on the antimaximum principle for the p laplacian with indefinite weight this paper is concerned with the antimaximum principle for the quasilinear prob 1 mathrm e mathrm m delta p u lambda m x u p 2 u h x delta p is the laplacian and m x is aweight function which may change sign we will in particular investigate the question of the uniformity of this principle and provide avariational characterization for the interval of uniformity an identity of picone s type for the mathrm p laplacian plays an important role in our approach

estimates on the spectral interval of validity of the anti maximum principle abstract the anti maximum principle for the homogeneous dirichlet problem to p u u p 2 u f x with positive f l states the existence of a critical value f 1 such that any solution of this problem with 1 f is strictly negative in this paper we give a variational upper bound for f and study its properties as an important supplementary result we investigate the branch of ground state solutions of the considered boundary value problem in 1 2

reduced aliasing formulations of the convective terms within the navier stokes equations for a compressible fluid the effect on aliasing errors of different formulations describing the cubically nonlinear convective terms within the discretized navier stokes equations is examined in the presence of a non trivial density spectrum fourier analysis shows that the existing skew symmetric forms of the convective term result in reduced aliasing errors relative to the conservation form several formulations of the convective term including a new formulation proposed for cubically nonlinear terms are tested in direct numerical simulation dns of decaying compressible isotropic turbulence both in chemically inert small density fluctuations and reactive cases large density fluctuations and for different degrees of resolution in the dns of reactive turbulent flow the new cubic skew symmetric form gives the most accurate results consistent with the spectral error analysis and at the lowest cost in marginally resolved dns and les poorly resolved by definition the new cubic skew symmetric form represents a robust convective formulation which minimizes both aliasing and computational cost while also allowing a reduction in the use of computationally expensive high order dissipative filters

entropy stable discontinuous galerkin approximation with summation by parts property for the incompressible navier stokes cahn hilliard system abstract we develop an entropy stable two phase incompressible navier stokes cahn hilliard discontinuous galerkin dg flow solver method the model poses the cahn hilliard equation as the phase field method a skew symmetric form of the momentum equation and an artificial compressibility method to compute the pressure we design the model so that it satisfies an entropy law including free and no slip wall boundary conditions with non zero wall contact angle we then construct a high order dg approximation of the model that satisfies the summation by parts simultaneous approximation term property with the help of a discrete stability analysis the scheme has two modes an entropy conserving approximation with central advective fluxes and the bassi rebay 1 br1 method for diffusion and an entropy stable approximation with an exact riemann solver for advection and interface stabilization added to the br1 method the scheme is applicable to and the stability proofs hold for three dimensional unstructured meshes with curvilinear hexahedral elements we test the convergence of the schemes on a manufactured solution and their robustness by solving a flow initialized from random numbers in the latter we find that a similar scheme that does not satisfy an entropy inequality had 30 probability to fail while the entropy stable scheme never does we also solve the static and rising bubble test problems and to challenge the solver capabilities we compute a three dimensional pipe flow in the annular regime

on the existence of multiple positive entire solutions for a quasilinear elliptic systems abstract in this paper our main purpose is to establish the existence of infinitely positive bounded entire solutions of second order quasilinear elliptic systems these extend a work of zuodong yang zuodong yang existence of positive bounded entire solutions of quasilinear elliptic equations appl math comput 156 2004 743 754 which considered the single quasilinear elliptic equation these extend also a work of kawano and kusano n kawano t kusano on positive entire solutions of a class of second order semilinear elliptic systems math zeitschrift 186 3 1984 287 297 which studied the semilinear elliptic equation

solvability of p laplace equations subject to three point boundary value problems in this paper the existences of solutions for p laplace equations subject to three point boundary value conditions at resonance and non resonance are studied by using degree theory and some known results are improved

the hierarchical basis multigrid method we derive and analyze the hierarchical basis multigrid method for solving discretizations of self adjoint elliptic boundary value problems using piecewise linear triangular finite elements the method is analyzed as a block symmetric gau seidel iteration with inner iterations but it is strongly related to 2 level methods to the standard multigridv cycle and to earlier jacobi like hierarchical basis methods the method is very robust and has a nearly optimal convergence rate and work estimate it is especially well suited to difficult problems with rough solutions discretized using highly nonuniform adaptively refined meshes

