an algebraic multigrid method for a class of elliptic differential systems the development numerical investigation and analysis of an algebraic multigrid amg method for the numerical solution of a representative class of elliptic differential systems are presented our amg scheme is based on the amg strategy of brandt ruge and stuben however the present approach may be implemented within other amg schemes by means of numerical experiments with model problems arising from some application areas it is demonstrated that the computational performance of the proposed amg scheme is comparable to that of amg applied to a single scalar equation an amg convergence theory is presented that extends convergence results for amg methods for scalar problems to the case of amg applied to elliptic systems some details of our amg implementation are given

amge coarsening strategies and application to the oseen equations we provide some extensions to the algebraic multigrid method based on element interpolation amge concerning the agglomeration process the application to nonconforming elements and the application to the mixed finite element discretization of the oseen linearized navier stokes equations this last point using amge for mixed finite elements becomes straightforward because of the availability of coarse level topologies we show this exemplarily for the crouzeix raviart element including a stability result the numerical results show that it really pays off to take a closer look at the agglomeration strategy a wrong choice can lead to insufficient convergence or even divergence of the overall multigrid method

a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

fully implicit mixed hybrid finite element discretization for general purpose subsurface reservoir simulation abstract we present a new fully implicit mixed hybrid finite element mhfe discretization scheme for general purpose compositional reservoir simulation the locally conservative scheme solves the coupled momentum and mass balance equations simultaneously and the fluid system is modeled using a cubic equation of state we introduce a new conservative flux approach for the mass balance equations for this fully implicit approach we discuss the nonlinear solution procedure for the proposed approach and we present extensive numerical tests to demonstrate the convergence and accuracy of the mhfe method using tetrahedral elements we also compare the method to other advanced discretization schemes for unstructured meshes and tensor permeability finally we illustrate the applicability and robustness of the method for highly heterogeneous reservoirs with unstructured grids

multicolor fourier analysis of the multigrid method for quadratic fem discretizations abstract quadratic finite element methods offer some advantages for the numerical solution of partial differential equations pdes due to their improved approximation properties in comparison to linear approaches the algebraic linear systems arising from the discretization of pdes by this kind of methods require an efficient resolution and multigrid methods provide a good way to solve this problem to design geometric multigrid methods local fourier analysis lfa is a very useful tool however lfa for quadratic finite element discretizations can not be performed in a standard way since the discrete operator is defined by different stencils depending on the location of the points in the grid in this work a multicolor local fourier analysis is presented to analyze multigrid solvers for this type of discretizations with the help of this analysis some point wise and line wise smoothers are analyzed some results showing the good correspondence between the two grid convergence factors predicted by the analysis and the experimentally computed asymptotic convergence factors are presented finally this analysis is applied to design a very efficient multigrid solver for semi structured triangular grids based on a hybrid smoother

an axiomatization of the algebra of petri net concatenable processes the concatenable processes of a petri net n can be characterized abstractly as the arrows of a symmetric monoidal category view the mathml source n however this is only a partial axiomatization since it is based on a concrete ad hoc chosen category of symmetries symn in this paper we give a completely abstract characterization of the category of concatenable processes of n thus yielding an axiomatic theory of the noninterleaving behaviour of petri nets

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

a necessary and sufficient condition for the positive definiteness of interval symmetric matrices a necessary and sufficient condition for the positive definiteness of interval symmetric matrices is obtained and a useful application is shown

multigrid methods for propagators in lattice gauge theories multigrid methods were invented for the solution of discretized partial differential equations in ordered systems the slowness of traditional algorithms is overcome by updates on various length scales in this overview article we discuss generalizations of multigrid methods for disordered systems in particular for propagators in lattice gauge theories a discretized nonabelian gauge theory can be formulated as a system of statistical mechanics where the gauge field degrees of freedom are su n matrices on the links of the lattice these su n matrices appear as random coefficients in dirac equations we aim at finding an efficient method by which one can solve dirac equations without critical slowing down if this could be achieved monte carlo simulations of quantum chromodynamics the theory of the strong interaction would be accelerated considerably in principle however the methods discussed can be used in arbitrary space time dimension and for arbitrary gauge group moreover there are applications in multigrid monte carlo simulations and for the definition of block spins and blocked gauge fields in monte carlo renormalization group studies as a central result it was found that the geometric multigrid method works in principle in arbitrarily disordered gauge fields finally an overview is given of other approaches to the propagator problem in lattice gauge theories

efficient nonlinear solvers for laplace beltrami smoothing of three dimensional unstructured grids the laplace beltrami system of nonlinear elliptic partial differential equations has utility in the generation of computational grids on complex and highly curved geometry discretization of this system using the finite element method accommodates unstructured grids but generates a large sparse ill conditioned system of nonlinear discrete equations the use of the laplace beltrami approach particularly in large scale applications has been limited by the scalability and efficiency of solvers this paper addresses this limitation by developing two nonlinear solvers based on the jacobian free newton krylov jfnk methodology a key feature of these methods is that the jacobian is not formed explicitly for use by the underlying linear solver iterative linear solvers such as the generalized minimal residual gmres method do not technically require the stand alone jacobian instead its action on a vector is approximated through two nonlinear function evaluations the preconditioning required by gmres is also discussed two different preconditioners are developed both of which employ existing algebraic multigrid amg methods further the most efficient preconditioner overall for the problems considered is based on a picard linearization numerical examples demonstrate that these solvers are significantly faster than a standard newton krylov approach a speedup factor of approximately 26 was obtained for the picard preconditioner on the largest grids studied here in addition these jfnk solvers exhibit good algorithmic scaling with increasing grid size

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

convergence analysis of the mimetic finite difference method for elliptic problems we propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes the a priori error analysis relies on the connection between the mimetic formulation and the lowest order raviart thomas mixed finite element method the theoretical results are confirmed by numerical experiments

the incomplete factorization multigraph algorithm we present a new family of multigraph algorithms ilu mg based upon an incomplete sparse matrix factorization using a particular ordering and allowing a limited amount of fill in while much of the motivation for multigraph comes from multigrid ideas ilu mg is distinctly different from algebraic multilevel methods the graph of the sparse matrix a is recursively coarsened by eliminating vertices using a graph model similar to gaussian elimination incomplete factorizations are obtained by allowing only the fill in generated by the vertex parents associated with each vertex multigraph is numerically compared with algebraic multigrid on some examples arising from discretizations of partial differential equations on unstructured grids

mimetic discretization of two dimensional magnetic diffusion equations abstract in case of non constant resistivity cylindrical coordinates and highly distorted polygonal meshes a consistent discretization of the magnetic diffusion equations requires new discretization tools based on a discrete vector and tensor calculus we developed a new discretization method using the mimetic finite difference framework it is second order accurate on arbitrary polygonal meshes and a consistent calculation of the joule heating is intrinsic within it the second order convergence rates in l 2 and l 1 norms were verified with numerical experiments

the non sequential behaviour of petri nets the idea of representing non sequential processes as partially ordered sets occurrence nets is applied to place transition nets petri nets based on the well known notion of process for condition event systems for occurrence nets some theorems relating k density cut finiteness and discreteness are proved with these theorems the result that a place transition net is bounded if and only if its processes are k dense is obtained

robust solution of singularly perturbed problems using multigrid methods we consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction diffusion type standard discretization techniques are not suitable for such problems and so specially tailored methods are required usually involving adapted or fitted meshes that resolve important features such as boundary and or interior layers in this study we consider classical finite difference schemes on the layer adapted meshes of shishkin and bakhvalov we show that standard direct solvers exhibit poor scaling behavior with respect to the perturbation parameter when solving the resulting linear systems we propose and prove optimality of a new block structured preconditioning approach that is robust for small values of the perturbation parameter and compares favorably with standard robust multigrid preconditioners for these linear systems we also derive stopping criteria which ensure that the potential accuracy

an application of multigrid methods for a discrete elastic model for epitaxial systems we apply an efficient and fast algorithm to simulate the atomistic strain model for epitaxial systems recently introduced by schindler et al phys rev b 67 075316 2003 the discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge in this paper we construct an algebraic multigrid method suitable for efficient solution of the large scale discrete strain model using this method simulations are performed for several representative physical problems including an infinite periodic step train a layered nanocrystal and a system of quantum dots the results demonstrate the effectiveness and robustness of the method and show that the method attains optimal convergence properties regardless of the problem size the geometry and the physical parameters the effects of substrate depth and of invariance due to traction free boundary conditions are assessed for a system of quantum dots the simulated strain energy density supports the observations that trench formation near the dots provides strain relief

