshes here broken grad div stabilization must be used in addition to the normal velocity jump penalization in order to get the desired pressure robustness effect the analysis is performed for the stokes equations and more complex flows and crouzeix raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings

the recovered space advection scheme for lowest order compatible finite element methods abstract we present a new compatible finite element advection scheme for the compressible euler equations unlike the discretisations described in cotter and kuzmin 2016 and shipton et al 2018 the discretisation uses the lowest order family of compatible finite element spaces but still retains second order numerical accuracy this scheme obtains this second order accuracy by first recovering the function in higher order spaces before using the discontinuous galerkin advection schemes of cotter and kuzmin 2016 as well as describing the scheme we also present its stability properties and a strategy for ensuring boundedness we then demonstrate its properties through some numerical tests before presenting its use within a model solving the compressible euler equations

high order taylor galerkin methods for linear hyperbolic systems a new family of high order taylor galerkin schemes is presented for the analysis of first order linear hyperbolic systems the schemes are unconditionally stable which makes them very attractive to use in conjunction with adaptive hp finite element methods for spatial approximation

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

the use of proper orthogonal decomposition pod meshless rbf fd technique to simulate the shallow water equations abstract the main aim of this paper is to develop a fast and efficient local meshless method for solving shallow water equations in one and two dimensional cases the mentioned equation has been classified in category of advection equations the solutions of advection equations have some shock thus especial numerical methods should be employed for example discontinuous galerkin and finite volume methods here based on the proper orthogonal decomposition approach we want to construct a fast meshless method to this end we consider shallow water models and obtain a suitable time discrete scheme based on the predictor corrector technique then by applying the proper orthogonal decomposition technique a new set of basis functions can be built for the solution space in which the size of new solution space is less than the original problem thus by employing the new bases the cpu time will be reduced some examples have been studied to show the efficiency of the present numerical technique

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

efficient assembly of h mathrm div and h mathrm curl conforming finite elements in this paper we discuss how to efficiently evaluate and assemble general finite element variational forms on h mathrm div and h mathrm curl the proposed strategy relies on a decomposition of the element tensor into a precomputable reference tensor and a mesh dependent geometry tensor two key points must then be considered the appropriate mapping of basis functions from a reference element and the orientation of geometrical entities to address these issues we extend here a previously presented representation theorem for affinely mapped elements to piola mapped elements we also discuss a simple numbering strategy that removes the need to contend with directions of facet normals and tangents the result is an automated efficient and easy to use implementation that allows a user to specify finite element variational forms on h mathrm div and h mathrm curl in close to mathematical notation

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

wave dispersion properties of compound finite elements mixed finite elements use different approximation spaces for different dependent variables certain classes of mixed finite elements called compatible finite elements have been shown to exhibit a number of desirable properties for a numerical weather prediction model in two dimensions the lowest order element of the raviartthomas based mixed element is the finite element equivalent of the widely used c grid staggering which is known to possess good wave dispersion properties at least for quadrilateral grids it has recently been proposed that building compound elements from a number of triangular raviartthomas sub elements such that both the primal and implied dual grid are constructed from the same sub elements would allow greater flexibility in the use of different advection schemes along with the ability to build arbitrary polygonal elements although the wave dispersion properties of the triangular sub elements are well understood those of the compound elements are unknown it would be useful to know how they compare with the non compound elements and what properties of the triangular sub grid elements are inherited here a numerical dispersion analysis is presented for the linear shallow water equations in two dimensions discretised using the lowest order compound raviartthomas finite elements on regular quadrilateral and hexagonal grids it is found that in comparison with the well known c grid scheme the compound elements exhibit a more isotropic dispersion relation with a small over estimation of the frequency for short waves compared with the relatively large underestimation for the c grid on a quadrilateral grid the compound elements are found to differ from the non compound raviartthomas quadrilateral elements even for uniform elements exhibiting the influence of the underlying sub elements this is shown to lead to small improvements in the accuracy of the dispersion relation the compound quadrilateral element is slightly better for gravity waves but slightly worse for inertial waves than the standard lowest order raviartthomas element

a primal dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes a new numerical method is presented for solving the shallow water equations on a rotating sphere using quasi uniform polygonal meshes the method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation the method relies on two novel features the first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes the second is the use of dual finite element spaces on the dual of the original mesh along with suitably defined discrete hodge star operators to map between the primal and dual meshes enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes the resulting method has the same mimetic properties as a finite volume method presented previously but is more accurate on a number of standard test cases

rbf nets for approximating an object s boundary by image random sampling abstract our aim is to discuss the problem of approximating the boundary of an object in a binary image in contrast to earlier papers on similar topics we avoid finding contours and then their approximations instead we sample an image in a relatively small number of points 1 3 of pixels at random the collected data is used for imposing constraints on parameters of a radial basis functions rbf neural net it is proved that if the rbf net structure is sufficiently rich to approximate an object boundary then estimates of rbf net parameters tend to their true values as the number of samples approaches to infinity then a seemingly linear algorithm of estimating linear weights is proposed and its consistency is also proved its implementation which is based on an iterative use of the linear programming is briefly discussed and the results of its testing are shortly reported