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

the fredholm alternative at the first eigenvalue for the one dimensionalp laplacian abstract in this work we study the range of the operatoru u p 2 u 1 u p 2 u u 0 u t 0 p 1 we prove that all functionsh c1 0 t satisfying t0 h t sinp pt t dt 0 lie in the range but that ifp 2 andh 0 the solution set is bounded here sin pt t is a first eigenfunction associated to 1 we also show that in this case the associated energy functionalu 1 p t0 u p 1 p t0 u p t0 huis unbounded from below if 1 2 onw1 p0 0 t 1corresponds precisely to the best constant in thelp poincare inequality moreover we show that unlike the linear casep 2 forp 2 the range contains a nonempty open set inl 0 t

nonresonance conditions for generalised ϕ laplacian problems with jumping nonlinearities abstract we consider the boundary value problem 0 1 x u x u x f x u x u x a e x 0 1 0 2 c 00 u 0 c 01 u 0 c 10 u 1 c 11 u 1 where c j 0 c j 1 0 for each j 0 1 and f 0 1 r 2 r are caratheodory functions with suitable additional properties the differential operator generated by the left hand side of 0 1 together with the boundary conditions 0 2 is a generalisation of the usual p laplacian and also of the so called laplacian which corresponds to x s t t with an odd increasing homeomorphism for the p laplacian problem and more particularly the semilinear case p 2 nonresonance conditions which ensure the solvability of the problem 0 1 0 2 have been obtained in terms of either eigenvalues for non jumping f or the fucik spectrum or half eigenvalues for jumping f of the p laplacian in this paper under suitable growth conditions on and f we extend these conditions to the general problem 0 1 0 2

existence and multiplicity of solutions for second order impulsive differential equations with dirichlet problems abstract in this paper we consider the existence and the multiplicity of solutions for second order impulsive differential equations with dirichlet boundary conditions some new existence results are obtained by critical point theory our results further improve some existing results

generalization of fredholm alternative for nonlinear differential operators on considere le probleme aux valeurs limites u t p 2 u t f t u t g t t 0 u 0 u 0 on demontre que si lim u inf f t u u p 2 u lim u sup f t u u p 2 u sont entre des valeurs propres du probleme aux valeurs propres u p 2 u u p 2 u sur 0 u 0 u 0 alors le probleme aux limites est resoluble pour tout g l 1

a noncooperative elliptic system with p laplacians that preserves positivity a nonlinear noncooperative elliptic system is shown to have a positivity preserving property that is there exists a uniform positive constant such that whenever the noncooperative part is bounded by this constant positivity of the source term implies that the solution is positive the model operator is the p laplacian with 1 p on a one dimensional domain the source term appears in one of the equations

existence of positive solutions for singular boundary value problems involving the one dimensional p laplacian this paper studies the existence of positive solutions for singular dirichlet boundary value problems these results are obtained by using the global continuation theorem fixed point index theory and approximate method

an adaptive finite element method for evolutionary convection dominated problems abstract we present an adaptive finite element method for evolutionary convection diffusion problems the algorithm is based on an a posteriori indicator of the size of the oscillations displayed by the finite element approximation the procedure is able to refine or coarsen dynamically the mesh adjusting it automatically to evolving layers the method produces nearly non oscillatory approximations in the convection dominated regime we check the performance of the adaptive method with some numerical experiments

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

sharp conditions for the existence of sign changing solutions to equations involving the one dimensional p laplacian abstract we consider the boundary value problem involving the one dimensional p laplacian u p 2 u a x f u 0 0 x 1 u 0 u 1 0 where p 1 we establish sharp conditions for the existence of solutions with prescribed numbers of zeros in terms of the ratio f s s p 1 at infinity and zero our argument is based on the shooting method together with the qualitative theory for half linear differential equations