a greedy strategy for coarse grid selection efficient solution of the very large linear systems that arise in numerical modeling of real world applications is often only possible through the use of multilevel techniques while highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considered much recent interest has been in the development of purely algebraic approaches that may be applied in many situations without problem specific tuning here we consider an algebraic approach to finding the fine coarse partitions needed in multilevel approaches the algorithm is motivated by recent theoretical analysis of the performance of two common multilevel algorithms multilevel block factorization and algebraic multigrid while no guarantee on the rate of coarsening is given the splitting is shown to always yield an effective preconditioner in the two level sense numerical performance of two level and multilevel variants of this approach is demonstrated in combination with both algebraic multigrid and multilevel block factorizations and the advantages of each of these two algorithmic settings are explored

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

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

sequential and concurrent behaviour in petri net theory abstract two ways of describing the behaviour of concurrent systems have widely been suggested arbitrary interleaving and partial orders sometimes the latter has been claimed superior because concurrency is represented in a true way on the other hand some authors have claimed that the former is sufficient for all practical purposes petri net theory offers a framework in which both kinds of semantics can be defined formally and hence compared with each other occurence sequences correspond to interleaved behaviour while the notion of a process is used to capture partial order semantics this paper aims at obtaining formal results about the relationship between various classes of processes and occurence sequences in net theory we shall compare an axiomatic approach to the definition of processes with an inductive approach which relates processes to occurence sequences and generalisations thereof we shall show that in general axiomatic process semantics is more powerful than inductive semantics using occurence sequences however we shall also show that the latter can be generalised or the former be restricted to yield various equivalence results the structure of the relation between sequences and processes will also be explored exhibiting two meaningful relations one on the sequences and one on the processes which correspond to each other bijectively we shall apply and simplify the theory to the practically important case of nets which are of finite synchronisation and 1 safe

imf an incomplete multifrontal lu factorization for element structured sparse linear systems we propose an incomplete multifrontal lu factorization imf preconditioner that extends supernodal multifrontal methods to incomplete factorizations it can be used as a preconditioner in a krylov subspace method to solve large scale sparse linear systems with an element structure e g those arising from a finite element discretization of a partial differential equation the fact that the element matrices are dense is exploited to increase the computational performance and the robustness of the factorization through efficient partial pivoting imf is compared with the multilevel arms2 the level of fill in ilu and the threshold based ilutp preconditioners our experiments indicate that imf is competitive with arms2 on saddle point problems arising in the solution of the steady state navier stokes equation experiments with element structured matrices arising from structural engineering applications found in the university of florida sparse matrix collection illustrate the robustness of imf final

algebraic multigrid for markov chains an algebraic multigrid amg method is presented for the calculation of the stationary probability vector of an irreducible markov chain the method is based on standard amg for nonsingular linear systems but in a multiplicative adaptive setting a modified amg interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums we show how the adoption of a previously described lumping technique maintains the irreducible singular m matrix character of the coarse level operators on all levels together these properties are sufficient to guarantee the well posedness of the algorithm numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems

specification and proof in membership equational logic this paper is part of a long term effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic basis on which efficient execution by rewriting and powerful theorem proving tools can be based in particular our rewriting techniques provide semantic foundations for maude s functional sublanguage where they have been efficiently implemented

the mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes we study the mimetic finite difference discretization of diffusion type problems on unstructured polyhedral meshes we demonstrate high accuracy of the approximate solutions for general diffusion tensors the second order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes on non matching meshes and even on meshes with irregular shaped polyhedra with flat faces we show that in general the meshes with non flat faces require more than one flux unknown per mesh face to get optimal convergence rates

effective dispersion in heterogeneous media under random transient flow conditions 1 under steady state flow conditions solute dispersion in heterogeneous porous media is much smaller in the transverse than in the longitudinal direction this holds particularly for effective dispersion of a plume originating from a point like injection the effective dispersion coefficient describes the actual dispersive mixing of solutes in the aquifer the lack of dispersive transverse mixing may limit considerably natural attenuation of certain contaminants temporal fluctuations of the flow direction enhance horizontal transverse dispersion this has been shown previously for uniform flow and for macrodispersion in stationary media we present a linear stochastic theory for effective dispersion under quasi steady state flow conditions with random temporal fluctuations of the mean flow direction as for macrodispersion effective transverse dispersion proves to be dominated by transient flow effects we compare semianalytical results derived from linear theory to those from particle tracking random walk simulations for a three dimensional test case the parameters of the test case are similar to those obtained at the borden site where the mean transverse flow component fluctuated approximately by 10 linear theory and particle simulations agree well

