Foundations of mimetic finite difference method

2014 ◽  
pp. 41-65
Author(s):  
Lourenço Beirão da Veiga ◽  
Konstantin Lipnikov ◽  
Gianmarco Manzini
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Mimetic Finite Difference Method

scholarly journals A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems

2021 ◽  
Vol 8 (1) ◽  
pp. 1-11
Author(s):  
Abdul Abner Lugo Jiménez ◽  
Guelvis Enrique Mata Díaz ◽  
Bladismir Ruiz
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Stationary Regime ◽  
Heat Transfer Fluid ◽  
Mimetic Finite Difference Method ◽  
Linear Differential

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

scholarly journals A mortar mimetic finite difference method on non-matching grids

2005 ◽  
Vol 102 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Markus Berndt ◽  
Konstantin Lipnikov ◽  
Mikhail Shashkov ◽  
Mary F. Wheeler ◽  
Ivan Yotov
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Mimetic Finite Difference Method

Mimetic finite differences for nonlinear and control problems

2014 ◽  
Vol 24 (08) ◽  
pp. 1457-1493 ◽  
Author(s):  
P. F. Antonietti ◽  
L. Beirão da Veiga ◽  
N. Bigoni ◽  
M. Verani
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Finite Differences ◽  
Stokes Equations ◽  
Free Boundary Problems ◽  
Quasilinear Elliptic Equations ◽  
Control Problems ◽  
Mimetic Finite Difference Method ◽  
Mimetic Finite Differences

In this paper we review some recent applications of the mimetic finite difference method to nonlinear problems (variational inequalities and quasilinear elliptic equations) and optimal control problems governed by linear elliptic partial differential equations. Several numerical examples show the effectiveness of mimetic finite differences in building accurate numerical approximations. Finally, driven by a real-world industrial application (the numerical simulation of the extrusion process) we explore possible further applications of the mimetic finite difference method to nonlinear Stokes equations and shape optimization/free-boundary problems.

Mimetic finite difference method

2014 ◽  
Vol 257 ◽  
pp. 1163-1227 ◽  
Author(s):  
Konstantin Lipnikov ◽  
Gianmarco Manzini ◽  
Mikhail Shashkov
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Mimetic Finite Difference Method
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