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

embedded discontinuous galerkin transport schemes with localised limiters motivated by finite element spaces used for representation of temperature in the compatible finite element approach for numerical weather prediction we introduce locally bounded transport schemes for partially continuous finite element spaces the underlying high order transport scheme is constructed by injecting the partially continuous field into an embedding discontinuous finite element space applying a stable upwind discontinuous galerkin dg scheme and projecting back into the partially continuous space we call this an embedded dg transport scheme we prove that this scheme is stable in l 2 provided that the underlying upwind dg scheme is we then provide a framework for applying limiters for embedded dg transport schemes standard dg limiters are applied during the underlying dg scheme we introduce a new localised form of element based flux correction which we apply to limiting the projection back into the partially continuous space so that the whole transport scheme is bounded we provide details in the specific case of tensor product finite element spaces on wedge elements that are discontinuous p1 q1 in the horizontal and continuous p2 in the vertical the framework is illustrated with numerical tests

inspection of hexagonal and triangular c grid discretizations of the shallow water equations c grid discretizations based on a hexagonal or triangular mesh can be investigated with the help of a planar trivariate coordinate system where the vector components are either defined tangentially hexagonal c grid or perpendicularly triangular c grid to the coordinate lines inspecting the helmholtz decomposition of a vector in case of the linearly dependent trivariate coordinate description reveals insights into the structure and stencil of the discretized divergence and vorticity on such grids from a vector laplacian which is consistent with the helmholtz decomposition a general formulation for the inner product operator at grid edges can be derived thus the vector reconstruction of the tangential wind for the coriolis term in the shallow water equations can be given even for a slightly distorted tesselation as present on icosahedral spherical grids furthermore a rigorous comparison of the triangular and the hexagonal c grid linear shallow water equations is performed from a theoretical and also from an experimental viewpoint it turns out that the additional degree of freedom in the height field in the triangular c grid case compared to the hexagonal c grid is responsible for the decoupling of divergence values on upward and downward directed triangles this problem occurs especially for small rossby deformation radii and practically requires additional explicit diffusion in contrast the hexagonal c grid discretization has remarkable similarity to the quadrilateral c grid case in the wave dispersion properties and eigenvector structure numerical experiments performed with that option proved resilent

simplicity immunity relativizations and nondeterminism a diagonalization method is discussed by which a recursive oracle a can be constructed such that np a has a set l a which is both simple and p a immune

rehabilitation of the lowest order raviart thomas element on quadrilateral grids a recent study d n arnold d boffi and r s falk siam j numer anal 42 2005 pp 2429 2451 reveals that convergence of finite element methods using h mathrm div omega compatible finite element spaces deteriorates on nonaffine quadrilateral grids this phenomena is particularly troublesome for the lowest order raviart thomas elements because it implies loss of convergence in some norms for finite element solutions of mixed and least squares methods in this paper we propose reformulation of finite element methods based on the natural mimetic divergence operator m shashkov conservative finite difference methods on general grids crc press boca raton fl 1996 which restores the order of convergence reformulations of mixed galerkin and least squares methods for the darcy equation illustrate our approach we prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator furthermore we prove that standard and reformulated versions of the mixed galerkin method lead to identical linear systems but the two versions of the least squares method are veritably different the surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial and that the lowest order raviart thomas elements are safe to use in this method however the breakdown in the least squares method is real and there one should use our proposed reformulation

relativizations comparing np and exponential time the possible relationships between np and expak c 0 dtime 2cnk relative to oracles are examined it is first shown that for every oracle including the empty set and any k npa expak then it is shown that all other relationships are possible under relativization that is for each k 0 oracles a b and c are constructed such that i panpaexpak ii expbknpb and iii npc and expck are incomparable with respect to inclusion the construction of the set a is especially intricate apparently requiring a finite injury priority argument in each case in the constructions when a possible inclusion is ruled out it is done in a very strong way namely by finding a language in one of the classes which is immune with respect to the other class

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

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