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

petri nets are moniods abstract petri nets are widely used to model concurrent systems however their composition and abstraction mechanisms are inadequate we solve this problem in a satisfactory way we start by remarking that place transition petri nets can be viewed as ordinary directed graphs equipped with two algebraic operations corresponding to parallell and sequential composition of transitions a distributive law between the two operations captures a basic fact about concurrency new morphisms are defined mapping single atomic transitions into whole computations thus relating system descriptions at different levels of abstraction categories equipped with products and coproducts corresponding to parallel and nondeterministic compositions are introduced for petri nets with and without initial markings petri net duality is expressed as a duality functor and several new invariants are introduced a tensor product is defined on nets and their category is proved to be symmetric monoidal closed this construction is generalized to a large class of algebraic theories on graphs these results provide a formal basis for expressing the semantics of concurrent languages in terms of petri nets they also provide a new understanding of concurrency in terms of algebraic structures over graphs and categories that should apply to other models besides petri nets and thus contribute to the conceptual unification of concurrency

homogenization of richards equation in permeability fields with different connectivities 1 large scale modeling of transient flow in the unsaturated zone is important for the estimation of the water budget and solute transport in the vadose zone upscaled flow models need to capture the impact of small scale heterogeneities which are not resolved by the model on large scale flow we perform upscaling of the richards equation in heterogeneous porous media with continuous distributions of the soil hydraulic parameters using homogenization theory and stochastic averaging techniques we restrict the analysis to flow regimes in which the capillary equilibrium assumption holds on the small scale in order to account for effects of capillary entry pressure we apply the brooks corey model for the soil retention and relative permeability curves and consider leverett scaling for the coupling of intrinsic permeability and entry pressure for this model we derive and analyze the ensemble averaged parameter functions for the macroscopic flow equations the effects of a definite entry pressure vanish with increasing variance of the log intrinsic permeability we compare the statistically averaged parameter functions to numerically calculated effective functions for parameter fields with different connectivity properties these results illustrate that soils with well connected coarse materials differ in the relative permeability from those with well connected fine materials or those without particular connectedness

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

adaptive algebraic multigrid efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort overcoming the limitations of geometric multigrid methods to simple geometries and differential equations algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix while this allows for efficient black box solution of the linear systems associated with discretizations of many elliptic differential equations it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices this paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positive definite linear systems

compatible relaxation and coarsening in algebraic multigrid we introduce a coarsening algorithm for algebraic multigrid amg based on the concept of compatible relaxation cr the algorithm is significantly different from standard methods most notably because it does not rely on any notion of strength of connection we study its behavior on a number of model problems and evaluate the performance of an amg algorithm that incorporates the coarsening approach finally we introduce a variant of cr that provides a sharper metric of coarse grid quality and demonstrate its potential with two simple examples

first order system least squares for incompressible resistive magnetohydrodynamics magnetohydrodynamics mhd is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles hence the equations that describe the plasma form a nonlinear system that couples navier stokes equations with maxwell s equations this paper shows that the first order system least squares fosls finite element method is a viable discretization for these large mhd systems to solve this system a nested iteration newton fosls amg approach is taken most of the work is done on the coarse grid including most of the linearizations we show that at most one newton step and a few v cycles are all that are needed on the finest grid here we describe how the fosls method can be applied to incompressible resistive mhd and how it can be used to solve these mhd problems efficiently a 3d steady state and a reduced 2d time dependent test problem are studied the latter equations can simulate a large aspect ratio tokamak the goal is to resolve as much physics from the test problems with the least amount of computational work we show that this is achieved in a few dozen work units or fine grid residual evaluations

a finite volume method for approximating 3d diffusion operators on general meshes a finite volume method is presented for discretizing 3d diffusion operators with variable full tensor coefficients this method handles anisotropic non symmetric or discontinuous variable tensor coefficients while distorted non matching or non convex n faced polyhedron meshes can be used for meshes of polyhedra whose faces have not more than four edges the associated matrix is positive definite and symmetric if the diffusion tensor is symmetric a second order resp first order accuracy is numerically observed for the solution resp gradient of the solution

a new algebraic multigrid approach for stokes problems standard discretizations of stokes problems lead to linear systems of equations in saddle point form making difficult the application of algebraic multigrid methods in this paper a new approach is proposed it consists in first transforming the system by pre and post multiplication with simple algebraic sparse block triangular matrices this is a form of pre conditioning in the literal sense designed to make sure that the transformed matrix is well adapted to multigrid in particular after transformation all the diagonal blocks are symmetric and positive definite and correspond to or resemble a discrete laplace operator then to each of these diagonal blocks is associated a prolongation that works well for it using any relevant algebraic or geometric multigrid method next a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a galerkin coarse grid matrix for this approach combined with damped jacobi smoothing a uniform two grid convergence bound is derived for the global system under the assumption that the two grid schemes for the different diagonal blocks are themselves uniformly convergent this result is illustrated by a few examples showing further that time dependent problems and variable viscosity can be handled in a natural way without requiring parameter adjustment a numerical comparison also shows that the new approach can be more effective than state of the art block preconditioning techniques

a higher order formulation of the mimetic finite difference method a new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes this approach provides a higher order accurate approximation to the flux of the exact solution in analogy with the original formulation a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability the scalar solution field is approximated by a piecewise constant function a computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable finally optimal convergence rates and accuracy improvement with respect to the lower order formulation are shown by numerical examples

accurate gradient approximation for complex interface problems in 3d by an improved coupling interface method most elliptic interface solvers become complicated for complex interface problems at those exceptional points where there are not enough neighboring interior points for high order interpolation such complication increases especially in three dimensions usually the solvers are thus reduced to low order accuracy in this paper we classify these exceptional points and propose two recipes to maintain order of accuracy there aiming at improving the previous coupling interface method 26 yet the idea is also applicable to other interface solvers the main idea is to have at least first order approximations for second order derivatives at those exceptional points recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point whenever this is possible recipe 2 is to flip domain signatures and introduce a ghost state so that a second order method can be applied this ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface the original state is recovered by a post processing using nearby states and jump conditions the choice of recipes is determined by a classification scheme of the exceptional points the method renders the solution and its gradient uniformly second order accurate in the entire computed domain numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions and a real molecule 1d63 which is double helix shape and composed of hundreds of atoms

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

algebraic multigrid for finite element elasticity equations determination of nodal dependence via edge matrices and two level convergence we study an algebraic multigrid amg method for solving elliptic finite element equations of linear elasticity problems in this method which has been proposed in kraus siam j sci comput 2008 30 505 524 the coarsening is based on the so called edge matrices which allows to generalize the concept of strong and weak connections as used in the classical amg to algebraic vertices that accumulate the nodal degrees of freedom in case of vector field problems the major contribution of this work is related to the investigation of a measure for the nodal dependence and on the generation of the edge matrices which are the basic building blocks of this method a natural measure is the cosine of the abstract angle between the two subspaces spanned by the basis functions corresponding to the respective algebraic vertices another original contribution of this work is a two level convergence analysis of the method the presented numerical results cover also problems with jumps in young s modulus of elasticity and orthotropic materials copyright 2010 john wiley sons ltd

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

a monotone finite volume scheme for diffusion equations on general non conforming meshes a nonlinear monotone finite volume scheme on general non conforming meshes for diffusion equations is introduced which deals with discontinuous tensor coefficients rigorously since the expression of normal flux depends on auxiliary unknowns defined at cell vertex including hanging nodes we propose a new method to eliminate vertex unknown by using primary unknowns at the centers of the cells sharing the vertex especially the unknowns defined on hanging nodes are eliminated by flux continuous conditions the resulting scheme is monotone and preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficient problems numerical results show that the convergent order of the monotone scheme by different methods of eliminating vertex unknowns will vary remarkably and our new method can assure that it has almost second order accuracy and more accurate than some existing methods

greedy coarsening strategies for nonsymmetric problems the solution of large scale linear systems in computational science and engineering requires efficient solvers and preconditioners often the most effective such techniques are those based on multilevel splittings of the problem in this paper we consider the problem of partitioning both symmetric and nonsymmetric matrices based solely on algebraic criteria a new algorithm is proposed that combines attractive features of two previous techniques proposed by the authors it offers rigorous guarantees of certain properties of the partitioning yet is naturally compatible with the threshold based dropping known to be effective for incomplete factorizations numerical results show that the new partitioning scheme leads to improved results for a variety of problems the effects of further matrix reordering within the fine scale block are also considered

petri nets event structures and domains part i abstract the general aim of this paper is to find a theory of concurrency combining the approaches of petri and scott and others in part i we introduce our formalisms to connect the abstract ideas of events and domains of information we show how casual nets induce certain kinds of domains where the information points are certain sets of events this allows translations between the languages of net theory and domain theory following the idea that events of causal nets are occurrences we generalise causal nets to occurrence nets by adding forwards conflict just as infinite flow charts unfold finite ones so transition nets can be unfolded into occurrence nets next we extend the above connections between nets and domains to these new nets event structures which are intermediate between nets and domains play an important part in all our work finally as an example of how concepts translate from one formalism to the other we show how petri s notion of confusion ties up with kahn and plotkin s concrete domains in part ii we shall continue the job of connecting up notions within net theory and the theory of domains in particular we shall examine the idea of states of computations

analysis and comparison of geometric and algebraic multigrid for convection diffusion equations the discrete convection diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes estimates of error reduction rates for both geometric multigrid gmg and algebraic multigrid amg are established on uniform rectangular meshes for a model problem our analysis shows that gmg with line gauss seidel smoothing and bilinear interpolation converges if h gg epsilon 2 3 and amg with the same smoother converges more rapidly than gmg if the interpolation constant beta in the approximation assumption of amg satisfies beta ll frac h sqrt epsilon alpha hs 1mm mbox where alpha big begin smallmatrix 1 h sqrt varepsilon 2 h ge sqrt varepsilon end smallmatrix on unstructured triangular meshes the performance of gmg and amg both as solvers and as preconditioners for gmres are evaluated numerical results show that gmres with amg preconditioning is a robust and reliable solver on both type of meshes

non negative mixed finite element formulations for a tensorial diffusion equation we consider the tensorial diffusion equation and address the discrete maximum minimum principle of mixed finite element formulations in particular we address non negative solutions which is a special case of the maximum minimum principle of mixed finite element formulations it is well known that the classical finite element formulations like the single field galerkin formulation and raviart thomas variational multiscale and galerkin least squares mixed formulations do not produce non negative solutions that is they do not satisfy the discrete maximum minimum principle on arbitrary meshes and for strongly anisotropic diffusivity coefficients in this paper we present two non negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques these proposed mixed formulations produce non negative numerical solutions on arbitrary meshes for low order i e linear bilinear and trilinear finite elements the first formulation is based on the raviart thomas spaces and the second non negative formulation is based on the variational multiscale formulation for the former formulation we comment on the effect of adding the non negative constraint on the local mass balance property of the raviart thomas formulation we perform numerical convergence analysis of the proposed optimization based non negative mixed formulations we also study the performance of the active set strategy for solving the resulting constrained optimization problems the overall performance of the proposed formulation is illustrated on three canonical test problems

an event structure semantics for general petri nets abstract in this paper we address the following question what type of event structures are suitable for representing the behaviour of general petri nets as a partial answer to this question we define a new class of event structures called local event structures and identify a subclass called ul event structures we propose that ul event structures are appropriate for capturing the behaviour of general petri nets our answer is a partial one in that in the proposed event structure semantics auto concurrency is filtered out from the behaviour of petri nets it turns out that this limited event structure semantics for petri nets is nevertheless a non trivial and conservative extension of the prime event structure semantics of 1 safe petri nets provided in nielsen et al 1981 we also show that the strong relationship between prime event structures and 1 safe petri nets established in a categorical framework in winskel 1987 can be extended to the present setting provided we restrict our attention to the subclass of petri nets whose behaviours do not exhibit any auto concurrency finally we show that winskel s general and stable event structures can be smoothly related to local event structures and that similarly prime event structures can be related to ul event structures

optimization of the multigrid convergence rate on semi structured meshes by local fourier analysis in this paper a local fourier analysis for multigrid methods on tetrahedral grids is presented different smoothers for the discretization of the laplace operator by linear finite elements on such grids are analyzed a four color smoother is presented as an efficient choice for regular tetrahedral grids whereas line and plane relaxations are needed for poorly shaped tetrahedra a novel partitioning of the fourier space is proposed to analyze the four color smoother numerical test calculations validate the theoretical predictions a multigrid method is constructed in a block wise form by using different smoothers and different numbers of pre and post smoothing steps in each tetrahedron of the coarsest grid of the domain some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm

a framework for the verification of infinite state graph transformation systems we propose a technique for the analysis of infinite state graph transformation systems based on the construction of finite structures approximating their behaviour following a classical approach one can construct a chain of finite under approximations k truncations of the winskel style unfolding of a graph grammar more interestingly also a chain of finite over approximations k coverings of the unfolding can be constructed the fact that k truncations and k coverings approximate the unfolding with arbitrary accuracy is formalised by showing that both chains converge in a categorical sense to the full unfolding we discuss how the finite over and under approximations can be used to check properties of systems modelled by graph transformation systems illustrating this with some small examples we also describe the augur tool which provides a partial implementation of the proposed constructions and has been used for the verification of larger case studies

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

on generalizing the algebraic multigrid framework we present a theory for algebraic multigrid amg methods that allows for general smoothing processes and general coarsening approaches the goal of the theory is to provide guidance in the development of new more robust amg algorithms in particular we introduce several compatible relaxation methods and give theoretical justification for their use as tools for measuring the quality of coarse grids

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

petr nets algebras morphisms and compositionality abstract it is shown how a category of petri nets can be viewed as a subcategory of two sorted algebras over multisets this casts petri nets in a familiar framework and provides a useful idea of morphism on nets different from the conventional definition the morphisms here respect the behaviour of nets the categorical constructions which result provide a useful way to synthesise nets and reason about nets in terms of their components for example various forms of parallel composition of petri nets arise naturally from the product in the category this abstract setting makes plain a useful functor from the category of petri nets to a category of spaces of invariants and provides insight into the generalisations of the basic definition of petri nets for instance the coloured and higher level nets of kurt jensen arise through a simple modification of the sorts of the algebras underlying nets further it provides a smooth formal relation with other models of concurrency such as milner s calculus of communicating systems ccs and hoare s communicating sequential processes csp though this is only indicated in this paper

a note on mgr methods by mgr we denote a class of special highly efficient multigrid methods for solving h discrete elliptic differential equations unlike standard multigrid methods 4 9 18 mgr methods are characterized by intermediate grids between the given h grid and the 2h grid and by special fine to coarse and coarse to fine grid transfer techniques the mgr idea has been conceived 13 7 by the second author in trying to extend the range of applicability of the total reduction method 15 16 to more general problems described in a somewhat different way methods of mgr type have in the meantime also been considered by braess 1 3 and meis 11 the convergence properties of the simplest mgr method mgr 0 are essentially improved if it is combined with one step of checkered gauss seidel iteration mgr ch for the model problem of poisson s equation in the unit square for example the spectral radius decreases from 12 to 227 by such a modification whereas the computational effort is not essentially enlarged for poisson like equations mgr ch yields the fastest iterative solver known so far other mgr variants are particularly suitable for anisotropic operators

algebraic multigrid for complex symmetric matrices and applications this paper is concerned with the numerical study of an algebraic multigrid preconditioner for complex symmetric system matrices we use several different krylov subspace methods as an outer iteration namely the qmr method proposed by freund and nachtigal the bicgcr method of clemens and the csym method of bunse gerstner and stoever in addition we compare the results with the standard jacobi preconditioner for complex symmetric problems we test our approach on the numerical simulation of high voltage insulators

mimetic finite difference methods for diffusion equations on non orthogonal non conformal meshes mimetic discretizations based on the support operators methodology are derived for non orthogonal locally refined quadrilateral meshes the second order convergence rate on non smooth meshes is verified with numerical examples

discontinuous galerkin and mimetic finite difference methods for coupled stokes darcy flows on polygonal and polyhedral grids we study locally mass conservative approximations of coupled darcy and stokes flows on polygonal and polyhedral meshes the discontinuous galerkin dg finite element method is used in the stokes region and the mimetic finite difference method is used in the darcy region dg finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces optimal convergence estimates for the numerical scheme are derived results from computational experiments supporting the theory are presented

sufficient condition for the asymptotic stability of interval matrices the stability of interval matrices has been studied by bialas 1983 but bialas s main theorem on the stability of interval matrices has been shown to be false by barmish 1984 a new sufficient condition for the asymptotic stability of interval matrices is given in the present paper

flux reconstruction and solution post processing in mimetic finite difference methods abstract we present a post processing technique for the mimetic finite difference solution of diffusion problems in mixed form our post processing method yields a piecewise linear approximation of the scalar variable that is second order accurate in the l2 norm on quite general polyhedral meshes including non convex and non matching elements the post processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables this technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied the post processing method is computationally inexpensive optimal performance is confirmed by numerical experiments

process versus unfolding semantics for place transition petri nets in the last few years the semantics of petri nets has been investigated in several different ways apart from the classical token game one can model the behaviour of petri nets via non sequential processes via unfolding constructions which provide formal relationships between nets and domains and via algebraic models which view petri nets as essentially algebraic theories whose models are monoidal categories in this paper we show that these three points of view can be reconciled in our formal development a relevant role is played by cdecocc a category of occurrence nets appropriately decorated to take into account the history of tokens the structure of decorated occurrence nets at the same time provides natural unfoldings for place transition pt nets and suggests a new notion of processes the decorated processes which induce on petri nets the same semantics as that of unfolding in addition we prove that the decorated processes of a net can be axiomatized as the arrows of a symmetric monoidal category which therefore provides the aforesaid unification

convergence of algebraic multigrid methods for symmetric positive definite matrices with weak diagonal dominance for symmetric positive definite matrices with weak diagonal dominance the algebraic sense of smooth errors is discussed and convergence of algebraic multigrid amg methods is proved it is found that the results of amg for symmetric positive definite m matrices with weak diagonal dominance also hold without the m matrix assumption the key result is the new formula x email protected ia i email protected j email protected email protected j a i j a j i x 2 email protected email protected j email protected email protected j a i j x email protected email protected ia i j a i j x j 2

cell centered finite volume methods with flexible stencils for diffusion equations on general nonconforming meshes a cell centered finite volume method is presented for discretizing diffusion operator on general nonconforming meshes the node values are accurately approximated using a new weighted interpolation formula in which the calculation of the weight is adaptive to both geometric parameters and diffusion coefficients it follows that an explicit expression composed of cell centered unknowns only is obtained for the discretization of normal flux numerical results demonstrate that linear solutions are reproduced exactly on the nonconforming random grids and that the convergence rate is close to second order for non linear or discontinuous problems

functorial models for petri nets we show that although the algebraic semantics of place transition petri nets under the collective token philosophy can be fully explained in terms of strictly symmetric monoidal categories the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality we introduce the notion of pre nets to overcome this obtaining a fully satisfactory categorical treatment where the operational semantics of nets yields an adjunction this allows us to present a uniform logical description of net behaviors under both the collective and the individual token philosophies in terms of theories and theory morphisms in partial membership equational logic moreover since the universal property of adjunctions guarantees that colimit constructions on nets are preserved in our algebraic models the resulting semantic framework has good compositional properties

iterative solution methods for modeling multiphase flow in porous media fully implicitly we discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell centered finite difference discretization of two and three phase porous media flow every outer nonlinear newton iteration requires solution of a nonsymmetric jacobian linear system two major types of preconditioners supercoarsening multigrid scmg and two stage are developed for the gmres iteration applied to the solution of the jacobian system the scmg reduces the three dimensional system to two dimensions using a vertical aggregation followed by a two dimensional multigrid the two stage preconditioners are based on decoupling the system into a pressure and concentration equations several pressure preconditioners of different types are described extensive numerical results are presented using the integrated parallel reservoir simulator ipars and indicate that these methods have low arithmetical complexity per iteration and good convergence rates

on the fly adaptive smoothed aggregation multigrid for markov chains a new adaptive algebraic multigrid scheme is developed for the solution of markov chains where the hierarchy of operators is adapted on the fly in a setup process that is interlaced with the solution process the setup process feeds the solution process with improved operators while the solution process provides the adaptive setup process with better approximations on which to base further improved operators the approach is demonstrated using petrov galerkin smoothed aggregation where only the prolongation operator is smoothed while the restriction remains of low order results show that the on the fly adaptive scheme can improve the performance of multigrid solvers that require extensive setup computations in both serial and parallel environments

multilevel solvers for unstructured surface meshes parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing such parameterization approaches give rise to large and exceedingly ill conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies since the underlying meshes are very fine to begin with such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy in this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature we introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner