on error estimates of the penalty method for unsteady navier stokes equations the penalty method has been widely used for numerical computations of the unsteady navier stokes equations however the best error estimates available to the author s knowledge were not optimal an

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

time discretization of stochastic 2 d navier stokes equations with a penalty projection method a time discretization of the stochastic incompressible navier stokes problem by penalty method is analyzed some error estimates are derived combined and eventually arrive at a speed of convergence in probability of order 1 4 of the main algorithm for the pair of variables velocity and pressure also using the law of total probability we obtain the strong convergence of the scheme for both variables

long time numerical solution of the navier stokes equations based on a sequential regularization formulation the sequential regularization method is a reformulation of the unsteady navier stokes equations from the viewpoint of constrained dynamical systems or the approximate helmholtz hodge projection in this paper we study the long time behavior of the sequential regularization formulation we give a uniform in time estimate between the solution of the reformulated system and that of the navier stokes equations we also conduct an error analysis for the temporal discrete system and show that the error bound is independent of time a couple of long time flow examples are computed to demonstrate this method

stability of some mixed finite element methods for stokesian flows in so called primitive variable formulations of problems of flow of viscous incompressible stokesian fluids two fields appear as unknowns the velocity field u and the pressure field p the latter representing a lagrange multiplier associated with the incompressibility constraint div u 0 finite element methods based on such formulations were first introduced over a decade ago 18 since the mid 1970s interest in these methods was rekindled by the appearance of several new techniques which provided for very efficient calculation of the element pressures these included mixed methods which employ pressure approximations which are discontinuous at interelement boundaries as well as the closely related mixed type methods which employ an exterior penalty approximation of the incompressibility condition and reduced integration of the penalty terms all of these methods have the attractive feature that the discontinuous element pressures can be eliminated element by element reducing the problem to one only involving velocities upon determining velocities element pressures can then be evaluated through a simple post processing operation methods of this type were developed and discussed by several authors and we mention in particular the papers of malkus 14 151 hughes 12 malkus and hughes 17 reddy 24 bercovier 2 the book of engleman and sani 6 and the references therein in 1980 however mathematical analyses indicated that some of the more popular discontinuouspressure mixed methods might be numerically unstable w 22 it was discovered that while certain of these methods perform well in problems with smooth solutions for which regular uniform meshes are employed serious oscillations in the pressure approximation can occur when the data or the mesh pattern are mildly irregular and these oscillations increase in amplitude as the mesh is refined oden kikuchi and song 22 attributed the deficiency of these unstable methods to their failure to satisfy a key stability criterion which they referred to as the lbb condition making reference to the work of ladyszhenskaya 13 on existence theorems of viscous flow problems and of babuska l and brezzi 3 on the approximation of elliptic problems with constraints the discrete lbb condition of oden kikuchi and song is basically the requirement that the discrete approximation bz of the transpose b of the constraint operator b div be bounded

a multi level stabilized finite element method for the stationary navier stokes equations abstract this paper proposes and analyzes a multi level stabilized finite element method for the two dimensional stationary navier stokes equations approximated by the lowest equal order finite element pairs the method combines the new stabilized finite element method with the multi level discretization under the assumption of the uniqueness condition the multi level stabilized finite element method consists of solving the nonlinear problem on the coarsest mesh and then performs one newton correction step on each subsequent mesh thus only solving one large linear systems the error analysis shows that the multi level stabilized finite element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary navier stokes equations on a fine mesh for an appropriate choice of mesh widths h j h j 1 2 j 1 j moreover the numerical illustrations agree completely with the theoretical expectations therefore the multi level stabilized finite element method is more efficient than the standard one level stabilized finite element method

frobenius splitting of equivariant closures of regular conjugacy classes let g denote a connected semisimple and simply connected algebraic group over an algebraically closed field k of positive characteristic and let g denote a regular element of g let x denote any equivariant embedding of g we prove that the closure of the conjugacy class of g within x is normal and cohen macaulay moreover when x is smooth we prove that this closure is a local complete intersection as a consequence the closure of the unipotent variety within x shares the same geometric properties

on pressure approximation via projection methods for nonstationary incompressible navier stokes equations projection methods are an efficient tool to approximate strong solutions of the incompressible navier stokes equations as a major deficiency these methods often suffer from reduced accuracy for pressure iterates caused by nonphysical boundary data going along with suboptimal error estimates for pressure iterates we verify a rigorous bound for arising boundary layers in chorin s scheme under realistic regularity assumptions in a second step the new chorin penalty method is proposed where optimal rate of convergence for pressure iterates is shown

multiple d3 instantons and mock modular forms i we study d3 instanton corrections to the hypermultiplet moduli space in type iib string theory compactified on a calabi yau threefold in a previous work consistency of d3 instantons with s duality was established at first order in the instanton expansion using the modular properties of the m5 brane elliptic genus we extend this analysis to the two instanton level where wall crossing phenomena start playing a role we focus on the contact potential an analogue of the kahler potential which must transform as a modular form under s duality we show that it can be expressed in terms of a suitable modification of the partition function of d4 d2 d0 bps black holes constructed out of the generating function of msw invariants the latter coincide with donaldson thomas invariants in a particular chamber modular invariance of the contact potential then requires that in case where the d3 brane wraps a reducible divisor the generating function of msw invariants must transform as a vector valued mock modular form with a specific modular completion built from the msw invariants of the constituents physically this gives a powerful constraint on the degeneracies of bps black holes mathematically our result gives a universal prediction for the modular properties of donaldson thomas invariants of pure two dimensional sheaves

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

asymptotic behavior and time discretization analysis for the non stationary navier stokes problem the asymptotic behavior and the euler time discretization analysis are presented for the two dimensional non stationary navier stokes problem if the data and f t satisfy a uniqueness condition corresponding to the stationary navier stokes problem we then obtain the convergence of the non stationary navier stokes problem to the stationary navier stokes problem and the uniform boundedness stability and error estimates of the euler time discretization for the non stationary navier stokes problem

a finite element penalty projection method for incompressible flows the penalty projection method for the solution of navier stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage namely the solution of the momentum balance equation to constrain the divergence of the predicted velocity field after a presentation of the scheme in the time semi discrete formulation then in fully discrete form for a finite element discretization we assess its behaviour against a set of benchmark tests including in particular prescribed velocity and open boundary conditions the results demonstrate that the augmentation always produces beneficial effects as soon as the augmentation parameter takes a significant value the projection method splitting error is reduced pressure boundary layers are suppressed and the loss of spatial convergence of the incremental projection scheme in case of open boundary conditions does not occur anymore for high values of the augmentation parameter the results of coupled solvers are recovered consequently in contrast with standard penalty methods there is no need for a dependence of the augmentation parameter with the time step and this latter can be kept to reasonable values to avoid to degrade too severely the conditioning of the linear operator associated to the velocity prediction step

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

a new stabilized finite element method for the transient navier stokes equations abstract this paper is concerned with the development and analysis of a new stabilized finite element method based on two local gauss integrations for the two dimensional transient navier stokes equations by using the lowest equal order pair of finite elements this new stabilized finite element method has some prominent features parameter free avoiding higher order derivatives or edge based data structures and stabilization being completely local at the element level an optimal error estimate for approximate velocity and pressure is obtained by applying the technique of the galerkin finite element method under certain regularity assumptions on the solution compared with other stabilized methods using the same pair of mixed finite elements for the two dimensional transient navier stokes equations through a series of numerical experiments it is shown that this new stabilized method has better stability and accuracy results

from sheaves on p 2 to a generalization of the rademacher expansion moduli spaces of stable coherent sheaves on a surface are of much interest for both mathe matics and physics yoshioka computed generating functions of poincare polynomials of such moduli spaces if the surface is p 2 and the rank of the sheaves is 2 motivated by physical arguments this pa per investigates the modular properties of these generating functions it is shown that these functions can be written in terms of the lerch sum and theta function based on this we prove a conjecture by vafa and witten which expresses the generating functions of euler numbers as a mixed mock mod ular form moreover we derive an exact formula of rademacher type for the fourier coefficients of this function this formula requires a generalization of the classical circle method this is the first example of an exact formula for the fourier coefficients of mixed mock modular forms which is of independent mathematical interest 1 introduction and statement of results in the past interactions between physics and mathematics have led to many interesting results motivated by strong weak coupling duality or s duality in physics this article considers various gen erating functions which appear in the study of moduli spaces of stable coherent sheaves on the projective plane p 2 we express the generating functions of poincare polynomials of moduli spaces of rank 2 sheaves in terms of the lerch sum and theta function which we will recall later using these expressions we prove a conjec ture by vafa and witten 45 for the generating functions of euler numbers these functions appear to be related to ramanujan s mock theta functions and therefore transform almost as weakly holomorphic modular forms i e meromorphic mod ular forms whose poles if there are any may only lie in cusps our second main result is an exact formula for the fourier coefficients of these generating functions that formally resembles the rademacher expansion for the coefficients of weakly holomorphic modular forms moduli spaces of coherent sheaves on a complex surface s receive much attention see for example 28 for an extensive work on such moduli spaces more specifically one is interested in the moduli space m r c1 c 2 of semi stable sheaves of rank r with first chern class c1 and second chern class c2 w e will consider topological invariants of m in particular the poincare polynomial p m s 2d imc m i 0 bi m s i and the euler number m p m 1 w here

investigations on two kinds of two level stabilized finite element methods for the stationary navier stokes equations abstract this article considers two kinds of two level stabilized finite element methods based on local gauss integral technique for the two dimensional stationary navier stokes equations approximated by the lowest equal order elements which do not satisfy the inf sup condition the two level methods consist of solving a small non linear system on the coarse mesh and then solving a linear system on the fine mesh the error analysis shows that the two level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the navier stokes equations on a fine mesh for a related choice of mesh widths therefore the two level methods are of practical importance in scientific computation finally the performance of two kinds of two level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments the conclusion is that simple two level stabilized method is a viable choice for the lowest equal order approximations of the stationary navier stokes problem

analysis of a sequential regularization method for the unsteady navier stokes equations the incompressibility constraint makes navier stokes equations difficult a reformulation to a better posed problem is needed before solving it numerically the sequential regularization method srm is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods in the paper we study the existence and uniqueness for the solution of the srm and provide a simple proof of the convergence of the solution of the srm to the solution of the navier stokes equations we also give error estimates for the time discretized srm formulation

flow rate boundary problems for an incompressible fluid in deformable domains formulations and solution methods in this paper we consider the numerical solution of the interaction of an incompressible fluid and an elastic structure in a truncated computational domain as well known in this case there is the problem of prescribing realistic boundary data on the artificial sections when only partial data are available this problem has been investigated extensively for the rigid case in this work we start considering the compliant case by focusing on the flow rate conditions for the fluid we propose three formulations of this problem different algorithms for its numerical solution and carry out several 2d numerical simulations with the aim of comparing the performances of the different algorithms

s duality and refined bps indices whenever available refined bps indices provide considerably more information on the spectrum of bps states than their unrefined version extending earlier work on the modularity of generalized donaldson thomas invariants counting d4 d2 d0 brane bound states in type iia strings on a calabi yau threefold mathfrak y we construct the modular completion of generating functions of refined bps indices supported on a divisor class although for compact mathfrak y the refined indices are not protected switching on the refinement considerably simplifies the construction of the modular completion furthermore it leads to a non commutative analogue of the tba equations which suggests a quantization of the moduli space consistent with s duality in contrast for a local cy threefold given by the canonical bundle over a complex surface s refined dt invariants are well defined and equal to vafa witten invariants of s our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank in cases where all reducible components of the divisor class are collinear which occurs e g when b 2 mathfrak y 1 or in the local case we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order in the local case it agrees with an earlier proposal by minahan et al for unrefined invariants and extends it to the refined level using the afore mentioned non commutative structure finally we show that these general predictions reproduce known results for u 2 and u 3 vafa witten theory on mathrm p 2 and make them explicit for u 4

an improved penalty method for power law stokes problems for the numerical approximation of fluid flow phenomena it is often highly desirable to decouple the equations defining conservation of momentum and conservation of mass by using a penalty function method the current penalty function methods for power law stokes fluids converge at a sublinear rate with respect to the penalty parameter in this article we show theoretically and numerically that a linear penalty function approximation to a power law stokes problem yields a higher order accuracy over the known nonlinear penalty method theoretically finite element approximation of the linear penalty function method is shown to satisfy an improved order of approximation with respect to the penalty parameter the numerical experiments presented in the paper support the theoretical results and satisfy a linear order of approximation

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

iterative penalty methods for the steady navier stokes equations abstract this paper presents one level and two level iterative penalty finite element methods to approximate the solutions of steady navier stokes equations first one level iterative penalty finite element method is applied to solve the steady navier stokes equations numerically and its h 1 and l 2 error estimates are derived then two level iterative penalty scheme is given and its error estimates are obtained for velocity and pressure finally the numerical results are displayed to verify the theoretical analysis

euler characteristics of su 2 instanton moduli spaces on rational elliptic surfaces recently minahan nemeschansky vafa and warner computed partition functions for n 4 topological yang mills theory on rational elliptic surfaces in particular they computed generating functions of euler characteristics of su 2 instanton moduli spaces in mathematics they are expected to coincide with those of gieseker compactifications in this paper we compute euler characteristics of these spaces and show that our results coincide with theirs we also check the modular property of z su 2 and z so 3 conjectured by vafa and witten

configurations in abelian categories iv invariants and changing stability conditions this is the last in a series on configurations in an abelian category a given a finite poset i an i configuration is a finite collection of objects j and morphisms j k or j k j k in a satisfying some axioms where j k are subsets of i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in a using artin stacks it showed well behaved moduli stacks obja m i a of objects and configurations in a exist when a is the abelian category coh p of coherent sheaves on a projective scheme p or mod kq of representations of a quiver q the second studied algebras of constructible functions and stack functions on obja the third introduced stability conditions t on a and showed the moduli space objss of semistable objects in class is a constructible subset in obja so its characteristic function ss is a constructible function it formed algebras h pa h to h pa h to of constructible and stack functions on obja and proved many identities in them in this paper if t and t are stability conditions on a we write ss in terms of the ss and deduce the algebras h pa h to are independent of t we study invariantsiss or iss i counting semistable objects or configurations in a which satisfy additive and multiplicative identities we compute them completely when a mod kq or a coh p for p a smooth curve we also find invariants with special properties when a coh p for p a smooth surface with kp 1 nef or a calabi yau 3 fold

penalty finite element method for stokes problem with nonlinear slip boundary conditions abstract the penalty finite element method for stokes problem with nonlinear slip boundary conditions based on the finite element subspace v h m h which satisfies the discrete inf sup condition is investigated in this paper since this class of nonlinear slip boundary conditions include the subdifferential property the weak variational formulation associated with stokes problem is variational inequality under some regularity assumptions we obtain the optimal h 1 and l 2 error estimates between u and u h and between u and u h e where the error orders are e h for h 1 error and e h 2 for l 2 error

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

frobenius splitting and geometry of g schubert varieties let x be an equivariant embedding of a connected reductive group g over an algebraically closed field k of positive characteristic let b denote a borel subgroup of g a g schubert variety in x is a subvariety of the form diag g v where v is a b b orbit closure in x in the case where x is the wonderful compactification of a group of adjoint type the g schubert varieties are the closures of lusztig s g stable pieces we prove that x admits a frobenius splitting which is compatible with all g schubert varieties moreover when x is smooth projective and toroidal then any g schubert variety in x admits a stable frobenius splitting along an ample divisors although this indicates that g schubert varieties have nice singularities we present an example of a nonnormal g schubert variety in the wonderful compactification of a group of type g2 finally we also extend the frobenius splitting results to the more general class of schubert varieties

error estimates for the spectral approximation of the nonstationary stokes problem this paper analyzes a discretization of the nonstationary stokes problem with periodic nonperiodic boundary conditions fourier legendre spectral approximations are considered with respect to the space variables while the time discretization is based on the second order crank nicolson scheme optimal convergence estimates are proved for the velocity field

two level variational multiscale finite element methods for navier stokes type variational inequality problem in this paper we present two level variational multiscale finite element method based on two local gauss integrations for navier stokes equations with friction boundary conditions which are of the form of navier stokes type variational inequality of the second kind we solve navier stokes type variational inequality problem on the coarse mesh and solve linearized navier stokes type variational inequality problem corresponding to newton iteration on the fine mesh the error estimates in h 1 norm for velocity and l 2 norm for pressure are derived meanwhile uzawa iteration schemes are constructed to solve the subproblems in this two level method finally the numerical results are displayed to support the theoretical analysis

vafa witten theory and iterated integrals of modular forms vafa witten vw theory is a topologically twisted version of mathcal n 4 supersymmetric yang mills theory s duality suggests that the partition function of vw theory with gauge group su n transforms as a modular form under duality transformations interestingly vafa and witten demonstrated the presence of a modular anomaly when the theory has gauge group su 2 and is considered on the complex projective plane mathbb p 2 this modular anomaly could be expressed as an integral of a modular form and also be traded for a holomorphic anomaly we demonstrate that the modular anomaly for gauge group su 3 involves an iterated integral of modular forms moreover the modular anomaly for su 3 can be traded for a holomorphic anomaly which is shown to factor into a product of the partition functions for lower rank gauge groups the su 3 partition function is mathematically an example of a mock modular form of depth two

on the penalty projection method for the navier stokes equations with the mac mesh we deal with the time dependent navier stokes equations nse with dirichlet boundary conditions on the whole domain or on a part of the domain and open boundary conditions on the other part it is shown numerically that combining the penalty projection method with spatial discretization by the marker and cell scheme mac yields reasonably good results for solving the above mentioned problem the scheme which has been introduced combines the backward difference formula of second order bdf2 namely gear s scheme for the temporal approximation the second order richardson extrapolation for the nonlinear term and the penalty projection to split the velocity and pressure unknowns similarly to the results obtained for other projection methods we estimate the errors for the velocity and pressure in adequate norms via the energy method

the stabilized extrapolated trapezoidal finite element method for the navier stokes equations abstract we consider a fully discrete stabilized finite element method for the navier stokes equations which is unconditionally stable and has second order temporal accuracy of o k 2 hk spatial error the method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti diffusion of its effects at the old time level lowering the cell reynolds number of the oseen problem to o 1 the method requires only the solution of one linear system arising from an oseen problem per time step the effectiveness of the method is illustrated in several numerical experiments

a stable weak galerkin finite element method for stokes problem abstract we study the weak galerkin finite element method for stokes problem a new weak galerkin finite element velocity pressure space pair is presented which satisfies the discrete inf sup condition based on this space pair we establish a stable weak galerkin approximation scheme without adding any stability term or penalty term then we further derive the optimal error estimates for velocity and pressure approximations respectively numerical experiments are provided to illustrate the theoretical analysis

an efficient moving mesh spectral method for the phase field model of two phase flows we develop in this paper a moving mesh spectral method for the phase field model of two phase flows with non periodic boundary conditions the method is based on a variational moving mesh pde for the phase function coupled with efficient semi implicit treatments for advancing the mesh function the phase function and the velocity and pressure in a decoupled manner ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method

on compactifications of the steinberg zero fiber abstract let g be a connected semisimple linear algebraic group over an algebraically closed field k of positive characteristic and let x denote an equivariant embedding of g we define a distinguished steinberg fiber n in g called the zero fiber and prove that the closure of n within x is normal and cohen macaulay furthermore when x is smooth we prove that the closure of n is a local complete intersection

a stabilized finite element method based on two local gauss integrations for the stokes equations this paper considers a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for the stokes equations approximated by the lowest equal order finite element pairs i e the p 1 p 1 and q 1 q 1 pairs this method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf sup condition optimal error estimates are obtained by applying the standard galerkin technique finally the numerical illustrations agree completely with the theoretical expectations

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

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

optimal error estimate of the penalty finite element method for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations the time discretization of the penalty navier stokes equations is based on the backward euler scheme the spatial discretization of the time discretized penalty navier stokes equations is based on a finite element space pair x h m h which satisfies some approximate assumption an optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters e at and h are sufficiently small

iterative methods in penalty finite element discretization for the steady mhd equations this paper characterizes one penalty finite element method for the incompressible mhd equations the method is an interesting combination of the classic iterative schemes stokes newton and oseen iterations with two different finite element pairs p1b p1 p1b and p1 p0 p1 moreover the rigorous analysis of stability and error estimate for the proposed methods are given finally the applicability and effectiveness of the presented schemes are illustrated in several numerical experiments

error estimate of the p 1 nonconforming finite element method for the penalized unsteady navier stokes equations we consider a finite element method for the penalty formulation of the time dependent navier stokes equations usually the improper choice of the finite element space will lead that the error estimate inversely depends on the penalty parameter epsilon we use the classical p 1 nonconforming finite element space for the spatial discretization optimal epsilon uniform error estimations for both velocity and pressure are obtained

wall crossing holomorphic anomaly and mock modularity of multiple m5 branes using wall crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two m5 branes wrapping a rigid divisor inside a calabi yau manifold the anomaly originates from restoring modularity of an indefinite theta function capturing the wall crossing of bps invariants associated to d4 d2 d0 brane systems we show the compatibility of this equation with anomaly equations previously observed in the context of mathcal n 4 topological yang mills theory on mathbbm p 2 and e strings obtained from wrapping m5 branes on a del pezzo surface the non holomorphic part is related to the contribution originating from bound states of singly wrapped m5 branes on the divisor we show in examples that the information provided by the anomaly is enough to compute the bps degeneracies for certain charges we further speculate on a natural extension of the anomaly to higher d4 brane charge

decoupled schemes for unsteady mhd equations ii finite element spatial discretization and numerical implementation in this article a decoupled fully discrete scheme for solving 2d magnetohydrodynamics mhd equations is proposed the decoupled scheme is used for time discretization and the finite element method is used for spatial discretization firstly the almost unconditional stability t c of this scheme is established then optimal l2 and h1 error estimates of numerical solution are provided finally a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme

a penalty finite element method based on the euler implicit explicit scheme for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations where the time discretization is based on the euler implicit explicit scheme with some implicit linear terms and an explicit nonlinear term and the finite element spatial discretization is based on the p 1b p 1 element pair which satisfies the discrete inf sup condition this method allows us to separate the computation of the velocity from the computation of the pressure with a larger time step size dt so that the numerical velocity u e h n and the pressure p e h n are easily computed an optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter e the time step size dt and the mesh size h satisfy the following stability conditions ec 1 1 dt k 1 1 and h 2 b 1 dt respectively for some positive constants c 1 k 1 and b 1 finally some numerical tests to confirm the theoretical results of the penalty finite element method are provided

numerical analysis of a penalty approach for the solution of a transient eddy current problem abstract the aim of this paper is to propose and analyze a numerical method to solve transient eddy current problems formulated in terms of the magnetic field intensity space discretization is based on nedelec edge elements while a backward euler scheme is used for time discretization the curl free constraint in the dielectric domain is imposed by means of a penalty strategy convergence of the penalized problem as the penalty parameter goes to zero is proved for the continuous and the discrete problems for the latter uniformly in the discretization parameters optimal order error estimates for the convergence of the discrete penalized problem with respect to the penalty and the discretization parameters are also proved finally some numerical tests are reported to assess the performance of this approach

on a grauert riemenschneider vanishing theorem for frobenius split varieties in characteristic p it is known that the grauert riemenschneider vanishing theorem is not valid in characteristic p 1 here we show that it may be restored in the presence of a suitable frobenius splitting the proof uses interchanging two projective limits one involving iterated frobenius maps cf 2 and 4 the other coming from grothendieck s theorem on formal functions that leads to the following general vanishing theorem which we then apply in the situation of the grauert riemenschneider theorem

black holes and higher depth mock modular forms by enforcing invariance under s duality in type iib string theory compactified on a calabi yau threefold we derive modular properties of the gene rating function of bps degeneracies of d4 d2 d0 black holes in type iia string theory compactified on the same space mathematically these bps degeneracies are the generalized donaldson thomas invariants counting coherent sheaves with support on a divisor cal d at the large volume attractor point for cal d irreducible this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping m5 brane on cal d and is therefore known to be modular instead when cal d is the sum of n irreducible divisors cal d i we show that the generating function acquires a modular anomaly we characterize this anomaly for arbitrary n by providing an explicit expression for a non holomorphic modular completion in terms of generalized error functions as a result the generating function turns out to be a mixed mock modular form of depth n 1

two level penalty finite element methods for the stationary incompressible magnetohydrodynamics problem in the paper we consider the penalty finite element method for the stationary incompressible magnetohydrodynamics mhd problem with a factor of penalty parameter stability and convergence of numerical solutions are established furthermore two level penalty methods are also developed for the mhd problem our methods consist of solving a nonlinear mhd problem by the usual penalty method on a coarse mesh with mesh size h and then a linearized mhd problem based on the stokes newton and oseen iterations on a fine mesh is solved by the penalty method with mesh size h h h respectively stability and error estimates of numerical solutions in two level penalty methods are presented finally some numerical tests are provided to demonstrate the effectiveness of the developed algorithms

high order time stepping for the navier stokes equations with minimal computational complexity in this paper we present extensions of the schemes proposed in guermond and minev 2015 that lead to a decoupling of the velocity components in the momentum equation the new schemes reduce the solution of the incompressible navier stokes equations to a set of classical scalar parabolic problems for each cartesian component of the velocity the pressure is explicitly recovered after the velocity is computed

geometry of b b orbit closures in equivariant embeddings abstract let x denote an equivariant embedding of a connected reductive group g over an algebraically closed field k let b denote a borel subgroup of g and let z denote a b b orbit closure in x when the characteristic of k is positive and x is projective we prove that z is globally f regular as a consequence z is normal and cohen macaulay for arbitrary x and arbitrary characteristics moreover in characteristic zero it follows that z has rational singularities this extends earlier results by the second author and m brion

on convergent schemes for diffuse interface models for two phase flow of incompressible fluids with general mass densities we are concerned with convergence results for fully discrete finite element schemes suggested in grun and klingbeil j comput phys in press doi 10 1016 j jcp 2013 10 028 they were developed for the diffuse interface model in h abels h garcke and g grun math models methods appl sci 22 2012 1150013 which describes two phase flow of immiscible incompressible viscous fluids we formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution we identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution in this case our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions our argument is particularly based on higher regularity results

a pressure poisson stabilized finite element method for the non stationary stokes equations to circumvent the inf sup condition in this article stabilized finite element methods are considered for the non stationary stokes equations based on some lowest equal order finite elements space pair x h m h which do not satisfy the discrete inf sup condition the stability of two kinds of methods is derived under some regularity assumptions then the convergence of the penalty method and the pressure poisson stabilized method is compared the result shows that the former error limits the order of approximation to o e h e and the latter yields the optimal error estimate o h

a finite element penalty method for the linearized viscoelastic oldroyd fluid motion equations abstract in this paper a fully discrete finite element penalty method is considered for the two dimensional linearized viscoelastic fluid motion equations arising from the oldroyd model for the non newton fluid flows with the finite element method for the spatial discretization and the backward euler scheme for the temporal discretization the velocity and pressure are decoupled in this method which leads to a large reduction of the computational scale under some realistic assumptions the unconditional stability of the fully discrete scheme is proved moreover the optimal error estimates are obtained which are better than the existing results finally some numerical results are given to verify the theoretical analysis the difference between the motion of the newton and non newton fluid is also observed

d instantons and twistors some exact results we present some results on instanton corrections to the hypermultiplet moduli space in calabi yau compactifications of type ii string theories previously using twistor methods only a class of d instantons d2 instantons wrapping a cycles was incorporated exactly and the rest was treated only linearly we go beyond the linear approximation and give a set of holomorphic functions which through a known procedure capture the effect of d instantons at all orders moreover we show that for a sector where all instanton charges have vanishing symplectic invariant scalar product the hypermultiplet metric can be computed explicitly

the boundary penalty method for the diffusion equation subject to the specification of mass in this paper we concern with a new boundary penalty finite element method for the one dimensional time dependent diffusion equation subject to the specification of mass instead of solving the diffusion equation directly we will use the finite element method to solve a boundary penalty problem which is derived from the diffusion equation and analyse the error estimate of the semi discrete and fully discrete approximations when the penalty parameter e is sufficiently small the numerical experiments show that our method is very effective and supports our analysis

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class