fundamental group schemes for stratified sheaves abstract we study stratified sheaves in positive characteristic algebraic geometry using the technique of tannakian categories we show that the associated tannakian group scheme is a perfect scheme we also prove that in the proper case the largest unipotent quotient of is pro etale special attention is paid to abelian varieties where a description of is obtained we finish with a discussion of the case where the ground field is valued

a study of solvers for nonlinear afc discretizations of convection diffusion equations abstract nonlinear discretizations are necessary for convection diffusion equations for obtaining accurate solutions that satisfy the discrete maximum principle the numerical solution of the arising nonlinear problems is often difficult this paper presents several approaches for solving the nonlinear problems of algebraic flux correction afc schemes for the kuzmin limiter and the bjk limiter comprehensive numerical studies are performed at examples that model the transport of energy from a body in a flow field in two and three dimensions it turns out that the most efficient approach from the point of view of computing times is a simple fixed point iteration because the iteration matrix possesses properties that can be exploited by the solvers of the arising linear systems of equations

an unstructured finite elements method for solving the compressible rans equations and the spalart allmaras turbulence model this paper deals with a stabilized finite element method for solving the compressible navier stokes equations combined with the spalart allmaras turbulence model the aim is to assess the ability of this formulation to solve high reynolds number turbulent flows over anisotropic meshes this formulation lies in the framework of the stream line upwind petrove galerkin method we propose a simple and efficient formulation for the stabilization matrix and develop a shock capturing operator numerical tests on the 3d boundary layer over a flat plate and on the onera m6 wing show the stability and robustness of the proposed method

an eulerian finite element formulation for modelling stationary finite strain elastic deformation processes a new finite element formulation aimed at the modelling of stationary finite strain elastic deformation processes is presented the new formulation is based on an eulerian description of motion and on the transport of the deformation gradient tensor copyright 2004 john wiley sons ltd

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

regular singular stratified bundles and tame ramification let x be a smooth variety over an algebraically closed field k of positive characteristic we define and study a general notion of reg ular singularities for stratified bundles i e ox coherent dx k modules on x without relying on resolution of singularities the main result is that the category of regular singular stratified bundles with finite mon odromy is equivalent to the category of continuous representations of the tame fundamental group on finite dimensional k vector spaces as a corollary we obtain that a stratified bundle with finite monodromy is regular singular if and only if it is regular singular along all curves mapping to x if x is a smooth connected complex variety then it is an elementary fact that a vector bundle with flat connection and finite monodromy at a closed point x x c is automatically regular singular this implies to gether with the riemann hilbert correspondence as developed in del70 that the category of flat connections with finite monodromy is equivalent to the category repf cont 1 x x of continuous representations of theetale fun damental group of x on finite dimensional complex vector spaces equipped with the discrete topology the goal of this article is to establish an analo gous statement over an algebraically closed field k of positive characteristic p

a geometrical method towards first integrals for dynamical systems we develop a method based on darboux s and liouville s works to find first integrals and or invariant manifolds for a physically relevant class of dynamical systems without making any assumption on these elements forms we apply it to three dynamical systems lotka volterra lorenz and rikitake

a note on constant free a posteriori error estimates in this note we look at constant free a posteriori error estimates from a different perspective we show that they can be interpreted as an alternative way of expressing the residual of a finite element approximation and thus fit into the same framework as other a posteriori error estimates such as residual error indicators our approach also reveals that when applied to singularly perturbed reaction diffusion or convection diffusion problems constant free a posteriori error estimates will not be fully robust unless extra measures are taken

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

adjoint weighted equation for inverse problems of incompressible plane stress elasticity the adjoint weighted equation awe formulation is applied to incompressible isotropic plane stress elasticity for the solution of inverse problems for the strong form of the problem to have a solution a restrictive strain compatibility condition must hold in the awe variational formulation on the other hand a much milder sufficient condition is that uniaxial stress states be precluded making the method more robust to noisy data we prove that the formulation leads to a stable converging numerical method and show through computations that the method performs well

invariant algebraic curves and conditions for a centre conditions for the existence of a centre in two dimensional systems are considered along the lines of darboux we show how these methods can be used in the search for maximal numbers of bifurcating limit cycles we also extend the method to include more degenerate cases such as are encountered in less generic systems these lead to new classes of integrals in particular the kukles system is considered and new centre conditions for this system are obtained