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

fourier analysis of a robust multigrid method for convection diffusion equations we consider a two grid method for solving 2d convection diffusion problems the coarse grid correction is based on approximation of the schur complement as a preconditioner of the schur complement we use the exact schur complement of modified fine grid equations we assume constant coefficients and periodic boundary conditions and apply fourier analysis we prove an upper bound for the spectral radius of the two grid iteration matrix that is smaller than one and independent of the mesh size the convection diffusion ratio and the flow direction i e we have a strong robustness result numerical results illustrating the robustness of the corresponding multigrid w cycle are given

nonresonance for one dimensional p laplacian with regular restoring abstract in this paper the existence of periodic solutions of the equation p x f t x 0 is proved where f 0 t r r has some nonlinear growth near x and the nonresonance condition sp which can be explained using the periodic eigenvalues concerning p laplacian is satisfied

periodic solutions for p laplacian neutral functional differential equation with deviating arguments by using the theory of coincidence degree we study a kind of periodic solutions to p laplacian neutral functional differential equation with multiple deviating arguments a new result on the existence of periodic solutions is obtained

a class of multi parameter eigenvalue problems for perturbed p laplacians this paper is devoted to multi parameter eigenvalue problems for one dimensional perturbed p laplacians modelling travelling waves for a class of nonlinear evolution pde dispersion relations between the eigen parameters the existence of eigenfunctions and positive eigenfunctions variational principles for eigenvalues and constructing solutions in the analytical and implicit forms are the main subject of this paper we use both variational and analytical methods

oscillation and interlacing for various spectra of the p laplacian abstract we consider the eigenvalue problem p u r q u p 1 sgn u a e on 0 b where b 0 p 1 p is the p laplacian q r l 1 0 b r 0 and r a variety of boundary conditions will be imposed at 0 and b but the main focus is on dirichlet neumann and periodic anti periodic conditions in the linear case p 2 there are well known oscillation and interlacing results for the eigenvalues of the above problem with these boundary conditions and we explore similarities and differences between this case and the nonlinear case with p 2 we will see that several new phenomena occur in the periodic anti periodic problems for example with separated boundary conditions the structure of the spectrum when p 2 is the same as when p 2 but this is not true in the periodic anti periodic cases we also consider the set of half eigenvalues and the fucik spectrum of the problem these concepts are useful for solving jumping nonlinearity problems and we show that new phenomena also appear here in addition to analogues of those occurring for the usual eigenvalues

a high order solver for unsteady incompressible navier stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping we report development of a high order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping the method falls under the class of methods now referred to as flux reconstruction correction procedure via reconstruction the governing equations employ chorin s classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems an implicit non linear lower upper symmetric gauss seidel scheme with backward euler discretization is used to efficiently march the solution in pseudo time while a second order backward euler discretization is used to march in physical time we verify and validate implementation of the high order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems the current implicit time stepping scheme is proven effective in satisfying the divergence free constraint on the velocity field in the artificial compressibility formulation within the context of the high order flux reconstruction method this compact high order method is very suitable for parallel computing and can easily be extended to moving and deforming grids

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

on the use of kinetic energy preserving dg schemes for large eddy simulation abstract recently element based high order methods such as discontinuous galerkin dg methods and the closely related flux reconstruction fr schemes have become popular for compressible large eddy simulation les element based high order methods with riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behavior for multi scale problems dispersion errors are very low for a broad range of scales while dissipation errors are very low for well resolved scales and are very high for scales close to the nyquist cutoff in some sense the inherent numerical dissipation caused by the interface riemann solver acts as a filter of high frequency solution components this observation motivates the trend that element based high order methods with riemann solvers are used without an explicit les model added only the high frequency type inherent dissipation caused by the riemann solver at the element interfaces is used to account for the missing sub grid scale dissipation due to under resolution of vortical dominated structures typical for les type setups element based high order methods suffer from stability issues caused by aliasing errors of the non linear flux terms a very common strategy to fight these aliasing issues and instabilities is so called polynomial de aliasing where interpolation is exchanged with projection based on an increased number of quadrature points in this paper we start with this common no model or implicit les iles dg approach with polynomial de aliasing and riemann solver dissipation and review its capabilities and limitations we find that the strategy gives excellent results but only when the resolution is such that about 40 of the dissipation is resolved for more realistic coarser resolutions used in classical les e g of industrial applications the iles dg strategy becomes quite inaccurate we show that there is no obvious fix to this strategy as adding for instance a sub grid scale models on top doesn t change much or in worst case decreases the fidelity even more finally the core of this work is a novel les strategy based on split form dg methods that are kinetic energy preserving the scheme offers excellent stability with full control over the amount and shape of the added artificial dissipation this premise is the main idea of the work and we will assess the les capabilities of the novel split form dg approach when applied to shock free moderate mach number turbulence we will demonstrate that the novel dg les strategy offers similar accuracy as the iles methodology for well resolved cases but strongly increases fidelity in case of more realistic coarse resolutions

