Journal of Computational Physics
Analysis of an exact fractional step method
Journal of Computational Physics
A numerical method for large-eddy simulation in complex geometries
Journal of Computational Physics
Difference schemes on uniform grids performed by general discrete operators
Applied Numerical Mathematics
Journal of Computational Physics
Discrete calculus methods for diffusion
Journal of Computational Physics
Discrete Orthogonal Decomposition and Variational Fluid Flow Estimation
Journal of Mathematical Imaging and Vision
Boundary value problems on weighted networks
Discrete Applied Mathematics
Some remarks on quadrilateral mixed finite elements
Computers and Structures
The convergence of mimetic discretization for rough grids
Computers & Mathematics with Applications
On variational methods for fluid flow estimation
IWCM'04 Proceedings of the 1st international conference on Complex motion
The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes
Journal of Computational Physics
A nine-point scheme with explicit weights for diffusion equations on distorted meshes
Applied Numerical Mathematics
Discrete orthogonal decomposition and variational fluid flow estimation
Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
A study of non-smooth convex flow decomposition
VLSM'05 Proceedings of the Third international conference on Variational, Geometric, and Level Set Methods in Computer Vision
Equivalent projectors for virtual element methods
Computers & Mathematics with Applications
Journal of Computational Physics
Mimetic finite difference method
Journal of Computational Physics
Symmetry-preserving discretization of Navier-Stokes equations on collocated unstructured grids
Journal of Computational Physics
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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.