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

asymptotic analysis of a semi linear elliptic system in perforated domains well posedness and correctors for the homogenization limit abstract in this study we prove results on the weak solvability and homogenization of a microscopic semi linear elliptic system posed in perforated media the model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media our interest lies in deriving homogenization limits upscaling for alike systems and particularly in justifying rigorously the obtained averaged descriptions essentially we prove the well posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations our techniques include moser like iteration techniques a variational formulation two scale asymptotic expansions as well as energy like estimates

a supra convergent finite difference scheme for the variable coefficient poisson equation on non graded grids we introduce a method for solving the variable coefficient poisson equation on non graded cartesian grids that yields second order accuracy for the solutions and their gradients we employ quadtree in 2d and octree in 3d data structures as an efficient means to represent the cartesian grid allowing for constraint free grid generation the schemes take advantage of sampling the solution at the nodes vertices of each cell in particular the discretization at one cell s node only uses nodes of two 2d or three 3d adjacent cells producing schemes that are straightforward to implement numerical results in two and three spatial dimensions demonstrate supra convergence in the l norm

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

a second order accurate projection method for the incompressible navier stokes equations on non graded adaptive grids we present an unconditionally stable second order accurate projection method for the incompressible navier stokes equations on non graded adaptive cartesian grids we employ quadtree and octree data structures as an efficient means to represent the grid we use the supra convergent poisson solver of c h min f gibou h ceniceros a supra convergent finite difference scheme for the variable coefficient poisson equation on fully adaptive grids cam report 05 29 j comput phys in press a second order accurate semi lagrangian method to update the momentum equation an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non graded grids we sample all the variables at the grid nodes producing a scheme that is straightforward to implement we propose two and three dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the l1 and l norms

a nonconforming virtual element method for the stokes problem on general meshes abstract in this paper a nonconforming virtual element method for the stokes problem is proposed and investigated this method gives a unified scheme for two and three dimensions simultaneously in addition it provides an attractive computational feature treatment of general elements including non convex and degenerate elements moreover by choosing a proper low order velocity and pressure pair we prove the discrete inf sup condition for this method to obtain its solvability and stability furthermore we show optimal energy norm error estimates for velocity and l 2 norm error estimates for both velocity and pressure finally a series of numerical experiments are performed to validate that this method has good stability and accuracy for the stokes problem

asymptotic analysis of perforated plates and membranes part 2 static and dynamic problems for large holes abstract static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi scale asymptotic homogenization method the problems of bending and vibration of perforated plates are considered using the asymptotic homogenization method the original boundary value problems are reduced to the combination of two types of problems first one is a recurrent system of unit cell problems with the conditions of periodic continuation and the second problem is a homogenized boundary value problem for the entire domain characterized by the constant effective coefficients obtained from the solution of the unit cell problems in the present paper the perforated plates with large holes are considered and the singular perturbation method is used to solve the pertinent unit cell problems matching of limiting solutions for small and large holes using the two point pade approximants is also accomplished and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained

random yield and random demand in a production system with downward substitution in this paper we present and solve a single period multiproduct downward substitution model our model has one raw material as the production input and produces n different products as outputs the demands and yields for the products are random we determine the optimal production input and allocation of the n products to satisfy demands the problem is modeled as a two stage stochastic program which we show can be decomposed into a parameterized network flow problem we present and compare three different solution methods a stochastic linear program a decomposition resulting in a series of network flow subproblems and a decomposition where the same network flow subproblems are solved by a new greedy algorithm

a stochastic dynamic programming approach based yield management with substitution and uncertainty in semiconductor manufacturing yield management is important and challengeable in semiconductor industry for the quality uncertainty of the final products the total yield rate of the semiconductor manufacturing process is uncertain each product is graded into one of several quality levels according to performance before being shipped a product originally targeted to satisfy the demand of one product may be used to satisfy the demand of other products when it conforms to their specifications at the same time the products depreciate in allocation periods which mainly results from technical progresses this paper studies the semiconductor yield management issue of a make to stock system with single input multi products multi demand periods upward substitution and periodic depreciation the whole time horizon of the system operation process can be divided into two stages the production stage and the allocation stage at the first stage the firm invests in raw materials before any actual demand is known and produces multiple types of products with random yield rates at the second stage products are classified into different classes by quality and allocated in numbers of periods the production and allocation problem are modeled as a stochastic dynamic program in which the objective is to maximize the profit of the firm we show that the pra parallel allocation first then upgrade allocation policy is the optimal allocation policy and the objective function is concave in production input an iterative algorithm is designed to find the optimal production input and numerical experiments are used to illustrate its effectiveness