perturbation of the p laplacian by vanishing nonlinearities in one dimension abstract we treat the quasi linear spectral problem p u p 2 u u p 2 u h x u 0 x a u 0 u a 0 at resonance with vanishing nonlinearity h x u g u f x g u 0 as u and f l 0 a f 0 satisfying certain orthogonality related hypotheses 1 p 3 p 2 the parameter r takes an arbitrary resonant value we first establish the boundedness of the set of all weak solutions in the sobolev space w 0 1 p 0 a which then enables us to obtain an existence result by the leray schauder degree theory the boundedness is obtained from a very precise asymptotic estimate valid for large solutions to p which can be applied thanks to a sufficiently fast rate of decay g u 0 as u

a high order discontinuous galerkin fourier incompressible 3d navier stokes solver with rotating sliding meshes we present the development of a sliding mesh capability for an unsteady high order order 3 h p discontinuous galerkin solver for the three dimensional incompressible navier stokes equations a high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh through the use of curved boundary elements and mixed triangular quadrilateral meshes a second order stiffly stable method is used to discretise in time the arbitrary lagrangian eulerian form of the incompressible navier stokes equations spatial discretisation is provided by the symmetric interior penalty galerkin formulation with modal basis functions in the x y plane allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques spatial discretisation in the z direction is provided by a purely spectral method that uses fourier series and allows computation of spanwise periodic three dimensional flows the developed solver is shown to provide high order solutions second order in time convergence rates and spectral convergence when solving the incompressible navier stokes equations on meshes where fixed and rotating elements coexist in addition an exact implementation of the no slip boundary condition is included for curved edges circular arcs and naca 4 digit airfoils where analytic expressions for the geometry are used to compute the required metrics the solver capabilities are tested for a number of two dimensional problems governed by the incompressible navier stokes equations on static and rotating meshes the taylor vortex problem a static and rotating symmetric naca0015 airfoil and flows through three bladed cross flow turbines in addition three dimensional flow solutions are demonstrated for a three bladed cross flow turbine and a circular cylinder shadowed by a pitching naca0012 airfoil

parameter convexity and concavity of generalized trigonometric functions abstract we study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter we show that p sin p y and p cos p y are log concave on the appropriate intervals while p tan p y is log convex we also prove similar facts about the generalized hyperbolic functions in particular our results settle a major part of the conjecture recently put forward in 4

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

extremal solutions for nonlinear functional ϕ laplacian impulsive equations abstract in this paper we deal with some nonlinear laplacian second order boundary value problems with impulses subject to a general type of boundary value conditions which cover dirichlet and periodic boundary data as particular cases our approach is that of upper and lower solutions together with growth restrictions of nagumo s type the impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear in both cases functional dependence on the solution is allowed

eigenvalue estimates for the p laplace operator on manifolds abstract in this paper some new geometric estimates for the first eigenvalue and the fundamental tone of the p laplacian on manifolds in terms of admissible vector fields are obtained also a new spectral invariant and its relation with the geometry of the manifold is defined

p laplacian problems with jumping nonlinearities we consider the p laplacian boundary value problem 1 p u x f x u x u x a e x 0 1 2 c00u 0 c01u 0 c10u 1 c11u 1 where p 1 is a fixed number p s s p 2s s r and for each j 0 1 cj0 cj1 0 the function f 0 1 r2 r is a caratheodory function satisfying for x s t 0 1 r2 x p s e x s t f x s t x p s e x s t s 0 where l1 0 1 and e has the form e x s t x e s t with l1 0 1 0 e 0 and limr e r r1 p 0 this condition allows the nonlinearity in 1 to behave differently as u such a nonlinearity is often termed jumping related to 1 2 is the problem 3 p u a p u b p u p u in 0 1 together with 2 where a b l1 0 1 r and u x max u x 0 for x 0 1 this problem is positively homogeneous and jumping values of for which 2 3 has a nontrivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to existence and nonexistence results for the problem 1 2 we also consider a related bifurcation problem and obtain a global bifurcation result similar to the well known rabinowitz global bifurcation theorem this then leads to a multiplicity result for 1 2 when the functions a and b are constant the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

