11,149 research outputs found
A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
We investigate the connections between several recent methods for the
discretization of anisotropic heterogeneous diffusion operators on general
grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite
Volume scheme and the Mixed Finite Volume scheme are in fact identical up to
some slight generalizations. As a consequence, some of the mathematical results
obtained for each of the method (such as convergence properties or error
estimates) may be extended to the unified common framework. We then focus on
the relationships between this unified method and nonconforming Finite Element
schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit
lifting operator close to the ones used in some theoretical studies of the
Mimetic Finite Difference scheme. We also show that for isotropic operators, on
particular meshes such as triangular meshes with acute angles, the unified
method boils down to the well-known efficient two-point flux Finite Volume
scheme
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
Numerical analysis for the pure Neumann control problem using the gradient discretisation method
The article discusses the gradient discretisation method (GDM) for
distributed optimal control problems governed by diffusion equation with pure
Neumann boundary condition. Using the GDM framework enables to develop an
analysis that directly applies to a wide range of numerical schemes, from
conforming and non-conforming finite elements, to mixed finite elements, to
finite volumes and mimetic finite differences methods. Optimal order error
estimates for state, adjoint and control variables for low order schemes are
derived under standard regularity assumptions. A novel projection relation
between the optimal control and the adjoint variable allows the proof of a
super-convergence result for post-processed control. Numerical experiments
performed using a modified active set strategy algorithm for conforming,
nonconforming and mimetic finite difference methods confirm the theoretical
rates of convergence
Numerical results for mimetic discretization of Reissner-Mindlin plate problems
A low-order mimetic finite difference (MFD) method for Reissner-Mindlin plate
problems is considered. Together with the source problem, the free vibration
and the buckling problems are investigated. Full details about the scheme
implementation are provided, and the numerical results on several different
types of meshes are reported
The curved Mimetic Finite Difference method: allowing grids with curved faces
We present a new mimetic finite difference method for diffusion problems that
converges on grids with \textit{curved} (i.e., non-planar) faces. Crucially, it
gives a symmetric discrete problem that uses only one discrete unknown per
curved face. The principle at the core of our construction is to abandon the
standard definition of local consistency of mimetic finite difference methods.
Instead, we exploit the novel and global concept of -consistency.
Numerical examples confirm the consistency and the optimal convergence rate of
the proposed mimetic method for cubic grids with randomly perturbed nodes as
well as grids with curved boundaries.Comment: Accepted manuscrip
Compatible finite element methods for numerical weather prediction
This article takes the form of a tutorial on the use of a particular class of
mixed finite element methods, which can be thought of as the finite element
extension of the C-grid staggered finite difference method. The class is often
referred to as compatible finite elements, mimetic finite elements, discrete
differential forms or finite element exterior calculus. We provide an
elementary introduction in the case of the one-dimensional wave equation,
before summarising recent results in applications to the rotating shallow water
equations on the sphere, before taking an outlook towards applications in
three-dimensional compressible dynamical cores.Comment: To appear in ECMWF Seminar proceedings 201
A mimetic finite difference method using Crank-Nicolson scheme for unsteady diffusion equation
n this article a new mimetic finite difference method to solve unsteady diffusionequation is presented. It uses Crank-Nicolson scheme to obtain time approximationsand second order mimetic discretizations for gradient and divergence operators inspace. The convergence of this new method is analyzed using Lax-Friedrichs equiv-alence theorem. This analysis is developed for one dimensional case. In addition tothe analytical work, we provide experimental evidences that mimetic Crank-Nicolsonscheme is better than standard finite difference because it achieves quadratic conver-gence rates, second order truncation errors and better approximations to the exactsolution.Keywords: mimetic scheme, finite difference method, unsteady diffusion equation,Lax-Friedrichs equivalence theorem
A Stable Mimetic Finite-Difference Method for Convection-Dominated Diffusion Equations
Convection-diffusion equations arise in a variety of applications such as
particle transport, electromagnetics, and magnetohydrodynamics. Simulation of
the convection-dominated regime for these problems, even with high-fidelity
techniques, is particularly challenging due to the presence of sharp boundary
layers and shocks causing jumps and discontinuities in the solution, and
numerical issues such as loss of the maximum principle in the discretization.
These complications cause instabilities, admitting large oscillations in the
numerical solution when using traditional methods. Drawing connections to the
simplex-averaged finite-element method (S. Wu and J. Xu, 2020), this paper
develops a mimetic finite-difference (MFD) discretization using
exponentially-averaged coefficients to overcome instability of the numerical
solution as the diffusion coefficient approaches zero. The finite-element
framework allows for transparent analysis of the MFD, such as proving
well-posedness and deriving error estimates. Numerical tests are presented
confirming the stability of the method and verifying the error estimates
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