algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations this article considers a posteriori error estimation and anisotropic mesh refinement for three dimensional laminar aerodynamic flow simulations the optimal order symmetric interior penalty discontinuous galerkin discretization which has previously been developed for the compressible navier stokes equations in two dimensions is extended to three dimensions symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model using duality arguments an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients furthermore residual based indicators as well as adjoint based indicators for goal oriented refinement are derived these refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous galerkin dg discretization two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint based error estimation are presented the performance of the proposed discretization error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows

level shift under the influence of relativistic point interaction potentials the level shift of dirac particles under the influence of point interaction potentials has been determined a green function method is used to obtain closed expressions for the level shift and perturbed wavefunctions provided that the corresponding eigenvalue problem without point interaction potentials can be solved exactly several applications are discussed in detail

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

the compact discontinuous galerkin cdg method for elliptic problems we present a compact discontinuous galerkin cdg method for an elliptic model problem the problem is first cast as a system of first order equations by introducing the gradient of the primal unknown or flux as an additional variable a standard discontinuous galerkin dg method is then applied to the resulting system of equations the numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level thus resulting in a global system that involves only the original unknown variable the proposed method is closely related to the local discontinuous galerkin ldg method b cockburn and c w shu siam j numer anal 35 1998 pp 2440 2463 but unlike the ldg method the sparsity pattern of the cdg method involves only nearest neighbors also unlike the ldg method the cdg method works without stabilization for an arbitrary orientation of the element interfaces the computation of the numerical interface fluxes for the cdg method is slightly more involved than for the ldg method but this additional complication is clearly offset by increased compactness and flexibility compared to the br2 f bassi and s rebay j comput phys 131 1997 pp 267 279 and ip j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 lecture notes in phys 58 springer berlin 1976 pp 207 216 methods which are known to be compact the present method produces fewer nonzero elements in the matrix and is computationally more efficient

preconditioning methods for local discontinuous galerkin discretizations a multilevel interior penalty method is used as an efficient preconditioner for the schur complement of the local discontinuous galerkin ldg discretization of a poisson problem the method is then used in a block triangular preconditioner of the ldg saddle point system the block preconditioner is of the same efficiency as the schur complement version finally the block preconditioner is extended to the discretization of the stokes problem by the ldg method again the preconditioned saddle point problem can be solved in about as many steps as the schur complement the influence of several parameters on the performance of these methods is investigated

high resolution viscous fingering simulation in miscible displacement using a p adaptive discontinuous galerkin method with algebraic multigrid preconditioner abstract high resolution simulation of viscous fingering can offer an accurate and detailed prediction for subsurface engineering processes involving fingering phenomena the fully implicit discontinuous galerkin dg method has been shown to be an accurate and stable method to model viscous fingering with high peclet number and mobility ratio in this paper we present two techniques to speedup large scale simulations of this kind the first technique relies on a simple p adaptive scheme in which high order basis functions are employed only in elements near the finger fronts where the concentration has a sharp change as a result the number of degrees of freedom is significantly reduced and the simulation yields almost identical results to the more expensive simulation with uniform high order elements throughout the mesh the second technique for speedup involves improving the solver efficiency we present an algebraic multigrid amg preconditioner which allows the dg matrix to leverage the robust amg preconditioner designed for the continuous galerkin cg finite element method the resulting preconditioner works effectively for fixed order dg as well as p adaptive dg problems with the improvements provided by the p adaptivity and amg preconditioning we can perform high resolution three dimensional viscous fingering simulations required for miscible displacement with high peclet number and mobility ratio in greater detail than before for well injection problems

two level fourier analysis of a multigrid approach for discontinuous galerkin discretization in this paper we study a multigrid mg method for the solution of a linear second order elliptic equation discretized by discontinuous galerkin dg methods and we give a detailed analysis of the convergence for different block relaxation strategies we find that pointwise block partitioning gives much better results than the classical cellwise partitioning both for the baumann oden method and for the symmetric dg method with and without interior penalty ip the block relaxation methods jacobi gauss seidel and symmetric gauss seidel give excellent smoothing procedures in a classical mg setting independent of the mesh size simple mg cycles give convergence factors of 0 075 0 4 per iteration sweep for the different discretization methods studied

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

robust smoothers for high order discontinuous galerkin discretizations of advection diffusion problems the multigrid method for discontinuous galerkin discretizations of advection diffusion problems is presented it is based on a block gauss seidel smoother with downwind ordering honoring the advection operator the cell matrices of the dg scheme are inverted in this smoother in order to obtain robustness for higher order elements employing a set of experiments we show that this technique actually yields an efficient preconditioner and that both ingredients downwind ordering and blocking of cell matrices are crucial for robustness discretizations of partial differential equations by higher order finite element methods or hp methods gained popularity during the last years after computing power and memory became readily available these methods yield much higher accuracy with similar degrees of freedom than standard schemes based on linear or bilinear polynomials nevertheless the utility of these schemes depends very much on the availability of efficient solvers for the discrete problems in this article we continue previous work and demonstrate that multilevel methods with suitably chosen smoothers can provide such solvers even in the presence of non self adjoint advection terms and locally refined meshes in 8 we devised a multilevel preconditioner for the symmetric interior penalty method and proved it s optimality these results were extended to other adjoint

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

anisotropic and dynamic mesh adaptation for discontinuous galerkin methods applied to reactive transport mesh adaptation strategies are proposed for discontinuous galerkin methods applied to reactive transport problems with emphasis on dynamic and anisotropic adaptation they include an anisotropic mesh adaptation scheme and two isotropic methods using an l 2 l 2 norm error estimator and a hierarchic error indicator these dynamic mesh adaptation approaches are investigated using benchmark cases numerical results demonstrate that the three approaches resolve time dependent transport adequately without slope limiting for both long term and short term simulations it is shown that these results apply to problems where either diffusion or advection dominates in different subdomains moreover for these schemes mass conservation is retained locally during dynamic mesh modification comparison studies indicate that the anisotropic mesh adaptation provides the most efficient meshes and has the least numerical diffusion among the three adaptation approaches 2005 elsevier b v all rights reserved

multigrid smoothers for ultraparallel computing this paper investigates the properties of smoothers in the context of algebraic multigrid amg running on parallel computers with potentially millions of processors the development of multigrid smoothers in this case is challenging because some of the best relaxation schemes such as the gauss seidel gs algorithm are inherently sequential based on the sharp two grid multigrid theory from r d falgout and p s vassilevski siam j numer anal 42 2004 pp 1669 1693 and r d falgout p s vassilevski and l t zikatanov numer linear algebra appl 12 2005 pp 471 494 we characterize the smoothing properties of a number of practical candidates for parallel smoothers including several c f polynomial and hybrid schemes we show in particular that the popular hybrid gs algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications provided that the problem size per processor is large enough this is encouraging news for the scalability of amg on ultraparallel computers we also introduce the more robust ell 1 smoothers which are always convergent and have already proven essential for the parallel solution of some electromagnetic problems t kolev and p vassilevski j comput math 27 2009 pp 604 623

a high order fully coupled upwind compact discontinuous galerkin method for modeling of viscous fingering in compressible porous media abstract we present a new approach for high fidelity compressible porous media simulations our method is based on a fully coupled upwind high order discontinuous galerkin formulation of the equations of miscible displacement transport a key feature of the proposed method is the high level of flexibility to compute complex subsurface geometries maintaining full order of accuracy the present formulation solves pressure and transport equations in a fully coupled fashion and as a result can capture the strong interaction between pressure and concentrations of injected components in highly compressible media the proposed method also shows very low sensitivity to mesh orientation in contrast with classical finite volume approximations used in porous media flow simulations the robustness and accuracy of the method are demonstrated in a number of challenging compressible and incompressible multiphase flow problems numerical simulations also reveal interesting phenomena associated with high pore compressibility such as the reduction of the growth rate of viscous fingers and the alteration of flow patterns near impermeable obstacles additional computations performed with an anisotropic permeability tensor reveal that somewhat against intuition viscous fingers have a preferential growth along the direction of low permeability

a fully coupled upwind discontinuous galerkin method for incompressible porous media flows high order computations of viscous fingering instabilities in complex geometry we present a new approach to the simulation of viscous fingering instabilities in incompressible miscible displacement flows in porous media in the past high resolution computational simulations of viscous fingering instabilities have always been performed using high order finite difference or fourier spectral methods which do not posses the flexibility to compute very complex subsurface geometries our approach instead by means of a fully coupled nonlinear implementation of the discontinuous galerkin method possesses a fundamental differentiating feature in that it maintains high order accuracy on fully unstructured meshes in addition the proposed method shows very low sensitivity to mesh orientation in contrast with classical finite volume approximation used in porous media flow simulations the robustness and accuracy of the method are demonstrated in a number of challenging computational problems

an analysis of the embedded discontinuous galerkin method for second order elliptic problems the embedded discontinuous galerkin methods are obtained from hybridizable discontinuous galerkin methods by a simple change of the space of the hybrid unknown in this paper we consider embedded methods for second order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous functions this change results in a significantly smaller stiffness matrix whose size and sparsity structure coincides with those of the stiffness matrix of the statically condensed continuous galerkin method it is shown that this computational advantage has to be balanced against the fact that the approximate solutions for the scalar variable and its flux lose each a full order of convergence indeed we prove that if polynomials of degree k ge1 are used for the original hybridizable discontinuous galerkin method its approximations to the scalar variable and its flux converge with order k 2 and k 1 respectively whereas those of the corresponding embedded discontinuous galerkin method converge with orders k 1 and k respectively only we also provide numerical results comparing the relative efficiency of the methods

unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g in this paper we study the geometric quantization of the symplectic quotient m g guillemin and sternberg invent math 67 515 538 1982 have shown under suitable regularity assumptions that there is a natural invertible map between the quantum hilbert space over m g and the g invariant subspace of the quantum hilbert space over m

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

toeplitz operators and hamiltonian torus actions this paper is devoted to semi classical aspects of symplectic reduction consider a compact prequantizable kahler manifold m with a hamiltonian torus action in the seminal paper v guillemin s sternberg geometric quantization and multiplicities of group representations invent math 67 3 1982 515 538 guillemin and sternberg introduced an isomorphism between the invariant part of the quantum space associated to m and the quantum space associated to the symplectic quotient of m provided this quotient is non singular we prove that this isomorphism is a fourier integral operator and that the toeplitz operators of m descend to toeplitz operators of the reduced phase space we also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced toeplitz operator a result related to the riemann roch kawasaki theorem

adaptive enriched galerkin methods for miscible displacement problems with entropy residual stabilization we present a novel approach to the simulation of miscible displacement by employing adaptive enriched galerkin finite element methods eg coupled with entropy residual stabilization for transport in particular numerical simulations of viscous fingering instabilities in heterogeneous porous media and hele shaw cells are illustrated eg is formulated by enriching the conforming continuous galerkin finite element method cg with piecewise constant functions the method provides locally and globally conservative fluxes which are crucial for coupled flow and transport problems moreover eg has fewer degrees of freedom in comparison with discontinuous galerkin dg and an efficient flow solver has been derived which allows for higher order schemes dynamic adaptive mesh refinement is applied in order to reduce computational costs for large scale three dimensional applications in addition entropy residual based stabilization for high order eg transport systems prevents spurious oscillations numerical tests are presented to show the capabilities of eg applied to flow and transport enriched galerkin approximations for coupled flow and transport system entropy residual stabilization for enriched galerkin to stabilize higher order transport scheme dynamic mesh adaptivity for viscous fingering in two and three dimensions

a discontinuous galerkin method for gravity driven viscous fingering instabilities in porous media we present a new approach to the simulation of gravity driven viscous fingering instabilities in porous media flow these instabilities play a very important role during carbon sequestration processes in brine aquifers our approach is based on a nonlinear implementation of the discontinuous galerkin method and possesses a number of key features first the method developed is inherently high order and is therefore well suited to study unstable flow mechanisms secondly it maintains high order accuracy on completely unstructured meshes the combination of these two features makes it a very appealing strategy in simulating the challenging flow patterns and very complex geometries of actual reservoirs and aquifers this article includes an extensive set of verification studies on the stability and accuracy of the method and also features a number of computations with unstructured grids and non standard geometries

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

constraints reduction and quantization theorems are proved that establish the unitary equivalence of the extended and reduced phase space quantizations of a constrained classical system with symmetry several examples are presented

analytical and variational numerical methods for unstable miscible displacement flows in porous media the miscible displacement of one fluid by another in a porous medium has received considerable attention in sub surface environmental and petroleum engineering applications when a fluid of higher mobility displaces another of lower mobility unstable patterns referred to as viscous fingering may arise their physical and mathematical study has been the object of numerous investigations over the past century the objective of this paper is to present a review of these contributions with particular emphasis on variational methods these algorithms are tailored to real field applications thanks to their advanced features handling of general complex geometries robustness in the presence of rough tensor coefficients low sensitivity to mesh orientation in advection dominated scenarios and provable convergence with fully unstructured grids this paper is dedicated to the memory of dr jim douglas jr for his seminal contributions to miscible displacement and variational numerical methods