symmetry preserving discretization of navier stokes equations on collocated unstructured grids a fully conservative discretization is presented in this paper the same principles followed by verstappen and veldman 2003 3 are generalized for unstructured meshes here a collocated mesh scheme is preferred over a staggered one due to its simpler form for such meshes the basic idea behind this approach remains the same mimicking the crucial symmetry properties of the underlying differential operators i e the convective operator is approximated by a skew symmetric matrix and the diffusive operator by a symmetric positive definite matrix a novel approach to eliminate the checkerboard spurious modes without introducing any non physical dissipation is proposed to do so a fully conservative regularization of the convective term is used the supraconvergence of the method is numerically showed and the treatment of boundary conditions is discussed finally the new discretization method is successfully tested for a buoyancy driven turbulent flow in a differentially heated cavity

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

the orthogonal decomposition theorems for mimetic finite difference methods accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations these methods mimic many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields i e they are divergence free and should satisfy a discrete version of the orthogonal decomposition theorem this theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models including the navier stokes equations and in electrodynamics we are deriving mimetic finite difference approximations of the divergence gradient and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators we first define the natural discrete divergence gradient and curl operators based on coordinate invariant definitions such as gauss s theorem for the divergence next we use the formal adjoints of these natural operators to derive compatible divergence gradient and curl operators with complementary domains and ranges of values in this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way satisfying a discrete analog of the formula vec a ggrad varphi curl vec b we also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field

optimization based remap and transport a divide and conquer strategy for feature preserving discretizations this paper examines the application of optimization and control ideas to the formulation of feature preserving numerical methods with particular emphasis on the conservative and bound preserving remap constrained interpolation and transport advection of a single scalar quantity we present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization based remap obr of mass density the latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity subject to a system of linear inequality constraints the approximation of an exact mass update operator defines the target quantity which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds the latter are enforced by the system of linear inequalities in so doing the generic obr formulation separates accuracy considerations from the enforcement of physical properties we proceed to show how the generic obr formulation yields the recently introduced flux variable flux target fvft 1 and mass variable mass target mvmt 2 formulations of remap and then follow with a formal examination of their relationship using an intermediate flux variable mass target fvmt formulation we show the equivalence of fvft and mvmt optimal solutions to underscore the scope and the versatility of the generic obr formulation we introduce the notion of adaptable targets i e target quantities that reflect local solution properties extend fvft and mvmt to remap on the sphere and use obr to formulate adaptable conservative and bound preserving optimization based transport algorithms for cartesian and latitude longitude coordinate systems a selection of representative numerical examples on two dimensional grids demonstrates the computational properties of our approach

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

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

analysis and computation of compatible least squares methods for div curl equations we develop and analyze least squares finite element methods for two complementary div curl elliptic boundary value problems the first one prescribes the tangential component of the vector field on the boundary and is solved using curl conforming elements the second problem specifies the normal component of the vector field and is handled by div conforming elements we prove that both least squares formulations are norm equivalent with respect to suitable discrete norms yield optimal asymptotic error estimates and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included

the rotation number of the linear schrodinger equation with discontinuous almost periodic potentials abstract in this paper we study a linear schrodinger equation with a stepanov almost periodic potential where the potential is discontinuous from r to r we first show the existence of rotation numbers for the equation we then give the optimal estimate of rotation numbers

approximation of boundary conditions for mimetic finite difference methods abstract the numerical solution of partial differential equations solved with finite difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results furthermore those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation unless special care is taken mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient curl and divergence operators at the computational boundary in this paper we describe how to incorporate boundary conditions into finite difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus the approach is valid for a wide class of partial differential equations of mathematical physics and will be described for poisson s equation with dirichlet neumann and robin boundary conditions we prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions

global minimization for continuous multiphase partitioning problems using a dual approach this paper is devoted to the optimization problem of continuous multi partitioning or multi labeling which is based on a convex relaxation of the continuous potts model in contrast to previous efforts which are tackling the optimal labeling problem in a direct manner we first propose a novel dual model and then build up a corresponding duality based approach by analyzing the dual formulation sufficient conditions are derived which show that the relaxation is often exact i e there exists optimal solutions that are also globally optimal to the original nonconvex potts model in order to deal with the nonsmooth dual problem we develop a smoothing method based on the log sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal dual model and suggests labelings with maximum entropy such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions an expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization numerical experiments also show that our method outperforms several competitive approaches in various aspects such as lower energies and better visual quality

isotropic finite volume discretization finite volume methods traditionally employ dimension by dimension extension of the one dimensional reconstruction and averaging procedures to achieve spatial discretization of the governing partial differential equations on a structured cartesian mesh in multiple dimensions this simple approach based on tensor product stencils introduces an undesirable grid orientation dependence in the computed solution the resulting anisotropic errors lead to a disparity in the calculations that is most prominent between directions parallel and diagonal to the grid lines in this work we develop isotropic finite volume discretization schemes which minimize such grid orientation effects in multidimensional calculations by eliminating the directional bias in the lowest order term in the truncation error explicit isotropic expressions that relate the cell face averaged line and surface integrals of a function and its derivatives to the given cell area and volume averages are derived in two and three dimensions respectively it is found that a family of isotropic approximations with a free parameter can be derived by combining isotropic schemes based on next nearest and next next nearest neighbors in three dimensions use of these isotropic expressions alone in a standard finite volume framework however is found to be insufficient in enforcing rotational invariance when the flux vector is nonlinear and or spatially non uniform the rotationally invariant terms which lead to a loss of isotropy in such cases are explicitly identified and recast in a differential form various forms of flux correction terms which allow for a full recovery of rotational invariance in the lowest order truncation error terms while preserving the formal order of accuracy and discrete conservation of the original finite volume method are developed numerical tests in two and three dimensions attest the superior directional attributes of the proposed isotropic finite volume method prominent anisotropic errors such as spurious asymmetric distortions on a circular reaction diffusion wave that feature in the conventional finite volume implementation are effectively suppressed through isotropic finite volume discretization furthermore for a given spatial resolution a striking improvement in the prediction of kinetic energy decay rate corresponding to a general two dimensional incompressible flow field is observed with the use of an isotropic finite volume method instead of the conventional discretization c 2014 elsevier inc all rights reserved

using a curvilinear grid to construct symmetry preserving discretizations for lagrangian gas dynamics the goal of this paper is to construct discretizations for the equations of lagrangian gas dynamics that preserve plane cylindrical and spherical symmetry in the solution of the original differential equations the new method uses a curvilinear grid that is reconstructed from a given logically rectangular distribution of nodes the sides of the cells of the reconstructed grid can be segments of straight lines or arcs of local circles our procedure is exact for straight lines and circles that is it reproduces rectangular and polar grids exactly we use the method of support operators to construct a conservative finite difference method that we demonstrate will preserve spatial symmetries for certain choices of the initial grid we also introduce a curvilinear version of artificial edge viscosity that also preserves symmetry we present numerical examples to demonstrate our theoretical considerations and the robustness of the new method

a shift transformation for fully conservative methods turbulence simulation on complex unstructured grids abstract operator transformations are presented that allow matrix operators for collocated variables to be transformed into matrix operators for staggered variables while preserving symmetries these shift transformations permit conservative skew symmetric convective operators and symmetric positive definite diffusive operators to be obtained for staggered variables using collocated operators shift transformations are not limited to uniform or structured meshes and this formulation leads to a generalization of the works of perot j comput phys 159 2000 58 and verstappen and veldman j comput phys 187 2003 343 a set of shift operators have been developed for and applied to a time adaptive cartesian mesh method with a fractional step algorithm the resulting numerical scheme conserves mass to machine error and conserves momentum and energy to second order in time a mass conserving interpolation is used for the variables during mesh adaptation the interpolation conserves momentum and energy to second order in space turbulent channel flow simulations were conducted at re 180 using direct numerical simulation dns the dns results from the adaptive method compare favourably with spectral dns results despite the use of a formally second order accurate scheme

a framework for developing a mimetic tensor artificial viscosity for lagrangian hydrocodes on arbitrary polygonal meshes we construct a new mimetic tensor artificial viscosity on general polygonal meshes the tensor artificial viscosity is based on discretization of coordinate invariant operators divergence of a tensor and gradient of a vector the focus of this paper is on the non symmetric form div email protected u of the tensor artificial viscosity the discretizations of this operator is derived for the case of a full tensor coefficient m however in the numerical experiments we only use scalar m we prove that the new tensor viscosity preserves spatial symmetry on special meshes we demonstrate performance of the new viscosity for the noh implosion sedov explosion and saltzman piston problems on a set of various polygonal meshes in both cartesian and axisymmetric coordinate systems

a fully conservative mimetic discretization of the navier stokes equations in cylindrical coordinates with associated singularity treatment we present a finite difference discretization of the incompressible navier stokes equations in cylindrical coordinates this currently is to the authors knowledge the only scheme available that is demonstrably capable of conserving mass momentum and kinetic energy in the absence of viscosity on both uniform and non uniform grids simultaneously we treat the inherent discretization issues that arise due to the presence of the coordinate singularity at the polar axis we demonstrate the validity of the conservation claims by performing a number of numerical experiments with the proposed scheme and we show that it is second order accurate in space using the method of manufactured solutions

the convergence of mimetic discretization for rough grids we prove that the mimetic finite difference discretizations of laplace s equation converges on rough logically rectangular grids with convex cells mimetic discretizations for the invariant operators divergence gradient and curl satisfy exact discrete analogs of many of the important theorems of vector calculus the mimetic discretization of the laplacian is given by the composition of the discrete divergence and gradient we first construct a mimetic discretization on a single cell by geometrically constructing inner products for discrete scalar and vector fields then constructing a finite volume discrete divergence and then constructing a discrete gradient that is consistent with the discrete divergence theorem this construction is then extended to the global grid we demonstrate the convergence for the two dimensional laplace equation with dirichlet boundary conditions on grids with a lower bound on the angles in the cell corners and an upper bound on the cell aspect ratios the best convergence rate to be expected is first order which is what we prove the techniques developed apply to far more general initial boundary value problems

discrete hodge operators many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms discrete differential forms provide a natural and canonical approach to their discretization however much freedom remains concerning the choice of discrete hodge operators that is discrete analogues of constitutive laws a generic discrete hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations we reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus

support operator finite difference algorithms for general elliptic problems abstract an algorithm is developed for discretizing boundary value problems given by a general linear elliptic second order partial differential equation with general mixed or robin boundary conditions in general logically rectangular grids the continuum problem can be written as an operator equation where the operator is self adjoint and positive definite the discrete approximations have the same property consequently the matrices for the discrete problem are symmetric and positive definite also the scheme has a nearest neighbor stencil consequently the most powerful linear solvers can be applied in smooth grids the algorithm produces second order accurate solutions it is the generality of the problem general matrix coefficients general boundary conditions general logically rectangular grids that makes finding such an algorithm difficult the algorithm which is a combination of the method of support operators and the mapping method overcomes certain difficulties of the individual methods producing a high quality algorithm for solving general elliptic problems

a spatially continuous max flow and min cut framework for binary labeling problems we propose and investigate novel max flow models in the spatially continuous setting with or without i priori defined supervised constraints under a comparative study of graph based max flow min cut we show that the continuous max flow models correspond to their respective continuous min cut models as primal and dual problems in this respect basic conceptions and terminologies from discrete max flow min cut are revisited under a new variational perspective we prove that the associated nonconvex partitioning problems unsupervised or supervised can be solved globally and exactly via the proposed convex continuous max flow and min cut models moreover we derive novel fast max flow based algorithms whose convergence can be guaranteed by standard optimization theories experiments on image segmentation both unsupervised and supervised show that our continuous max flow based algorithms outperform previous approaches in terms of efficiency and accuracy

direct discretization of planar div curl problems a control volume method is proposed for planar div curl systems the method is independent of potential and least squares formulations and works directly with the div curl system the novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way round a discrete vector field theory comes quite naturally from this idea and is developed in the paper error estimates are proved for the method and other ramifications investigated

constrained interpolation remap of divergence free fields abstract a novel constrained interpolation algorithm for remapping of solenoidal face finite element vector fields is presented the algorithm is based on explicit recovery postprocessing and interpolation of a potential for the original vector field and a subsequent application of a curl operator to obtain the desired divergence free finite element field on the new mesh the use of interpolation instead of advection in the remap process offers valuable computational advantages old and new meshes are neither required to have the same connectivity nor to be close to each other slope limiting and upwinding which can be sensitive to grid structure are avoided and replaced by local optimization to control energy of the remapped field the new method is validated using a suite of cyclic remap problems on random and tensor product mesh sequences a comparison with a local remapper based on a constrained transport advection algorithm is also included

a discrete duality finite volume approach to hodge decomposition and div curl problems on almost arbitrary two dimensional meshes we define discrete differential operators such as gradient divergence and curl on general two dimensional nonorthogonal meshes these discrete operators verify discrete analogues of usual continuous theorems discrete green formulae discrete hodge decomposition of vector fields and vector curls have a vanishing divergence and gradients have a vanishing curl we apply these ideas to discretize div curl systems we give error estimates based on the reformulation of these systems into equivalent equations for the potentials numerical results illustrate the use of the method on several types of meshes some of which are degenerating triangular meshes and nonconforming locally refined meshes

the sensitivity and accuracy of fourth order finite difference schemes on nonuniform grids in one dimension abstract we construct local fourth order finite difference approximations of first and second derivatives on nonuniform grids in one dimension the approximations are required to satisfy symmetry relationships that come from the analogous higher dimensional fundamental operators the divergence the gradient and the laplacian for example we require that the discrete divergence and gradient be negative adjoint of each other div grad and the discrete laplacian is defined as lap divgrad the adjointness requirement on the divergence and gradient guarantees that the laplacian is a symmetric negative operator the discrete approximations we derive are fourth order on smooth grids but the approach can be extended to create approximations of arbitrarily high order we analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids

fast finite volume simulation of 3d electromagnetic problems with highly discontinuous coefficients we consider solving three dimensional electromagnetic problems in parameter regimes where the quasi static approximation applies and the permeability permittivity and conductivity may vary significantly the difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing maxwell s equations in the frequency domain the present article extends methods we proposed earlier for constant permeability e haber u ascher d aruliah and d oldenburg j comput phys 163 2000 pp 150 171 d aruliah u ascher e haber and d oldenburg math models methods appl sci to appear to handle also problems in which the permeability is variable and may contain significant jump discontinuities in order to address the problem of slow convergence we reformulate maxwell s equations in terms of potentials applying a helmholtz decomposition to either the electric field or the magnetic field the null space of the curl operators can then be annihilated by adding a stabilizing term using a gauge condition and thus obtaining a strongly elliptic differential operator a staggered grid finite volume discretization is subsequently applied to the reformulated pde system this scheme works well for sources of various types even in the presence of strong material discontinuities in both conductivity and permeability the resulting discrete system is amenable to fast convergence of ilu preconditioned krylov methods we test our method using several numerical examples and demonstrate its robust efficiency we also compare it to the classical yee method using similar iterative techniques for the resulting algebraic system and we show that our method is significantly faster especially for electric sources

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

on glm curl cleaning for a first order reduction of the ccz4 formulation of the einstein field equations abstract in this paper we propose an extension of the generalized lagrangian multiplier method glm of munz et al 52 30 which was originally conceived for the numerical solution of the maxwell and mhd equations with divergence type involutions to the case of hyperbolic pde systems with curl type involutions the key idea here is to solve an augmented pde system in which curl errors propagate away via a maxwell type evolution system the new approach is first presented on a simple model problem in order to explain the basic ideas subsequently we apply it to a strongly hyperbolic first order reduction of the ccz4 formulation fo ccz4 of the einstein field equations of general relativity which is endowed with 11 curl constraints several numerical examples including the long time evolution of a stable neutron star in anti cowling approximation are presented in order to show the obtained improvements with respect to the standard formulation without special treatment of the curl involution constraints the main advantages of the proposed glm approach are its complete independence of the underlying numerical scheme and grid topology and its easy implementation into existing computer codes however this flexibility comes at the price of needing to add for each curl involution one additional 3 vector plus another scalar in the augmented system for homogeneous curl constraints and even two additional scalars for non homogeneous curl involutions for the fo ccz4 system with 11 homogeneous curl involutions this means that additional 44 evolution quantities need to be added

lattice electromagnetic theory from a topological viewpoint the language of differential forms and topological concepts are applied to study classical electromagnetic theory on a lattice it is shown that differential forms and their discrete counterparts cochains provide a natural bridge between the continuum and the lattice versions of the theory allowing for a natural factorization of the field equations into topological field equations i e invariant under homeomorphisms and metric field equations the various potential sources of inconsistency in the discretization process are identified distinguished and discussed a rationale for a consistent extension of the lattice theory to more general situations such as to irregular lattices is considered

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

mimetic finite difference operators for second order tensors on unstructured grids abstract we use the support operators method to derive discrete approximations for the gradient of a vector and divergence of a tensor on unstructured grids in two dimensions these discrete operators satisfy discrete analogs of the integral identities of the differential operators on unstructured grids where vector functions are defined at the grid points and tensor functions are defined as tangential projections to the zone edges or as normal projections to the median mesh we evaluate the accuracy of the discrete operators by determining the order of convergence of the truncation error on structured and unstructured grids and show that the truncation error of the method is between first and second order depending on the smoothness of the grid in a test problem on a highly nonuniform grid we confirm that the convergence rate is between first and second order

splines in higher order tv regularization splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces in this paper we show in a strictly discrete setting that splines of degree m 1 solve also a minimization problem with quadratic data term and m th order total variation tv regularization term in contrast to problems with quadratic regularization terms involving m th order derivatives the spline knots are not known in advance but depend on the input data and the regularization parameter more precisely the spline knots are determined by the contact points of the m th discrete antiderivative of the solution with the tube of width 2 around the m th discrete antiderivative of the input data we point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the sobolev space w 2 0 m from this point of view the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines

global well posedness for compressible viscoelastic fluids near equilibrium in this paper we prove local well posedness for compressible viscoelastic fluids of the oldroyd model under the assumption that the initial density is bounded away from zero and global well posedness near equilibrium the proof of global well posedness relies on some intrinsic properties of viscoelastic fluids and on a uniform estimate for a linearized hyperbolic parabolic system with convection terms

the effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods abstract the support operators method of discretizing partial differential equations produces discrete analogs of continuum initial boundary value problems that exactly satisfy discrete conservation laws analogous to those satisfied by the continuum system thus the stability of the method is assured but currently there is no theory that predicts the accuracy of the method on nonuniform grids in this paper we numerically investigate how the accuracy particularly the accuracy of the fluxes depends on the definition of the inner product for discrete vector fields we introduce two different discrete inner products the standard inner product that we have used previously and a new more accurate inner product the definitions of these inner products are based on interpolation of the fluxes of vector fields the derivation of the new inner product is closely related to the use of the piola transform in mixed finite elements computing the formulas for the new accurate inner product requires a nontrivial use of computer algebra from the results of our numerical experiments we can conclude that using more accurate inner product produces a method with the same order of convergence as the standard inner product but the constant in error estimate is about three times less however the method based on the standard inner product is easier to compute with and less sensitive to grid irregularities so we recommend its use for rough grids

simultaneous higher order optical flow estimation and decomposition we study the estimation and decomposition of optical flows from highly nonrigid motions to this end recent methods from image decomposition into structural and textural parts are combined with variational optical flow estimation the approaches we suggest amount to minimizing discrete convex functionals using second order cone programming higher order regularization is necessary in order to accurately recover important flow structure like vortices and to incorporate key physical properties such as vanishing divergence for proper discretization we apply the finite mimetic difference method which preserves the identities fulfilled by the continuous differential operators numerical examples demonstrate the feasibility of the complex approaches

wave equation simulation on manifold by unconditional stable schemes abstract to predict the wave propagation in a given region over time it is often necessary to find the numerical solution for wave equation with the techniques of discrete differential calculus we propose two unconditional stable numerical schemes for simulation wave equation on the space manifold and the time the analysis of their stability and error is also accomplished

on least squares finite element methods for the poisson equation and their connection to the dirichlet and kelvin principles least squares finite element methods for first order formulations of the poisson equation are not subject to the inf sup condition and lead to stable solutions even when all variables are approximated by equal order continuous finite element spaces for such elements one can also prove optimal convergence in the energy norm equivalent to a norm on h 1 omega times h omega mbox rm div for all variables and optimal l2 convergence for the scalar variable however showing optimal l2 convergence for the flux has proven to be impossible without adding the redundant curl equation to the first order system of partial differential equations in fact numerical evidence strongly suggests that nodal continuous flux approximations do not posses optimal l2 accuracy in this paper we show that optimal l2 error rates for the flux can be achieved without the curl constraint provided that one uses the div conforming family of brezzi douglas marini or brezzi douglas duran fortin elements then we proceed to reveal an interesting connection between a least squares finite element method involving h omega mbox rm div conforming flux approximations and mixed finite element methods based on the classical dirichlet and kelvin principles we show that such least squares finite element methods can be obtained by approximating through an l2 projection the hodge operator that connects the kelvin and dirichlet principles our principal conclusion is that when implemented in this way a least squares finite element method combines the best computational properties of finite element methods based on each of the classical principles

a generalized mac scheme on curvilinear domains we propose a simple finite difference scheme for navier stokes equations in primitive formulation on curvilinear domains with proper boundary treatment and interplay between covariant and contravariant components the spatial discretization admits exact hodge decomposition and energy identity as a result the pressure can be decoupled from the momentum equation with explicit time stepping no artificial pressure boundary condition is needed in addition it can be shown that this spatially compatible discretization leads to uniform inf sup condition which plays a crucial role in the pressure approximation of both dynamic and steady state calculations numerical experiments demonstrate the robustness and efficiency of our scheme

a compatible and conservative spectral element method on unstructured grids we derive a formulation of the spectral element method which is compatible on very general unstructured three dimensional grids here compatible means that the method retains discrete analogs of several key properties of the divergence gradient and curl operators the divergence and gradient are anti adjoints the negative transpose of each other the curl is self adjoint and annihilates the gradient operator and the divergence annihilates the curl the adjoint relations hold globally and at the element level with the inclusion of a natural discrete element boundary flux term we then discretize the shallow water equations on the sphere using the cubed sphere grid and show that compatibility allows us to locally conserve mass energy and potential vorticity conservation is obtained without requiring the equations to be in conservation form the conservation is exact assuming exact time integration

rehabilitation of the lowest order raviart thomas element on quadrilateral grids a recent study d n arnold d boffi and r s falk siam j numer anal 42 2005 pp 2429 2451 reveals that convergence of finite element methods using h mathrm div omega compatible finite element spaces deteriorates on nonaffine quadrilateral grids this phenomena is particularly troublesome for the lowest order raviart thomas elements because it implies loss of convergence in some norms for finite element solutions of mixed and least squares methods in this paper we propose reformulation of finite element methods based on the natural mimetic divergence operator m shashkov conservative finite difference methods on general grids crc press boca raton fl 1996 which restores the order of convergence reformulations of mixed galerkin and least squares methods for the darcy equation illustrate our approach we prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator furthermore we prove that standard and reformulated versions of the mixed galerkin method lead to identical linear systems but the two versions of the least squares method are veritably different the surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial and that the lowest order raviart thomas elements are safe to use in this method however the breakdown in the least squares method is real and there one should use our proposed reformulation

formulation of an unstructured grid model for global ocean dynamics a conservative discretization of the ocean primitive equations for global ocean dynamics is formulated on an unstructured grid the grid consists of triangular cells with a c type staggering of variables where scalar variables are located at centers of grid cells and normal components of velocity are placed at cell boundaries reconstructions necessitated by the staggering of variables are chosen from a set of admissible reconstructions such that the discrete equations obey a weighted weak form which guarantees discrete conservation properties at the same time a spurious mode specific to the triangular c grid is controlled in a way that is compatible with conservation properties the conservation properties of the discrete model are verified numerical simulations are presented ranging from idealized geophysical flows to an ocean simulation at eddy permitting resolution in order to substantiate the statement that global ocean dynamics can be formulated on triangular c grids

local reconstruction of a vector field from its normal components on the faces of grid cells we compare two local least squares approximations for the reconstruction of the cartesian components of a vector field at the nodes of a logically rectangular grid when this vector field is given by its components normal to the faces edges in 2d of the cells of the grid such a problem appears naturally for example for lagrangian gas dynamics codes based on godunov s method where the normal component of a vector on an edge between two cells is computed from the solution of a 1d riemann problem 1 but the cartesian components are needed at nodes in order to compute the nodal motion this kind of reconstruction is also interesting in the framework of discrete vector analysis 2 as a map between different spaces of vector functions reconstructed vector fields are often used to compute discrete analogs of differential operators such as divergence and curl therefore to estimate the quality of the reconstruction we need to evaluate not only the accuracy of the vector itself but also the accuracy of the approximate discrete div and curl more specifically we are given the exact values at the edge centroids of the projection of a vector function on the directions normal to the sides of the cells see fig 1a for notation our goal is to reconstruct the cartesian components ax i j a y i j of the vector a at the node i j the zero order reconstruction algorithm described in 1 is based on choosing ai j to minimize the local quadratic functional

application of vector calculus to numerical simulation of continuum mechanics problems abstract it is well known that vector tensor notation is a compact and natural language for the mathematical formulation of continuum mechanics problems here we describe the application of vector technique to numerical simulation starting with a mathematical formulation we provide an efficient numerical scheme and furnish an implementation as a computer program as a result in comparison with traditional component form the two last steps are significantly simplified especially for multidimensional problems with various boundary conditions in irregular geo metries where nonorthogonal meshes are applied therefore more attention can be focused on the physical nature of problems apart from the cleaner syntax vector notation conserves the structure of traditional numerical algorithms and solves multidimensional problems with minimum additional programming effort complex practical applications of this technique are described as well

an algebraic multigrid approach based on a compatible gauge reformulation of maxwell s equations with the rise in popularity of compatible finite element finite difference and finite volume discretizations for the time domain eddy current equations there has been a corresponding need for fast solvers of the resulting linear algebraic systems however the traits that make compatible discretizations a preferred choice for the maxwell s equations also render these linear systems essentially intractable by truly black box techniques we propose an algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid amg technique for this reformulated problem the reformulation process takes advantage of a discrete hodge decomposition on cochains to replace the discrete eddy current equations by an equivalent 2 times2 block linear system whose diagonal blocks are discrete hodge laplace operators acting on 1 cochains and 0 cochains respectively while this new amg technique requires somewhat specialized treatment on the finest mesh the coarser meshes can be handled using standard methods for laplace type problems our new amg method is applicable to a wide range of compatible methods on structured and unstructured grids including edge finite elements mimetic finite differences covolume methods and yee like schemes we illustrate the new technique using edge elements in the context of smoothed aggregation amg and present computational results for problems in both two and three dimensions