linear and non linear high order accurate residual distribution schemes for the discretization of the steady compressible navier stokes equations a robust and high order accurate residual distribution rd scheme for the discretization of the steady navier stokes equations is presented the proposed method is very flexible it is formulated for unstructured grids regardless the shape of the elements and the number of spatial dimensions a continuous approximation of the solution is adopted and standard lagrangian shape functions are used to construct the discrete space as in finite element methods the traditional technique for designing rd schemes is adopted evaluate for any element a total residual split it into nodal residuals sent to the degrees of freedom of the element solve the non linear system that has been assembled and then iterate up to convergence the main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries this is no longer true since the gradient of the state solution appears in the flux hence continuity is lost when using standard finite element approximations naive solution methods lead to very poor accuracy to cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution preserving the optimal accuracy of the method linear and non linear schemes are constructed and their accuracy is tested with the method of the manufactured solutions the numerical method is also used for the discretization of smooth and shocked laminar flows in two and three spatial dimensions we propose a very high order accurate methods for the approximation of the compressible navier stokes equations with residual distribution schemes the memory footprint is lower than for discontinuous galerkin like methods we show uniform accuracy for the whole range of reynolds numbers the solution technique is described in details and we show how to reach excellent convergence

invariant algebraic surfaces of the rikitake system in this paper we use the method of characteristic curves for solving linear partial differential equations to study the invariant algebraic surfaces of the rikitake system x y z y x z z xy our main results are the following first we show that the cofactor of any invariant algebraic surface is of the form rz c where r is an integer second we characterize all invariant algebraic surfaces moreover as a corollary we characterize all values of the parameters for which the rikitake system has a rational or algebraic first integral

analysis of a new stabilized higher order finite element method for advection diffusion equations abstract we consider a singularly perturbed advection diffusion two point boundary value problem whose solution has a single boundary layer based on piecewise polynomial approximations of degree k 1 a new stabilized finite element method is derived in the framework of a variation multiscale approach the method coincides with the supg method for k 1 but differs from it for k 2 estimates for the error to an appropriate interpolant are given in several norms on different types of meshes for k 1 enhanced accuracy is achieved by superconvergence postprocessing guarantees the same estimates for the error to the solution itself

free surface fluid dynamics on moving domains this paper presents a lagrangian approach for the simulation of two dimensional free surface flows along with a systematic validation fluid flow is traditionally modeled using an eulerian description in association with finite differences and more recently finite elements free surface flow introduces new difficulties in connection with the time dependence of the domain the proposed approach sets a framework for meshfree and finite element formulations it is extended to two phase media in preisig and zimmermann m preisig th zimmermann two phase free surface fluid dynamics on moving domains j comput phys 229 7 2010 2740 2758 in this paper triangular finite elements and a remeshing procedure to avoid excessive element distortion are chosen for their robustness convergence of the lagrangian description is analyzed and compared to results obtained with an eulerian description the equivalence of the two formulations for confined flows is demonstrated numerically illustrations include stratified flow flow over a backward facing step propagation of a solitary wave and dam break meshfree formulations will be addressed in separate papers c 2010 elsevier b v all rights reserved

stabilized residual distribution for shallow water simulations we propose a stabilized residual distribution rd scheme for the simulation of shallow water flows the final discretization is obtained combining the stabilized rd approach proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 with the conservative formulation already used in ricchiuto et al j comp phys 222 2007 to simulate shallow water flows the scheme proposed is a nonlinear variant of a lax friedrichs type discretization it is well balanced it actually yields second order of accuracy in smooth areas and it preserves the positivity of the height of the water in presence of dry areas this is made possible by the residual character of the discretization by properly adapting the stabilization operators proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 and thanks to the positivity preserving character of the underlying lax friedrichs scheme we demonstrate the properties of the discretization proposed on a wide variety of tests

tannakian duality over dedekind rings and applications we establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor called tannakian lattice both defined over a dedekind ring we use this duality and the known tannakian duality due to saavedra to study morphisms between flat affine group schemes next we apply our new duality to the category of stratified sheaves on a smooth scheme over a dedekind ring r to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers when r is a complete dvr of equal characteristic we show that this category is tannakian in the sense of saavedra