a cell local finite difference discretization of the low order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes we present a quasidiffusion qd method for solving neutral particle transport problems in cartesian xy geometry on unstructured quadrilateral meshes including local refinement capability neutral particle transport problems are central to many applications including nuclear reactor design radiation safety astrophysics medical imaging radiotherapy nuclear fuel transport storage shielding design and oil well logging the primary development is a new discretization of the low order qd loqd equations based on cell local finite differences the accuracy of the loqd equations depends on proper calculation of special non linear qd eddington factors from a transport solution in order to completely define the new qd method a proper discretization of the transport problem is also presented the transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic parabolic representation of the angular flux on incoming faces analytic and numerical tests are used to test the accuracy and spatial convergence of the non linear method all tests exhibit o h 2 convergence of the scalar flux on orthogonal random and multi level meshes

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

elimination of adaptive grid interface errors in the discrete cell centered pressure equation classical cell centered discretization of the reservoir simulation pressure equation on anh adaptive grid results in ano 1 h leading truncation error at the grid interface a new flux continuous finite volume correction is presented together with an improved version of a previously proposed correction while both corrections eliminate the leading error the new correction exhibits the best convergence rates and has the following properties the resulting matrix is in general symmetric positive definite diagonally dominant for locally isotropic spatially varying coefficients convergence too h is demonstrated and support of the standard approximation is retained ensuring a fully implicit implementation results computed by the uncorrected classical scheme and both correction schemes are compared for two phase flow simulations with multilevel dynamic local grid refinement

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

vibrations of thin plates with small clamped patches eigenvalues of fourth order elliptic operators feature prominently in stability analysis of elastic structures this paper considers out of plane modes of vibration of a thin elastic plate perforat

asymptotic analysis of perforated plates and membranes part 1 static problems for small holes abstract static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi scale homogenization method using the asymptotic homogenization method the original boundary value problem is reduced to the combination of two types of problems first one is a recurrent system of unit cell problems with the conditions of periodic continuation and the second problem is a homogenized boundary value problem for the entire domain characterized by the constant effective coefficients obtained from the solution of the unit cell problems the combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems taking into the account small size of holes the method of perturbation of the shape of the boundary and the schwarz alternating method are used the problems of torsion of a rod with perforated cross section deflection of the perforated membrane loaded by a normal load and bending of perforated plates with circular and square holes are considered consecutively the error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated

multilevel quasidiffusion method with mixed order time discretization for multigroup thermal radiative transfer problems abstract in this paper we present a numerical method for solving multigroup thermal radiative transfer trt problems in 2d cartesian geometry it is based on the quasidiffusion aka variable eddington factor method and defined by the multilevel system of multigroup high order radiative transfer rt equations and multigroup and grey low order equations for moments of the intensity with the exact closures we apply time integration schemes of different orders of accuracy to approximate the high order and low order equations the first order scheme is used for the high order rt equations the second order scheme is applied to the low order equations this improves the accuracy of the trt solution while using robust and relatively inexpensive scheme for the high order rt equations the solution of the low order equations is non monotonic because the hyperbolic low order quasidiffusion qd equations are discretized by the second order scheme to reduce non monotonicity of the low order solution we apply a monotonization procedure to the discretized time dependent low order equations based on the limited trapezoidal method numerical results of the fleck cummings test are presented to demonstrate performance of the developed mixed order time integration scheme for the multilevel system of high order and low order qd equations we use this trt test problem to analyze the convergence in time of the mixed order scheme

an octree multigrid method for quasi static maxwell s equations with highly discontinuous coefficients in this paper we develop an octree discretization for maxwell s equations in the quasi static regime we then use this discretization in order to develop a multigrid method for maxwell s equations with highly discontinuous coefficients we test our algorithms and compare it to other multilevel algorithms

hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems we present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems under a saturation assumption the global reliability and efficiency of the proposed a posteriori estimator are proved several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies finally we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures

deterministic approximations to co production problems with service constraints and random yields we study production planning problems where multiple item categories are produced simultaneously the items have random yields and are used to satisfy the demands of many products these products have specification requirements that overlap an item originally targeted to satisfy the demand of one product may be used to satisfy the demand of other products when it conforms to their specifications customers demand must be satisfied from inventory 100 of the time we formulate the problem with service constraints and provide near optimal solution to the problem with fixed planning horizon we also propose simple heuristics for the problem solved with a rolling horizon some of the heuristics performed very well over a wide range of parameters

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

a high order mimetic method on unstructured polyhedral meshes for the diffusion equation abstract we present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes these mimetic discretizations are built to satisfy local consistency and stability conditions the consistency condition is an exactness property i e the mimetic schemes are exact when the solution is a polynomial of an assigned degree the stability condition ensures the well posedness of the method the degrees of freedom are the solution moments on mesh faces and inside mesh cells higher order schemes are built using higher order moments the developed schemes are verified numerically on diffusion problems with constant and spatially variable possibly discontinuous tensorial coefficients

a framework for the upscaling of the electrical conductivity in the quasi static maxwells equations electromagnetic simulations of complex geologic settings are computationally expensive one reason for this is the fact that a fine mesh is required to accurately discretize the electrical conductivity model of a given setting this conductivity model may vary over several orders of magnitude and these variations can occur over a large range of length scales using a very fine mesh for the discretization of this setting leads to the necessity to solve a large system of equations that is often difficult to deal with to keep the simulations computationally tractable coarse meshes are often employed for the discretization of the model such coarse meshes typically fail to capture the fine scale variations in the conductivity model resulting in inaccuracies in the predicted data in this work we introduce a framework for constructing a coarse mesh or upscaled conductivity model based on a prescribed fine mesh model rather than using analytical expressions we opt to pose upscaling as a parameter estimation problem by solving an optimization problem we obtain a coarse mesh conductivity model the optimization criterion can be tailored to the survey setting in order to produce coarse models that accurately reproduce the predicted data generated on the fine mesh this allows us to upscale arbitrary conductivity structures as well as to better understand the meaning of the upscaled quantity we use 1d and 3d examples to demonstrate that the proposed framework is able to emulate the behavior of the heterogeneity in the fine mesh conductivity model and to produce an accurate description of the desired predicted data obtained by using a coarse mesh in the simulation process

analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems the maximum principle is one of the most important properties of solutions of partial differential equations its numerical analog the discrete maximum principle dmp is one of the most difficult properties to achieve in numerical methods especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic violation of the dmp may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one in this work we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference mfd method on two and three dimensional meshes these conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element efficient solutions are devised for meshes consisting of simplexes parallelograms and parallelepipeds and orthogonal locally refined elements as those used in the amr methodology on simplicial meshes it turns out that the mfd method coincides with the mixed hybrid finite element methods based on the low order raviart thomas vector space thus in this case we recover the well established conventional angle conditions of such approximations instead in the other cases a suitable design of the mfd method allows us to formulate a monotone discretization for which the existence of a dmp can be theoretically proved moreover on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the multi point flux approximation mpfa methods numerical experiments confirm the effectiveness of the considered monotonicity conditions

a cell functional minimization scheme for parabolic problem we are interested in a robust and accurate finite volume scheme for 2 d parabolic problems derived from the cell functional minimization approach the scheme has a local stencil is locally conservative treats discontinuity rigorously and leads to a symmetric positive definite linear system since the scheme has both cell centered unknowns and cell edge unknowns the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach the interface condition is of the dirichlet robin type and has a parameter l by choosing this parameter properly the convergence of the iteration process could be sped up numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes

a second order discretization of maxwell s equations in the quasi static regime on octree grids in this study we consider adaptive mesh refinement for the solution of maxwell s equations in the quasi static or diffusion regime we propose a new finite volume octree discretization for the problem and show how to construct second order stencils on yee grids extending the known first order discretization stencils we then develop an effective preconditioner to the problem we show that our preconditioner performs well for discontinuous conductivities as well as for a wide range of frequencies

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

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

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

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques