To help solve physical and engineering problems, mimetic or compatible algebraic discretization methods employ discrete constructs to mimic the continuous. In addition, they describe how their mimetic methods toolkit mtkavailable onlinecan be used for the computational implementation of mimetic discretization methods. Higher order mimetic methods for unstructured meshes, j. Latex file which generates a typeset bibliography from the mimetic. In the mimetic discretization, the discrete gradient and divergence operators are negative adjoint to each other. Discretization is usually performed prior to the learning process and it can be broken into two tasks. Mimetic discretization methods jose castillo, san diego state university mimetic discretizations or compatible discretizations have been a recurrent search in the history of numerical methods for solving partial differential equations with variable degree of success. Mimetic methods for the discretization of di usion problems in mixed form are presented in 40, 41, 25, 28, 26, 27.
Derive the other discrete operator from the discrete duality formula. Save up to 80% by choosing the etextbook option for isbn. Mimetic discretization methods 1, castillo, jose e. Highorder mimetic finite difference methods 349 defined by 2 df dfdx 1 this approach transforms the problem of approximating a derivative on a nonuniform grid to approximating two derivatives, d and dx on a uniform grid. Mimetic methods for the discretization of diffusion problems in mixed form are presented in. This page provides information about mimetic discretizations of continuum mechanics problems. Based on the castillogrone operators, this simple mimetic discretization method is invariably valid for spatial dimensions no greater than three. Mimetic finite difference method for elliptic problems.
The interpolation methods extend bilinear and areaweighted interpolation, which are widely used in earth sciences, to work with vector fields essentially discretized versions of differential 1forms and 2forms. Mimetic discretization methods kindle edition by castillo, jose e. The mimetic finite difference discretization of diffusion problem on. We explore some of the common approaches to the choice of form of the pde and the spacetime discretization, leaving discussion of the grids for a later chapter. Mimetic discretizations, compatible spatial discretizations. The mfd method for elliptic problems with staggered discretizations of diffusion coefficients. Search for library items search for lists search for contacts search for a library. The space discretization of a hamiltonian system is usually performed by one of the following techniques. To help solve physical and engineering problems, mimetic or compatible algebraic discretization methods employ discrete constructs to mimic the continuous identities and theorems found in vector calculus. Geometric sparsity and mimetic discretization lekheng lim program in scienti. On the other hand, finite difference methods and finite volume. Abstract pdf 3454 kb 2010 a mimetic discretization of the stokes problem with selected edge bubbles. The highorder mixed mimetic finite difference method for.
Interpolation methods for vector fields whose components are staggered on horizontal arakawa c or d grids are presented. Global discretization handles discretization of each numeric attribute as a preprocessing step, i. Elsevieracademic press, amsterdam, second edition, 2003. In the mimetic discretization, the discrete gradient and divergence operators are negativeadjoint toeachother. Consequently, these three methods are members of the same family of discretization techniques. The family of mimetic discretizations contains the classical mixed finite element discretizations. Computers and internet mathematics differential equations, partial research mathematical research partial differential equations tensors mathematics. Past workshop on mimetic discretization methods computational science research center san diego state university july 911, 2003. The same technique can be used to construct an approx imation of the second derivative by using the chain rule. These methods mimic many fundamental properties of the underlying physical problem including conservation laws.
The mimetic finite di erence discretization of di usion. Mimetic finite di erence methods an introduction andrea cangiani. Mimetic discretization methods 1st edition by jose e. In the original mimetic framework, we discretize simultaneously pairs of adjoint differential op. The discretization methodology is based on the support operator approach see 7 and references therein. The book also presents a numerical method for obtaining corresponding discrete. Finite element methods begin by converting the pdes into an equiv alent variational. This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advectiondiffusion, stokes, elasticity, magnetostatics and plate. Then, we construct the discrete generalized divergence operator and derive the discrete ux operator as the adjoint to it. Convectiondiffusion problems are considered in, while the problem of modeling flows in porous media is. Mimetic advection 4 central box method 1d advection. Loosely speaking, mimetic or compatible algebraic methods have discrete.
Mimetic finite difference methods in image processing. Tensor formulation of 3d mimetic finite differences and. This family of methods is introduced by brezzi et al. Mimetic discretization methods 1st edition 97814665433. In this work we introduce the mimetic methods toolkit mtk, an object oriented application programming interface for the implementation of mimetic discretization methods in. Report by electronic transactions on numerical analysis. Pdf convergence of a mimetic finite difference method. We demonstrate how to apply the framework for compatible discretization for two scalar versions of the hodge laplacian. More precisely, the gdm starts by defining a gradient discretization gd, which is a triplet. This meeting is being supported by the csrc and nsf. Pdes such as conservation laws, exact identities, solution.
Design principles of the mimetic finite difference schemes. The mimetic finite difference method for elliptic problems. 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 numerical solutions ensure the satisfaction of. The primal form of the mfd method is introduced and analyzed in. Mimetic finite difference method and nodal mimetic finite difference method.
For this purpose, we present the mimetic methods toolkit. The goal is to introduce the reader to various forms of discretization and to illustrate the numerical performance of. Mimetic discretization methods crc press book to help solve physical and engineering problems, mimetic or compatible algebraic discretization methods employ discrete constructs to mimic the continuous identities and theorems found in vector calculus. Work supported by nserc canada national science and engineering research council grants ogp0008781 and ogp0036901. The first workshop on mimetic discretization methods will be held at the computational science research center in june or july of 2003. In this section, we describe the mimetic discretization over the polyhedron e.
Mimetic discretization methods focuses on the to help solve physical and engineering problems, mimetic or compatible algebraic discretization methods employ discrete constructs to mimic the continuous identities and theorems found in vector calculus. First, we introduce vector spaces of discrete functions and inner products on them. Mimetic finite difference methods in image processing scielo. Mimetic nite di erence discretization h partition of into polygonal polyhedral elements.
Fvm can be related to other discretization methods such as the finite di. The following can be cast as compatible spatial discretizations. Incomplete list of various compatible discretization methods and frameworks includes cell method compatible discrete operators covolume method summation by parts hybrid fv, mixed fv, discrete duality fv mixed fe, weak galerkin, vem, kuznetsovrepin exterior calculus konstantin lipnikov design principles of the mimetic finite di erence schemes. Siam journal on numerical analysis siam society for. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and. Construction of mimetic numerical methods blair perot. Convergence analysis of the mimetic finite difference method for. The family of semidiscrete mimetic methods is proved theoretically to be energystable as the corresponding continuous problem. Mimetic discretization methods focuses on the recent mimetic discretization method codeveloped by the first author.
A generalized mimetic finite difference method and two. Conference organizers, nsf grant dms 01064, gene golub 1. The distinction between global and local discretization methods is dependent on when discretization is performed 28. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often.
Mimetic finite difference method department of mathematics. Som staggered edgeface natural neighbors mimetic methods all numerical methods. The family of mimetic discretizations contains the classical mixed. In mixed finite element the constitutive equation is discretized by using an explicit. Convergence of a mimetic finite difference method for static diffusion equation. Mimetic finite difference methods for diffusion equations. Mimetic discretization methods books pics download new. This introduces certain freedom, and by carefully picking the parameters that are needed to fully specify a method, one can construct mimetic methods that coincide with other known methods, or reduce to these methods e. Tensor formulation of 3d mimetic finite differences and applications to elliptic problems. Use features like bookmarks, note taking and highlighting while reading mimetic discretization methods. Mimetic discretization methods request pdf researchgate.
The convergence of mimetic discretization for rough grids james m. Mimetic interpolation of vector fields on arakawa cd. The mimetic methods toolkit mtk is an api for the implementation of cgmbased mfds for the resolution of pdes, yielding numerical solutions that guarantee uniform order of accuracy, all along the modeled physical domain. The mimetic finite difference mfd method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of. A unified approach for handling convection terms in finite. Principles of mimetic discretizations of differential operators.
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