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

developments of entropy stable residual distribution methods for conservation laws i this paper presents preliminary developments of entropy stable residual distribution methods for scalar problems controlling entropy generation is achieved by formulating an entropy conserved signals distribution coupled with an entropy stable signals distribution numerical results of the entropy stable residual distribution methods are accurate and comparable with the classic residual distribution methods for steady state problems high order accurate extensions for the new method on steady state problems are also demonstrated moreover the new method preserves second order accuracy on unsteady problems using an explicit time integration scheme the idea of the multi dimensional entropy stable residual distribution method is generic enough to be extended to the system of hyperbolic equations which will be presented in the sequel of this paper

a petrov galerkin modified operator formulation for convection diffusion problems a new petrov galerkin method to deal with convection diffusion problems is presented the formulation is derived from the concept of using a modifying function to make the differential operator self adjoint the so called optimal upwind parameter arises naturally from the process of approximating the modifying function transient and steady state examples on uniform and non uniform meshes are shown

residual based stabilized formulation for the solution of inverse elliptic partial differential equations abstract we consider an inverse problem where the forward problem is that of linear plane stress elasticity or equivalently that of linear heat hydraulic conduction we demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem in particular it is stable for the response variable and the lagrange multiplier but not for the material property field this lack of stability implies that we are unable to prove optimal convergence with mesh refinement for the overall problem we overcome this difficulty by adding to the saddle point problem a residual based term that provides sufficient stability and prove optimal convergence in an energy like norm we verify these estimates through simple numerical examples we note that while we have considered a specific model for an inverse elliptic problem in this manuscript similar ideas could be developed for a broad class of inverse elliptic problems

on uniformly high order accurate residual distribution schemes for advection diffusion we discuss preliminary results on the construction of uniformly high order residual distribution rd type discretizations for steady advection diffusion on unstructured grids a properly designed scaling of the rd upwind stabilization with the physical viscosity allows to obtain schemes with uniform and arbitrary accuracy on a very compact stencil second and third order examples are given to illustrate the potential of the approach

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

a fully eulerian formulation for fluid structure interaction problems in this work we present a fully eulerian framework for fluid structure interaction fsi problems coupling the incompressible navier stokes equations with a hyperelastic solid the fully eulerian framework is a monolithic implicit variational formulation for the coupled problem in contrast to the well established arbitrary lagrangian eulerian ale coordinates the fully eulerian framework formulates both subproblems fluid and solid in eulerian coordinates this concept circumvents various difficulties connected to ale coordinates since no artificial domain mapping is used the formulation is an interface capturing method and uses an extension of the solid s deformation the initial point set to detect the interface location by construction very large deformation as well as topology changes like contact of the solid to the domain boundary or other solid parts are possible

third order residual distribution schemes for the navier stokes equations we construct a third order multidimensional upwind residual distribution scheme for the system of the navier stokes equations the underlying approximation is obtained using standard p2 lagrange finite elements to discretise the inviscid component of the equations each element is divided in sub elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub element the residuals are distributed to the nodes of each sub element in a multi dimensional upwind way to obtain a discretisation of the viscous terms consistent with this multi dimensional upwind approach we make use of a petrov galerkin analogy the analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms the performance of this high order method is tested on flows with high and low reynolds number

adaptive spacetime discontinuous galerkin method for hyperbolic advection diffusion with a non negativity constraint summary applications where the diffusive and advective time scales are of similar order give rise to advection diffusion phenomena that are inconsistent with the predictions of parabolic fickian diffusion models non fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in fickian models in this work we implement a modified frame invariant form of cattaneo s hyperbolic diffusion relation within a spacetime discontinuous galerkin advection diffusion model an h adaptive spacetime meshing procedure supports an asynchronous patch by patch solution procedure with linear computational complexity in the number of spacetime elements this localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non negative concentration solution in contrast to some previous methods we do not modify the numerical fluxes to enforce non negative concentrations thus the element wise conservation properties that are intrinsic to discontinuous galerkin models are defined with respect to physically meaningful riemann fluxes on the element boundaries we present numerical examples that demonstrate the effectiveness of the proposed model and we explore the distinct features of hyperbolic advection diffusion response in subcritical and supercritical flows copyright 2015 john wiley sons ltd