split form nodal discontinuous galerkin schemes with summation by parts property for the compressible euler equations fisher and carpenter 2013 12 found a remarkable equivalence of general diagonal norm high order summation by parts operators to a subcell based high order finite volume formulation this equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux we show that besides the construction of entropy stable high order schemes a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms thus by changing the subcell finite volume flux to a specific choice we are able to generate in a systematic way all common split forms of the compressible euler advection terms such as the ducros splitting and the kennedy and gruber splitting although these split forms are not entropy stable we present a systematic way to prove which of those split forms are at least kinetic energy preserving with this we construct a unified high order split form dg framework we investigate with three dimensional numerical simulations of the inviscid taylorgreen vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows in fact we show that for certain test cases the novel split form discontinuous galerkin schemes are more robust than the discontinuous galerkin scheme with over integration

on an adaptive multigrid solver for convection dominated problems the objective of this paper is to develop and describe an adaptive multigrid scheme for the numerical solution of convection diffusion reaction equations with dominating convection or reaction term specifically we propose a systematic construction of problem and scale dependent restriction prolongation and smoothing operators in combination with an adaptive mesh refinement strategy in particular adaptivity is not only to reduce computational complexity but also to stabilize standard galerkin discretizations so that adding artificial viscosity can be avoided the approach is suited for any choice of underlying multiresolution sequences in particular for standard finite element discretizations on nested but possibly nonuniform meshes moreover all concepts are in principle independent of the spatial dimension and avoid any special flow dependent enumeration of unknowns the performance of the scheme is illustrated by various examples in one and two spatial dimensions they are chosen so as to exhibit possibly many types of adverse or singular behavior in connection with constant and varying convection or the interplay between convective and reactive terms in the light of these tests those issues are identified that require further research

nonlinear functionals of wavelet expansions adaptive reconstruction and fast evaluation summary this paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process in particular evaluation covers here the computation of inner products of such expressions with wavelets which arise for instance in the context of galerkin or petrov galerkin schemes for the solution of differential equations the central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion i e the number of nonzero wavelet coefficients as little as possible the following issues are addressed first motivated by previous treatments of the subject we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of besov norms the principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets the analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal we conclude with a simple numerical example

uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain n via a generalisation of serrin s sweeping principle in the case when is an annulus it is shown that the solution is radially symmetric

a free energy stable nodal discontinuous galerkin approximation with summation by parts property for the cahn hilliard equation abstract we present a nodal discontinuous galerkin dg scheme for the cahn hilliard equation that satisfies the summation by parts simultaneous approximation term sbp sat property the latter permits us to show that the discrete free energy is bounded and as a result the scheme is provably stable the scheme and the stability proof are presented for general curvilinear three dimensional hexahedral meshes we use the bassi rebay 1 br1 scheme to compute interface fluxes and a first order implicit explicit imex scheme to integrate in time we provide a semi discrete stability study and a fully discrete proof subject to a positivity condition on the solution lastly we test the theoretical findings using numerical cases that include two and three dimensional problems

nonlinear resonance in asymmetric oscillations in this paper we study the periodic problem f p x email protected f p x email protected f p x f x t where f p u u p 2u p 1 a 1 email protected 1 p 2n 1 email protected n by computing the associated topological degree we prove existence of periodic solutions and stability of large amplitude oscillations for equations with a small damping term

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

existence of three solutions for a quasilinear two point boundary value problem abstract in this paper we investigate a quasilinear second order differential equation with dirichlet boundary conditions the existence of an open interval of parameters which ensures this problem admits at least three solutions is determined by using the critical point theory

unilateral global bifurcation phenomena and nodal solutions for p laplacian in this paper we establish a dancer type unilateral global bifurcation result for one dimensional p laplacian problem p u m t p u g t u t 0 1 u 0 u 1 0 under some natural hypotheses on the perturbation function g 0 1 r r r we show that k p is a bifurcation point of the above problem and there are two distinct unbounded continua ck and ck consisting of the bifurcation branch ck from k p 0 where k p is the k th eigenvalue of the linear problem corresponding to the above problem as the applications of the above result we study the existence of nodal solutions for the following problem p u f t u 0 t 0 1 u 0 u 1 0 moreover based on the bifurcation result of girg and takac 2008 13 we prove that there exist at least a positive solution and a negative one for the following problem div p u f x u in u 0 on

existence of positive entire solutions for singular and non singular quasi linear elliptic equation we consider the quasilinear elliptic equation div u p 2 u m x u 0 in domain rn where 0 p 1 under several hypotheses on the m x we obtain the existence of positive entire solutions

an identity for a quasilinear ode and its applications to the uniqueness of solutions of bvps abstract the following boundary value problem 1 1 p u a x f u 0 x 0 x x 1 1 2 u x 0 u x 1 0 is considered where p s s p 2 s p 1 a c 1 x 0 x 1 a x 0 for x x 0 x 1 and f c 1 r an identity for solutions of 1 1 and its linearized equation is derived some applications of the identity to uniqueness of solutions of problem 1 1 1 2 are presented non uniqueness examples for problem 1 1 1 2 are also established moreover the results obtained here are applied to the study of radially symmetric solutions of the dirichlet problem for elliptic equations in annular domains

strongly nonlinear second order differential inclusions with generalized boundary conditions abstract in this paper we study second order differential inclusions in r n with a maximal monotone term and generalized boundary conditions the nonlinear differential operator need not be necessary homogeneous and incorporates as a special case the one dimensional p laplacian the generalized boundary conditions incorporate as special cases well known problems such as the dirichlet picard neumann and periodic problems as application to the proven results we obtain existence theorems for both convex and nonconvex problems when the maximal monotone term a is defined everywhere and when not case of variational inequalities

principal eigenvalues and an anti maximum principle for homogeneous fully nonlinear elliptic equations we study the fully nonlinear elliptic equation 0 1 f d2u du u x f in a smooth bounded domain under the assumption that the nonlinearity f is uniformly elliptic and positively homogeneous recently it has been shown that such operators have two principal half eigenvalues and that the corresponding dirichlet problem possesses solutions if both of the principal eigenvalues are positive in this paper we prove the existence of solutions of the dirichlet problem if both principal eigenvalues are negative provided the second eigenvalue is positive and generalize the anti maximum principle of clement and peletier p clement l a peletier an anti maximum principle for second order elliptic operators j differential equations 34 2 1979 218 229 to homogeneous fully nonlinear operators

the fredholm alternative for the one dimensional p laplacian abstract in this paper we consider the solvability of the boundary value problem p u 1 p u f u h t u 0 u t 0 where p 1 p u u p 2 u by using generalized polar coordinates transformation method we improve and generalize some results obtained recently in j differential equations 151 1999 386

eigenvalue analysis for the p laplacian convective perturbation abstract in this paper a detailed analysis of the eigenvalue problem under convection u p 2 u c u p 2 u u p 2 u 0 u 0 u 1 0 d d x is performed the analysis is based on a complete study of the phase space of the family of equations u p 2 u a u p 2 u b u p 2 u 0 a b constants

convergence of an adaptive semi lagrangian scheme for the vlasov poisson system an adaptive semi lagrangian scheme for solving the cauchy problem associated to the periodic 1 1 dimensional vlasov poisson system in the two dimensional phase space is proposed and analyzed a key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow the accuracy of the scheme is monitored by a prescribed tolerance parameter e which represents the local interpolation error at each time step in the l metric the numerical solutions are proved to converge in l towards the exact ones as e and t tend to zero provided the initial data is lipschitz and has a finite total curvature or in other words that it belongs to w 1 infty cap w 2 1 the rate of convergence is mathcal o delta t 2 varepsilon delta t which should be compared to the results of besse who recently established in siam j numer anal 42 1 350 382 2004 similar rates for a uniform semi lagrangian scheme but requiring that the initial data are in mathcal c 2 several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations

the bassi rebay 1 scheme is a special case of the symmetric interior penalty formulation for discontinuous galerkin discretisations with gauss lobatto points abstract in the discontinuous galerkin dg community several formulations have been proposed to solve pdes involving second order spatial derivatives e g elliptic problems in this paper we show that when the discretisation is restricted to the usage of gauss lobatto points there are important similarities between two common choices the bassi rebay 1 br1 method and the symmetric interior penalty sip formulation this equivalence enables the extrapolation of properties from one scheme to the other a sharper estimation of the minimum penalty parameter for the sip stability compared to the more general estimate proposed by shahbazi 1 more efficient implementations of the br1 scheme and the compactness of the br1 method for straight quadrilateral and hexahedral meshes

positive solutions and eigenvalue regions of two delay singular systems with a twin parameter abstract the paper is concerned with a two delay singular differential system with a twin parameter applying fixed point index theory we show the relationship between the asymptotic behaviors of nonlinearities at zero and infinity and the open regions eigenvalue regions of parameters which are correlated with delays such that the system has zero one and two positive solution s

an extension of mawhin s continuation theorem and its application to boundary value problems with a p laplacain abstract in order to study boundary value problems bvps with a p laplacian an extension of mawhin s continuation theorem is given then some existence theorems for bvps with a p laplacian at resonance are proved

lower bounds for the first eigenvalue of the p laplacian on compact manifolds with positive ricci curvature abstract we give some lower bounds for the first eigenvalue of the p laplace operator on compact riemannian manifolds with positive or non negative ricci curvature in terms of diameter of the manifolds for compact manifolds with boundary we consider the dirichlet eigenvalue with some proper geometric hypothesis

nonuniform nonresonance of semilinear differential equations abstract consider the dirichlet problem of nonlinear differential equations with the principal part the p laplacian when the nonlinearity satisfies some semilinearity conditions the usual nonuniform nonresonance conditions are obtained by comparing nonlinear equations with the classical eigenvalues in this article we will introduce some weighted eigenvalues the nonuniform nonresonance conditions proved in this article using weighted eigenvalues will improve the usual ones significantly

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

double positive solutions for a nonlinear four point boundary value problem with a p laplacian operator in this paper we study the existence of positive solutions for a nonlinear four point boundary value problem with a p laplacian operator by using a three functionals fixed point theorem in a cone the existence of double positive solutions for the nonlinear four point boundary value problem with ap laplacian operator is obtained this is different than previous results

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

periodic problems for strongly nonlinear second order differential inclusions in this paper we study periodic problems for second order differential inclusions in rn with a maximal monotone term the nonlinear differential operator is not necessarily homogeneous does not obey any growth condition and incorporates as a special case the one dimensional p laplacian using techniques from multivalued analysis and the theory of operators of monotone type we prove the existence of solutions for both the convex and nonconvex problems when the maximal monotone term a is defined everywhere and when it is not defined everywhere case of variational inequalities

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

existence of periodic solutions for quasilinear differential equations in this paper we find some sufficient conditions for the existence of periodic solutions to a class of p laplacian by using a fixed point theorem by ding chin ann math 2 1981 281

on a pseudo fuccik spectrum for strongly nonlinear second order odes and an existence result abstract we study here the existence of solutions to the nonlinear dirichlet problem p u f t u q t a e for t a b u a u b 0 where is an increasing odd homeomorphism of r f a b r r satisfies the caratheodory assumptions and q l 1 a b r with a view towards defining a pseudo fucik spectrum pfs we study using a time mapping approach the eigenvalue like problem u a u b u 0 u a 0 u b where a 0 b 0 we show here that this pfs which we denote by l consists of a set of curves in r 2 resembling the classical fucik spectrum i e when s s our main existence result which deals with nonresonance with respect to the pfs can be roughly stated as follows if for s 0 sufficiently large and for almost every t a b the pair f t s s f t s s lies in a compact rectangle contained in an open component of r 2 l which intersects the diagonal then problem p has at least a solution

an entropy stable discontinuous galerkin approximation for the incompressible navier stokes equations with variable density and artificial compressibility abstract we present a provably stable discontinuous galerkin spectral element method for the incompressible navier stokes equations with artificial compressibility and variable density stability proofs which include boundary conditions that follow a continuous entropy analysis are provided we define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressibility and derive its associated entropy conservation law the latter allows us to construct a provably stable split form nodal discontinuous galerkin dg approximation that satisfies the summation by parts simultaneous approximation term sbp sat property the scheme and the stability proof are presented for general curvilinear three dimensional hexahedral meshes we use the exact riemann solver and the bassi rebay 1 br1 scheme at the inter element boundaries for inviscid and viscous fluxes respectively and an explicit low storage runge kutta rk3 scheme to integrate in time we assess the accuracy and robustness of the method by solving a manufactured solution the kovasznay flow a lid driven cavity the inviscid taylor green vortex and the rayleigh taylor instability

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance