moment maps and riemannian symmetric pairs we study hamiltonian actions of a compact lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold the fixed point set of the involution on the manifold is a lagrangian submanifold we investigate its image under the moment map and conclude that the intersection with the weyl chamber is an easily described subpolytope of the kirwan polytope of special interest is the integral kahler case where much stronger results hold in particular we obtain convexity theorems for closures of orbits of the noncompact dual group in the sense of the theory of symmetric pairs in the abelian case these results were obtained earlier by duistermaat we derive explicit inequalities for polytopes associated with real flag varieties

trust region based solver for nonlinear transport in heterogeneous porous media we describe a new nonlinear solver for immiscible two phase transport in porous media where viscous buoyancy and capillary forces are significant the flux fractional flow function f is a nonlinear function of saturation and typically has inflection points and can be non monotonic the non convexity and non monotonicity of f are major sources of difficulty for nonlinear solvers of coupled multiphase flow and transport in natural porous media we describe a modified newton algorithm that employs trust regions of the flux function to guide the newton iterations the flux function is divided into saturation trust regions delineated by the inflection unit flux and end points the updates are performed such that two successive iterations cannot cross any trust region boundary if a crossing is detected the saturation value is chopped back to the appropriate trust region boundary the proposed trust region newton solver which is demonstrated across the parameter space of viscous buoyancy and capillary effects is a significant extension of the inflection point strategy of jenny et al jcp 2009 5 for viscous dominated flows we analyze the discrete nonlinear transport equation obtained using finite volume discretization with phase based upstream weighting then we prove convergence of the trust region newton method irrespective of the timestep size for single cell problems numerical results across the full range of the parameter space of viscous gravity and capillary forces indicate that our trust region scheme is unconditionally convergent for 1d transport that is for a given choice of timestep size the unique discrete solution is found independently of the initial guess for problems dominated by buoyancy and capillarity the trust region newton solver overcomes the often severe limits on timestep size associated with existing methods to validate the effectiveness of the new nonlinear solver for large reservoir models with strong heterogeneity we compare it with state of the art nonlinear solvers for two phase flow and transport using the spe 10 model compared with existing nonlinear solvers our trust region solver results in superior convergence performance and achieves reduction in the total newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

implicit hybrid upwinding for two phase flow in heterogeneous porous media with buoyancy and capillarity abstract we consider the numerical solution of the partial differential equations governing multiphase flow in porous media for highly nonlinear problems the temporal discretization of choice is often the unconditionally stable fully implicit method however the nonlinear systems often solved with newton s method are difficult to solve thus the computational cost is strongly dependent on the nonlinear convergence rate and enhancing this convergence property is key to speed up subsurface flow simulation we focus on the case of spatially discontinuous capillary pressure between rock regions to efficiently and accurately simulate the flow dynamics in heterogeneous porous media the flux computation combines implicit hybrid upwinding with transmission conditions between different rock regions this leads to a scheme that correctly represents the trapping mechanisms while improving the nonlinear convergence we extend our previous results hamon et al 2016 18 by generalizing the scheme to fully implicit coupled flow and transport to address realistic problems in multiple dimensions the generalized scheme is supported by an analysis of its mathematical properties our multidimensional numerical examples which range from buoyancy driven flow with capillary barriers to viscous dominated flow demonstrate that the implicit hybrid upwinding scheme improves the accuracy compared to the standard phase based upwinding scheme while leading to significant reductions in the number of nonlinear iterations in multiple dimensions

monotonicity considerations for saturated unsaturated subsurface flow it is demonstrated that monotonicity is a sufficient condition to ensure that no new nonphysical local maxima and minima can be produced in the discrete nonlinear unsaturated flow equation monotonicity conditions are derived for various types of weighting for the mobility term the basic discretization is of finite element type but the results can be extended to finite volume discretizations central weightings are only conditionally monotone while upstream weightings are unconditionally monotone sample computations are given for highly heterogeneous problems

a deconvolution enhancement of the navier stokes αβ model a deconvolution enhancement of the email protected b model for turbulent flow is introduced the energy and energy dissipation rate for the enhanced model are derived it is also shown that the consistency error relative to the navier stokes equations and the microscale of the enhanced model are less than those of the email protected b model the proposed model is used to simulate the taylor green vortex problem and results show a qualitatively improved representation of the mean square vorticity when compared to the email protected b model numerical studies of the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor for three dimensional turbulent flows with re 200 are used to explore the utility of the model a benchmark problem of a two dimensional channel flow over a step for re 600 also indicates that this model can be applied to more general flows than those involving periodic boundary conditions

monotone difference approximations of bv solutions to degenerate convection diffusion equations we consider consistent conservative form monotone difference schemes for nonlinear convection diffusion equations in one space dimension since we allow the diffusion term to be strongly degenerate solutions can be discontinuous and in general are not uniquely determined by their data here we choose to work with weak solutions that belong to the bv in space and time class and in addition satisfy an entropy condition a recent result of wu and yin northeastern math j 5 1989 pp 395 422 states that these so called bv entropy weak solutions are unique the class of equations under consideration is very large and contains to mention only a few the heat equation the porous medium equation the two phase flow equation and hyperbolic conservation laws the difference schemes are shown to converge to the unique bv entropy weak solution of the problem in view of the classical theory for monotone difference approximations of conservation laws the main difficulty in obtaining a similar convergence theory in the present context is to show that the strongly degenerate discrete diffusion term is sufficiently smooth we provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes finally we make some concluding remarks about monotone difference schemes for multidimensional equations

random walks on symmetric spaces and inequalities for matrix spectra abstract using harmonic analysis on symmetric spaces we reduce the singular spectral problem for products of matrices to the recently solved spectral problem for sums of hermitian matrices this proves r c thompson s conjecture matrix spectral inequalities johns hopkins university press baltimore md 1988

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

vertex approximate gradient discretization preserving positivity for two phase darcy flows in heterogeneous porous media abstract in this article a new nodal discretization is proposed for two phase darcy flows in heterogeneous porous media the scheme combines the vertex approximate gradient vag scheme for the approximation of the gradient fluxes with an hybrid upwind hu approximation of the mobility terms in the saturation equation the discretization in space incorporates naturally nodal interface degrees of freedom d o f allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves it is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3d large discrete fracture matrix dfm models

convergence of implicit monotone schemes with applications in multiphase flow in porous media phase based upstreaming which is a commonly used spatial discretization for multiphase flow in reservoir simulation naturally gives rise to implicit monotone schemes when implicit time stepping is used the nonlinear gauss seidel and jacobi algorithms are shown to converge to a unique bounded solution when applied to the resulting system of equations thus for one dimensional problems we obtain an alternate constructive proof that such schemes are well defined and converge to the entropy solution of the conservation law for arbitrary cfl numbers the accuracy of phase based upstream solutions is studied for various spatial and temporal grids revealing the importance of unconditional stability when nonuniform grids and or variable porosity is involved

ideal triangles in euclidean buildings and branching to levi subgroups abstract let g denote a connected reductive group defined and split over z and let m g denote a levi subgroup in this paper we study varieties of geodesic triangles with fixed vector valued side lengths in the bruhat tits buildings associated to g along with varieties of ideal triangles associated to the pair m g the ideal triangles have a fixed side containing a fixed base vertex and a fixed infinite vertex such that other infinite side containing has fixed ideal length and the remaining finite side has fixed length we establish an isomorphism between varieties in the second family and certain varieties in the first family the pair and the triple satisfy a certain relation we apply these results to the study of the hecke ring of g and the restriction homomorphism r g r m between representation rings we deduce some new saturation theorems for constant term coefficients and for the structure constants of the restriction homomorphism

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

nonlinear analysis of multiphase transport in porous media in the presence of viscous buoyancy and capillary forces nonlinear convergence problems in numerical reservoir simulation can lead to unacceptably large computational time and are often the main impediment to performing simulation studies of large scale problems we analyze the nonlinearity of the discrete transport mass conservation equation for immiscible incompressible two phase flow in porous media in the presence of viscous buoyancy and capillary forces although simulation problems are multi dimensional with large numbers of cells and variables we find that the essence of the nonlinear behavior can be understood by studying the discretized numerical flux function for the interface between two cells the numerical flux is expressed in terms of the saturations of the two cells discontinuities in the first order derivative of the flux function referred to as kinks and inflection lines are identified as the cause of convergence difficulty these critical features kinks and inflections change the curvature of the numerical flux function abruptly and can lead to overshoots oscillations or divergence in newton iterations based on our understanding of the nonlinearity a nonlinear solver is developed referred to as the numerical trust region ntr solver the solver is able to guide the newton iterations safely and efficiently through the different saturation trust regions delineated by the kinks and inflections specifically overshoots and oscillations that often lead to convergence failure are avoided numerical examples demonstrate that our ntr solver has superior convergence performance compared with existing methods in particular convergence is achieved for a wide range of timestep sizes and courant friedrichs lewy cfl numbers spanning several orders of magnitude in addition a discretization scheme is proposed for handling heterogeneities in capillary pressure saturation relationship the scheme has less degree of nonlinearity compared with the standard single point phase based upstream weighting scheme leading to an improved nonlinear convergence performance especially when used together with our ntr solver our proposed numerical solution strategy that is based on the numerical flux and handles capillarity extends the previous work by jenny et al 2009 6 and wang and tchelepi 2013 7 significantly the nonlinearity of multiphase transport in porous media is analyzed the analysis considers viscous buoyancy and capillary forces the critical features that cause nonlinear convergence failures are identified a nonlinear solver that converges without oscillations and divergence is developed the solver converges faster than existing methods and allows large timesteps

upstream differencing for multiphase flow in reservoir simulation upstream weighting for multiphase flow in reservoir simulation is analyzed the associated numerical flux is shown to be well defined monotone lipschitz continuous and consistent in the case of a two phase flow the corresponding numerical flux is compared to that of godunov and engquist and osher finally simple way to obtain a higher order scheme is outlined

a robust nitsche s formulation for interface problems with spline based finite elements summary the extended finite element method x fem has proven to be an accurate robust method for solving embedded interface problems with a few exceptions the x fem has mostly been used in conjunction with piecewise linear shape functions and an associated piecewise linear geometrical representation of interfaces in the current work the use of spline based finite elements is examined along with a nitsche technique for enforcing constraints on an embedded interface to obtain optimal rates of convergence we employ a hierarchical local refinement approach to improve the geometrical representation of curved interfaces we further propose a novel weighting for the interfacial consistency terms arising in the nitsche variational form with b splines a qualitative dependence between the weights and the stabilization parameters is established with additional element level eigenvalue calculations an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of large heterogeneities as well as elements with arbitrarily small volume fractions we demonstrate the accuracy and robustness of the proposed method through several numerical examples copyright 2015 john wiley sons ltd

implicit hybrid upwind scheme for coupled multiphase flow and transport with buoyancy abstract numerical simulation of coupled multiphase flow and transport in porous media is used in many scientific and industrial applications such as groundwater management and oil recovery to solve the governing partial differential equations we focus on the fully implicit backward euler finite volume method fim in this numerical scheme large algebraic systems must be solved at each time step using newton s method this can be quite expensive especially for highly nonlinear problems with buoyancy in this work we present a new first order approximation of the flux in the fim leading to improved nonlinear convergence rate and radius this approximation is based on implicit hybrid upwinding and relies on separate treatments of the viscous and buoyancy parts of the numerical flux moreover in the viscous part the total velocity discretization is key as it controls the coupling between the elliptic flow problem and the hyperbolic multiphase transport problem to achieve robustness and efficiency we propose a differentiable total velocity discretization that adapts to the balance of forces at each interface we analyze the theoretical properties of the resulting numerical scheme and prove that saturations remain between physical bounds challenging two and three phase test cases illustrate that our numerical scheme brings a significant reduction in the number of nonlinear iterations compared to a finite volume discretization relying on the widely used phase per phase upstream weighting our numerical scheme therefore reduces the simulation cost while providing similar accuracy

analysis of hybrid upwinding for fully implicit simulation of three phase flow with gravity the systems of algebraic equations arising from implicit backward euler finite volume discretization of the conservation laws governing multiphase flow in porous media are quite challenging for nonlinear solvers in the presence of countercurrent flow due to buoyancy the numerical flux obtained with single point phase potential upwinding ppu is not differentiable which causes convergence difficulties for nonlinear solvers recently lee efendiev and tchelepi adv water resour 82 2015 pp 27 38 proposed a hybrid upwinding strategy for two phase flow that gives rise to a differentiable numerical flux across the entire viscous gravity parameter space here we first present an implicit hybrid upwinding ihu scheme for hyperbolic conservation laws extending the work of lee efendiev and tchelepi to an arbitrary number of fluid phases we show that the numerical flux obtained with the ihu is consistent and a monotone function of its own saturation it is also a differentiable function of t

stability of measures on kahler manifolds abstract let m be a kahler manifold and let k be a compact group that acts on m in a hamiltonian fashion we study the action of k c on probability measures on m first of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability semi stability and polystability next we apply this setting to the action of k c on measures we get various stability criteria for measures on kahler manifolds the same circle of ideas gives a very general surjectivity result for a map originally studied by hersch and bourguignon li yau

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

a posteriori analysis of a b spline based finite element method for the stationary quasi geostrophic equations of the ocean abstract we provide a posteriori error analysis of a b spline based finite element method for the streamfunction formulation of the stationary quasi geostrophic equations to model the large scale wind driven ocean circulation the analysis gives rise to a posteriori error estimator for more efficient and accurate modeling of a strong western boundary layer via adaptive local refinement of the computational geometries to verify the capability of the error estimator we use hierarchical b splines in conjunction with dorfler marking strategy numerical studies are performed for several benchmark examples on rectangular and l shape ocean geometries

mathematical structure of the black oil model for petroleum reservoir simulation this model incorporates compressibility and general mass transfer between phases it consists of the conditions of thermodynamic equilibrium an equation of state for the volume balance between the fluidand the rock void darcy s law for the volumetric flow rates and component conservation equations these relations are manipulated to form a pressure equation and a modified system of component conservation equations it is shown that the pressure equation is parabolic and that in the absence of diffusive forces such as capillary pressure and mixing the component conservation equations are hyperbolic subject to technical conditions on the relative permeabilities this sequential formulation of the flow equations forms types of wave structures that occur for the system

harmonic maps to buildings and singular perturbation theory the notion of a uni versal building associated with a point in the hitchin base is introduced the universal building is a building equipped with a harmonic map from the universal cover of the given riemann surface that is initial among harmonic maps which induce the given cameral cover of the riemann surface the notion of a versal building is obtained by relaxing the uniqueness condition in the definition of an initial object in the rank one case the universal building is the leaf space of the quadratic differential defining the point in the hitchin base the main conjectures of this paper are 1 a versal building always exists 2 the asymptotics of the riemann hilbert correspondence and the non abelian hodge correspondence are controlled by the harmonic map associated to any versal building 3 spectral networks arise as inverse images of singularities of the versal building and 4 versal buildings encode the data of a 3d calabi yau category whose space of stability conditions has a connected component that contains the hitchin base the main theorem establishes the existence of the universal building conjecture 3 as well as the riemann hilbert part of conjecture 2 in the case of the rank two example introduced in the seminal work of berk nevins roberts on higher order stokes phenomena it is also shown that the asymptotics of the riemann hilbert correspondence are always controlled by a harmonic map to a certain building which is constructed as the asymptotic cone of a symmetric space

a combination theorem for anosov subgroups we prove an analogue of klein combination theorem for anosov subgroups by using a local to global principle for morse quasigeodesics

numerical approximation of a two phase flow problem in a porous medium with discontinuous capillary forces we consider a simplified model of a two phase flow through a heterogeneous porous medium focusing on the capillary forces motion a nonlinear degenerate parabolic problem is approximated in a domain shared in two homogeneous parts each of them being characterized by its relative permeability and capillary curves functions of the phase saturations we first give a weak form of the conservation equations on the whole domain with a new general expression of the conditions at the interface between the two regions we then propose a finite volume scheme for the approximation of the solution which is shown to converge to a weak solution in one two or three dimensional domains we conclude with some numerical tests