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

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

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

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

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

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

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

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

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

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

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

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

a family of linearity preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids a family of cell centered finite volume schemes are proposed for anisotropic diffusion problems on arbitrary polyhedral grids with planar facets the derivation of the schemes is done under a general framework through a certain linearity preserving approach the key ingredient of our algorithm is to employ solely the so called harmonic averaging points located at the cell interfaces to define the auxiliary unknowns which not only makes the interpolation procedure for auxiliary unknowns simple and positivity preserving but also reduces the stencil of the schemes the final schemes are cell centered with a small stencil of 25 point on the structured hexahedral grids moreover the schemes satisfy the local conservation condition treat discontinuity exactly and allow for a simple stability analysis a second order accuracy in the l2 norm and a first order accuracy in the h1 norm are observed numerically on general distorted meshes in case that the diffusion tensor is anisotropic and discontinuous

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

an improved monotone finite volume scheme for diffusion equation on polygonal meshes we construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes the new scheme uses the cell edge unknowns instead of cell vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes specially numerical results show that the new scheme is robust and more accurate than the existing monotone scheme on some kinds of meshes

high order mimetic finite difference method for diffusion problems on polygonal meshes the mimetic finite difference mfd methods mimic important properties of physical and mathematical models as a result conservation laws solution symmetries and the fundamental identities of the vector and tensor calculus are held for discrete models the mfd methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non convex and degenerate elements the existing mfd methods for solving diffusion type problems are second order accurate for the conservative variable temperature pressure energy etc and only first order accurate for its flux we developed new high order mfd methods which are second order accurate for both scalar and vector variables the second order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes

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 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 displacement based finite element formulation for general polyhedra using harmonic shape functions summary a displacement based continuous galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics the polyhedra can have an arbitrary number of vertices or faces the faces of the polyhedra can have an arbitrary number of edges and can be nonplanar the polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex averaged centroid conforming shape functions are constructed using harmonic functions defined on the undeformed configuration thus requiring the use of a total lagrangian finite element formulation in large deformation applications for nonlinear applications with computationally intensive constitutive models it is important to minimize the number of element quadrature points for this reason an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices as a first step toward verifying the element behavior in the general nonlinear setting several linear verification examples are presented using both random voronoi meshes and distorted hexahedral meshes for the hexahedral meshes results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation the element behavior in the nearly incompressible regime is also examined copyright 2013 john wiley sons ltd

an extension of the crouzeix raviart space to general meshes with application to quasi incompressible linear elasticity and stokes flow in this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of the standard crouzeix raviart space namely the continuity of mean values at interfaces and the existence of an interpolator which preserves the mean value of the gradient inside each element the construction borrows ideas from both cell centered galerkin and hybrid finite volume methods the discrete function space is defined from cell and face unknowns by introducing a suitable piecewise affine reconstruction on a fictitious pyramidal subdivision of the original mesh two applications are considered in which the discrete space plays an important role namely i the design of a locking free primal as opposed to mixed method for quasi incompressible planar elasticity on general polygonal meshes ii the design of an inf sup stable method for the stokes equations on general polygonal or polyhedral meshes in this context we also propose a general modification applicable to any suitable discretization which guarantees that the velocity approximation is unaffected by the presence of large irrotational body forces provided a helmholtz decomposition of the right hand side is available the relation between the proposed methods and classical finite volume and finite element schemes on standard meshes is investigated finally similar ideas are exploited to mimic key properties of the lowest order raviart thomas space on general polygonal or polyhedral meshes

mimetic finite difference methods for hamiltonian wave equations in 2d abstract in this paper we consider the numerical solution of the hamiltonian wave equation in two spatial dimensions we construct a two step procedure in which we first discretize the space by the mimetic finite difference mfd method and then we employ a standard symplectic scheme to integrate the semi discrete hamiltonian system derived the main characteristic of the mfd methods when applied to stationary problems is to mimic important properties of the continuous system this approach yields a full numerical procedure suitable to integrate hamiltonian problems a complete theoretical analysis of the method and some numerical simulations are developed in the paper

relationship between the vertex centered linearity preserving scheme and the lowest order virtual element method for diffusion problems on star shaped polygons abstract as a generalization of the finite element method the virtual element method vem has made remarkable achievements in the algorithm and analysis of simulating various problems recently vertex centered linearity preserving scheme vlps wu et al 2016 is a linear finite volume scheme which is currently only applied to the numerical simulation of diffusion problems in this paper we study the relationship between vlps and the lowest order vem for diffusion problems on star shaped polygonal meshes from an algebraic point of view the global stiffness matrix of vlps with a special stabilization term coincides with that of the lowest order vem while the load terms are generally different specifically we find that the global stiffness matrices of the two methods can be split as the consistency parts and the stability parts and the consistency parts are the same while the stability parts coincide under some assumptions as a by product a new stability term is obtained for vlps besides a post processing procedure is suggested for the lowest order vem to preserve the positivity of the cell centered unknowns and maintain at the same time the local conservation on the primary meshes the positivity existence and uniqueness of the cell centered unknowns are studied numerical experiments confirm the theoretical findings and demonstrate the efficiency of the post processing procedure on various polygonal meshes

algebraic multigrid for linear systems obtained by explicit element reduction we consider sparse linear systems where a set of interior degrees of freedom have been eliminated in order to reduce the problem size this elimination is assumed to be local so the interior principal submatrix is block diagonal and the resulting schur complement is still sparse for it to be beneficial the elimination process should lead to reduced memory requirements but equally importantly it should also result in an algebraic problem that can be solved efficiently in this paper we propose a general element reduction approach and show how the elimination process can exploit a particular subzonal discretization to maintain the sparsity of the schur complement we also investigate algebraic multigrid amg solution algorithms applied to the reduced problem and we discuss the influence of the local elimination on solver related properties of the matrix such as near nullspace preservation and the availability of stable subspace decompositions we focus on boomeramg a parallel variant of classical ruge stuben amg applied to scalar diffusion problems v henson and u yang appl numer math 41 2002 pp 155 177 and the auxiliary space maxwell solver ams for electromagnetic diffusion applications t kolev and p vassilevski j comput math 27 2009 pp 604 623 in the electromagnetic case we establish algebraically a reduced version of the hiptmair xu decomposition from r hiptmair and j xu siam j numer anal 45 2007 pp 2483 2509 and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void zero conductivity regions for scalar diffusion problems our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements we present a number of two dimensional three dimensional and axisymmetric numerical experiments which demonstrate that the combination of an appropriately chosen local elimination with the use of the boomeramg and ams solvers can lead to significant improvements in the overall solution time

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

innovative mimetic discretizations for electromagnetic problems in this paper we introduce a discretization methodology for maxwell equations based on mimetic finite differences mfd following the lines of the recent advances in mfd techniques see brezzi et al 2007 14 and the references therein and using some of the results of brezzi and buffa 2007 12 we propose mimetic discretizations for several formulations of electromagnetic problems both at low and high frequency in the time harmonic regime the numerical analysis for some of the proposed discretizations has already been developed whereas for others the convergence study is an object of ongoing research

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 properties and conservation laws in mimetic finite difference methods for pdes the mimetic finite difference mfd methods for pdes mimic crucial properties of mathematical systems duality and self adjointness of differential operators conservation laws and properties of the solution on general polytopal meshes in this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed in addition the numerical approximation of parabolic equations is discussed where the mfd approach is used in the space discretization while implicit method and explicit runge kutta chebyshev schemes are used in time discretization moreover we will show how the numerical solution preserves certain conservation laws of the theoretical solution

effects of time delayed position feedback on a van der pol duffing oscillator abstract the mechanism for the action of time delay in a non autonomous system is investigated in this paper the original mathematical model under consideration is a van der pol duffing oscillator with excitation a delayed system is obtained by adding both linear and nonlinear time delayed position feedbacks to the original system functional analysis is used to change the delayed system into a functional differential equation fde the time delay is taken as a variable parameter to investigate its effect on the dynamics of the system such as the stability and bifurcation of an equilibrium point phase locked periodic and phase shifting solution period doubling quasi periodic motion and chaos a periodic solution expressed in the closed form with the aid of the center manifold and averaging theorem is found to be in good agreement with that obtained by numerical simulation two routes to chaos are found namely period doubling bifurcation and torus breaking the results obtained in this paper show that time delay may be used as a simple but efficient switch to control motions of a system either from order motion to chaos or from chaotic motion to order for different applications

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

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

fuzzy complex analysis i differentiation abstract we consider functions mapping real numbers into the set of generalized complex fuzzy numbers c we define two derivatives of such functions which are generalizations of the derivative of real fuzzy mappings presented by dubois and prade 4 5 sufficient conditions on the function are given so that derivative will be a function mapping real numbers into c if the functional value is always a rectangular complex fuzzy number made up to lr real fuzzy numbers then we show that the derivative is easily computed from the derivatives of the lr fuzzy numbers other properties of the derivatives are presented

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

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

convergence of the multipoint flux approximation l method on general grids we prove first order convergence of the flux and the pressure for the multipoint flux approximation mpfa method known as the mpfa l method the proof is based on that presented for the mimetic finite difference paper by brezzi lipnikov and shashkov in 2005 particular to this procedure is that no velocity interpolation is needed inside the element hence the volume integration of the velocity field is substituted with a quadrature involving only the fluxes the proof is valid on polyhedra but limits the possibility of the mpfa l method to choose a stencil to each and every half face as is the usual procedure the assumption needed for convergence is local and poses limits on grid deformation and the anisotropy of the permeability tensor this assumption is then used to compare the convergence properties of the mpfa l method with those of the mpfa o method and the mimetic finite difference method