Supra-convergent schemes on irregular grids
Mathematics of Computation
Support-operator finite-difference algorithms for general elliptic problems
Journal of Computational Physics
Solving diffusion equations with rough coefficients in rough grids
Journal of Computational Physics
The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials
Journal of Computational Physics
Applied Numerical Mathematics
Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry
SIAM Journal on Scientific Computing
A local support-operators diffusion discretization scheme for quadrilateral r-z meshes
Journal of Computational Physics
The construction of compatible hydrodynamics algorithms utilizing conservation of total energy
Journal of Computational Physics
Mimetic discretizations for Maxwell's equations
Journal of Computational Physics
Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient
Applied Numerical Mathematics
A tensor artificial viscosity using a mimetic finite difference algorithm
Journal of Computational Physics
The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Large sparse linear systems arising from mimetic discretization
Computers & Mathematics with Applications
Boundary value problems on weighted networks
Discrete Applied Mathematics
Journal of Computational Physics
Mimetic finite difference method
Journal of Computational Physics
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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.