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

Authors

  • Abdul Abner Lugo Jiménez Instituto Superior de Formaci´on Docente Salomé Ureña, Recinto Félix Evaristo Mejía. Santo Domingo, República Dominicana.
  • Guelvis Enrique Mata Díaz Universidad de Los Andes, Facultad de Ciencias. Mérida, Venezuela.
  • Bladismir Ruiz Universidad Técnica de Manabí, Instituto de Ciencias Básicas. Portoviejo, Ecuador.

DOI:

https://doi.org/10.17268/sel.mat.2021.01.01

Keywords:

Mimetic method, Finite element method, Finite difference method, Conservative methods, Convergence

Abstract

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.

References

Calderón G, Lugo A. Estimación del Error y Adaptatividad en Esquemas Miméticos para Problemas de Contorno. Boletín de la Asociación Matemática Venezolana. 2015; 22(2):109–124.

Castillo JE, Grone RD. A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law. SIAM J Matrix Anal Appl. 2003; 25(1):128–142.

Hyman JM, Shashkov M. The Approximation of Boundary Conditions for Mimetic Finite Difference Methods. Computers Math Applic. 1998; 36(5):79–99.

Hyman JM, Shashkov M, Steinberg S. Mimetic Finite Difference Methods for Diffusion Equations. Computers Math Applic. 2002; 6(3-4):333–352.

Shashkov M, Steinberg S. Support-Operator Finite-Difference Algorithms for General Elliptic Problems. Journal of Computational

Physics. 1995; 118(1):131–151.

Freites MA. Un estudio comparativo de los métodos miméticos para la ecuación estacionaria de difusión; 2004. Tesis de grado, Facultad de Ciencias, UCV.

Guevara JM, Freites M, Castillo JE. A New Second Order Finite Difference Conservative Scheme. Divulgaciones Matem´aticas. 2005;13(1):107–122.

Li BQ. Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. 5th ed. London: Springer-Verlag; 2006.

Rivière B. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. 5th ed. Philadelphia: SIAM; 2008.

Cordero F, Díez P. XFEM+: una modificación de XFEM para mejorar la precisión de los flujos locales en problemas de difusión con conductividades muy distintas. Revista Internacional Métodos Numéricos para Cálculo y Diseño en Ingeniería. 2010; 26(2):121–133.

Arteaga J, Guevara JM. A Conservative Finite Difference Scheme for Static Diffusion Equation. Divulgaciones Matemáticas. 2008; 16(1):39–54.

Guevara JM. Sobre los Esquemas Miméticos de Diferencias Finitas para la Ecuación Estática de Difusión. Caracas, Venezuela: Facultad de Ciencias, UCV; 2005.

Becker EB, Carey GF, Oden JT. Finite Elements: An Introduction. New Jersey 07632: Prentice-Hall, Inc.; 1981.

Solín P. Partial Differential Equations and Finite Element Method. 5th ed. New Jersey: John Wiley & Sons, Ltd.; 2006.

Calderón G, Gallo R. Introducción al Método de los Elementos Finitos: un Enfoque Matemático. Caracas, Venezuela: IVIC; 2011.

Strikwerda JC. Finite Difference Schemes and Partial Differential Equations. 2nd ed. Philadelphia: SIAM, Ltd.; 2004.

Céa J. Approximation variationnelle des probl`emes aux limites. Annales de l’institut Fourier. 1964; 14(2):345–444.

Batista ED, Castillo JE. Mimetic Schemes on Non-Uniform Structured Meshes. Electronic Transactions on Numerical Analysis. 2009; 34(1):152–162.

Corbino J, Castillo JE. High-order mimetic finite-difference operators satisfying the extended Gauss divergence theorem. Journal of Computational and Applied Mathematics. 2020; 364:112326.

Boada Paolini C, Castillo JE. High-order mimetic finite differences for anisotropic elliptic equations. Computers & Fluids. 2020; 213:104746.

Abouali M, Castillo JE. Solving Poisson equation with Robin boundary condition on a curvilinear mesh using high order mimetic discretization methods. Mathematics and Computers in Simulation. 2017; 139:23–36.

Published

2021-07-29

How to Cite

Lugo Jiménez, A. A., Mata Díaz, G. E., & Ruiz, B. (2021). A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems. Selecciones Matemáticas, 8(01), 1 - 11. https://doi.org/10.17268/sel.mat.2021.01.01