the shifted boundary method for embedded domain computations part ii linear advection diffusion and incompressible navier stokes equations abstract we propose a new embedded finite element method for the linear advection diffusion equation and the laminar and turbulent incompressible navier stokes equations the proposed method belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates we include the full analysis of stability and convergence of the method in the linear advection diffusion equation and a battery of tests for the case of laminar and turbulent incompressible navier stokes equations we also discuss the conservation properties of the proposed method in all cases

a posteriori optimization of parameters in stabilized methods for convection diffusion problems part i abstract stabilized finite element methods for convection dominated problems require the choice of appropriate stabilization parameters from numerical analysis often only their asymptotic values are known this paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional exemplarily this framework is applied to the supg finite element method and the minimization of a residual based error estimator an error indicator and a functional including the crosswind derivative of the computed solution benefits of the basic approach are demonstrated by means of numerical results

on integrality of p adic iterated integrals abstract the purpose of this paper is to prove integrality for certain p adic iterated coleman integrals as underlying geometry we will take the complement of a divisor d x with good reduction where x is the projective line or an elliptic curve over the witt vectors of a perfect characteristic p field as a corollary we prove a lower bound for the valuations of p adic multiple zeta values

adaptive techniques in sold methods we present new results where free parameters in spurious oscillations at layers diminishing sold method are adaptively chosen provided numerical results are for conforming piecewise linear finite element space with free parameters from piecewise constant finite element space we focus on a higher order convergence we discovered in previous paper by lukas 2015

numerical computation of nonlinear normal modes in mechanical engineering this paper reviews the recent advances in computational methods for nonlinear normal modes nnms different algorithms for the computation of undamped and damped nnms are presented and their respective advantages and limitations are discussed the methods are illustrated using various applications ranging from low dimensional weakly nonlinear systems to strongly nonlinear industrial structures

convergence of a residual based artificial viscosity finite element method we present a residual based artificial viscosity finite element method to solve conservation laws the galerkin approximation is stabilized by only residual based artificial viscosity without any least squares supg or streamline diffusion terms we prove convergence of the method applied to a scalar conservation law in two space dimensions toward an unique entropy solution for implicit time stepping schemes

galois theory for iterative connections and nonreduced galois groups this article presents a theory of modules with iterative connection this theory is a generalisation of the theory of modules with connection in characteristic zero as given by n katz to modules over rings of arbitrary characteristic we show that these modules with iterative connection and also the modules with integrable iterative connection form a tannakian category assuming some nice properties for the underlying ring and we show how this generalises to modules over schemes stratifications on modules as introduced by a grothendieck generalise the notion of integrable ordinary connections in another way however over smooth rings we obtain an equivalence of stratifications and integrable iterative connections furthermore over a regular ring in positive characteristic we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by d gieseker in the second part of this article we set up a picard vessiot theory for fields of solutions for such a picard vessiot extension we obtain a galois correspondence which takes into account even nonreduced closed subgroup schemes of the galois group scheme on one hand and inseparable intermediate extensions of the picard vessiot extension on the other hand finally we compare our galois theory with the galois theory for purely inseparable field extensions given by s chase

darboux theory of integrability in the sparse case darboux s theorem and jouanolou s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field these results are given in terms of the degree of the polynomial vector field here we show that we can get the same kind of results if we consider the size of a newton polytope associated to the vector field furthermore we show that in this context the bound is optimal

modeling of steep layers in singularly perturbed diffusion reaction equation via flexible fine scale basis abstract we present a new stabilized method for the diffusion reaction equation which develops sharp boundary and or internal layers for the reaction dominated case i e singularly perturbed case the method relies on an improved expression for the stabilization parameter that is derived via the fine scale variational formulation facilitated by the variational multiscale vms framework the proposed fine scale basis consists of enrichment functions which may be nonzero at element edges the derived stabilization parameter enjoys spatial variation over element interiors and along inter element boundaries that helps model the rapid variation of the residual of the euler lagrange equations over the domain this feature also facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects the method is able to better satisfy the maximum principle as compared to other existing methods we present a priori mathematical analysis of the stability and convergence of the method for the diffusion reaction equation optimal convergence on meshes comprised of linear triangles and bilinear quadrilateral elements are presented for smooth problems as well as for problems with steep boundary layers stability and accuracy features of the method for problems with discontinuous forcing function internal layers and boundary layers are shown and its performance on unstructured and distorted meshes comprised of quadrilateral and triangular elements is highlighted

finite element formulation of general boundary conditions for incompressible flows abstract we study the finite element formulation of general boundary conditions for incompressible flow problems distinguishing between the contributions from the inviscid and viscid parts of the equations we use nitsche s method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter including the limit case of the euler equations in order to control the discrete kinetic energy additional consistent terms are introduced we treat the limit case as a degenerate system of hyperbolic equations using a balanced spectral decomposition of the flux jacobian matrix in analogy with compressible flows then following the theory of friedrich s systems the natural characteristic boundary condition is generalized to the considered physical boundary conditions several numerical experiments including standard benchmarks for viscous flows as well as inviscid flows are presented

invariant algebraic surfaces of the fitzhugh nagumo system abstract in this paper we characterize all the irreducible darboux polynomials and polynomial first integrals of the fitzhugh nagumo system by the called characteristic curves method which involves the theory of weight homogeneous polynomials it has been ever widely used to give a complete classification of darboux polynomials of some systems however it does not work for the fitzhugh nagumo system thus we introduce an assistant system and classify its invariant algebraic surfaces finally

analysis of numerical errors in large eddy simulation we consider the question of numerical errors in large eddy simulation it is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the reynolds number and dependence only on the resolution delta of the eddies sought this report considers this question analytically is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on delta but not re we consider the most common simplest and most mathematically tractable model and the most mathematically clear discretization in two cases we prove such an error estimate and try to explain why our technique of proof fails in the most general case our analysis aims to assume as little time regularity on the true solution as possible

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

simply connected projective manifolds in characteristic p 0 have no nontrivial stratified bundles we show that simply connected projective manifolds in characteristic p 0 have no nontrivial stratified bundles this gives a positive answer to a conjecture by d gieseker ann sc norm super pisa 4 ser 2 1 1 31 1975 the proof uses hrushovski s theorem on periodic points

darboux integrability and algebraic invariant surfaces for the rikitake system in this paper we study the darboux integrability of the rikitake system x x yz y y x z a z 1 xy more precisely we characterize all the invariant algebraic surfaces the exponential factors and the polynomial rational and darboux first integrals of this system according to the values of its parameters a and

on tannakian duality over valuation rings in this paper we generalize the tannakian theory which gives a correspondence between groupoids and tannakian categories over a field k to the case where k is a valuation ring we give a general theorem how to reconstruct groupoids in arbitrary categories from their category of representations and we show that this theorem can be applied to groupoids over dedekind rings we also give a partial answer how to see whether a category is the representation category of a groupoid over a valuation ring

algebraic structures and invariant manifolds of differential systems algebraic tools are applied to find integrability properties of odes bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations since all equations in this class are related to a canonical quadratic differential system the lotka volterra system these algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given their subalgebras are associated to nonlinear invariant manifolds in the phase space these manifolds are calculated explicitly more general algebraic invariant surfaces are also obtained by combining a theorem of walcher and the lotka volterra canonical form applications are given for lorenz model lotka may leonard and rikitake systems

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

a local projection stabilization continuous galerkin petrov method for incompressible flow problems abstract a local projection stabilization lps method in space is considered to approximate the evolutionary oseen equations optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous in time case for both the velocity and pressure approximation in addition the fully discrete case in combination with higher order continuous galerkin petrov cgp methods is studied error estimates of order k 1 are proved where k denotes the polynomial degree in time assuming that the convective term is time independent numerical results show that the predicted order is also achieved in the general case of time dependent convective terms

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented

the stack of formal groups in stable homotopy theory we construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of m hovey and n strickland on comodule categories for landweber exact algebras this leads to a geometric understanding of their results as well as to a generalisation

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