Approximating second-order vector differential operators on distorted meshes in two space dimensions

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 7, 2008, Pages 1065-1089

  Hermeline, Francois ^a  

^a DIF   (France)

Author keywords

Distorted mesheg; Elasticity; Finite volume method; Lagrangian mesh; Stokes equations; Vector elliptic equation

Indexed keywords

ELASTICITY; FINITE VOLUME METHOD; LAGRANGE MULTIPLIERS; PARTIAL DIFFERENTIAL EQUATIONS; VECTORS;

DISTORTED MESHEG; LAGRANGIAN MESH; LINEAR SOLVERS; LOCKING-FREE; NUMERICAL EXPERIMENTS; NUMERICAL RESULTS; QUADRILATERAL MESHES; SECOND ORDERS; STOKES EQUATIONS; SYMMETRIC POSITIVE DEFINITE; TWO SPACE DIMENSIONS; VECTOR DIFFERENTIAL OPERATORS; VECTOR PARTIAL DIFFERENTIAL EQUATIONS;

MATHEMATICAL OPERATORS;

EID: 60949089534     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2356     Document Type: Article

References (41)

1. Hermeline F. Une méthode de volumes finis pour les équations elliptiques du second ordre (in French). Comptes Rendus de l'Academie des Sciences Paris, Series I 1998; 326:1433-1436.
2. Hermeline F. A finite volume method for the approximation of diffusion operators on distorted meshes. Journal of Computational Physics 2000; 160:481-499.
3. Hermeline F. Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Computer Methods in Applied Mechanics and Engineering 2003; 192:1939-1959.
4. Hermeline F. A finite volume method for solving Maxwell equations in inhomogeneous media on arbitrary meshes. Comptes Rendus de l'Academie des Sciences Paris, Series I 2004; 339:893-898.
5. Domelevo K, Omnes P. A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. Mathematical Modelling and Numerical Analysis 2005; 39:1203-1249.
6. Andreianov B, Boyer F, Hubert F. Duplex finite volume schemes for nonlinear elliptic problems on general 2D meshes. Finite Volumes for Complex Applications, vol. 4, Benkhaldoun F, Ouazar D, Raghay S (eds). Hermes Science Publishing, 2005; 365-375.
7. Delcourte S, Domelevo K, Omnes P. Discrete duality finite volume method for second order elliptic problems. Finite Volumes for Complex Applications, vol. 4, Benkhaldoun F, Ouazar D, Raghay S (eds). Hermes Science Publishing, 2005; 447-458.
8. Pierre C. Modélisation et simulation de l'activité électrique du cœur dans le thorax, analyse numérique et méthodes de volumes finis (in French). Ph.D. Thesis, University of Nantes, France, 2005.
9. Andreianov B, Boyer F, Hubert F. Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numerical Methods for Partial Differential Equations 2007; 23(1):145-195.
10. Yuan G, Sheng Z. Analysis of accuracy of a finite volume scheme for diffusion equations on distorted meshes. Journal of Computational Physics 2007; 220:751-771.
11. Hermeline F. Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes. Computer Methods in Applied Mechanics and Engineering 2007; 196:2497-2526.
12. Delcourte S, Domelevo K, Omnes P. A discrete duality finite volume approach to Hodge decomposition and div-curl problems on almost arbitrary two-dimensional meshes. SIAM Journal on Numerical Analysis 2007; 45(3):1142-1174.
13. Chainais-Hillairet C. Discrete duality finite volume schemes for two-dimensional drift-diffusion and energy-transport models. International Journal for Numerical Methods in Fluids 2007; DOI: 10.1002/fld.1393.
14. Delcourte S. Développement de méthodes de volumes finis pour la mécanique des fluides (in French). Ph.D. Thesis, University of Toulouse, France, 2007.
15. Boyer F, Hubert F. Finite volume method for nonlinear transmission problems. Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science, vol. 60. Springer: Berlin, Heidelberg, 2008; 443-450.
16. Lebedev VI. Difference analogues of orthogonal decompositions, fundamental differential operators and certain boundary-value problems of mathematical physics (in Russian). Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 1964; 4:449-465.
17. Harlow FH, Welch JE. Numerical calculations of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids 1965; 8:2182-2189.
18. Nicolaides RA. Flow discretization by complementary volume techniques. Proceedings of the 9th AIAA CFD Meeting, Buffalo, 1989; AIAA Paper 89-1978.
19. Choudhury S, Nicolaides RA. Discretization of incompressible vorticity-velocity equations on triangular meshes. International Journal for Numerical Methods in Fluids 1990; 11:823-833.
20. Hall CA, Cavendish JC, Frey WH. The dual variable method for solving fluid flow difference equations on Delaunay triangulations. Computers and Fluids 1991; 20(2):145-164.
21. Cavendish JC, Hall CA, Porsching TA. Solution of incompressible Navier-Stokes equations on unstructured grids using dual tesselations. International Journal of Numerical Methods for Heat and Fluid Flow 1992; 2:483-502.
22. Hall CA, Porsching TA, Hu P. Covolume-dual variable method for thermally expandable flows on unstructured triangular grids. Computational Fluid Dynamics 1994; 2:111-139.
23. Dubois F. Une formulation tourbillon-vitesse-pression pour le problème de Stokes (in French). Comptes Rendus de l'Academie des Sciences Paris, Series I 1992; 314:277-280.
24. Girault V, Lopez H. Finite-element error estimates for the MAC scheme. IMA Journal of Numerical Analysis 1996; 16:347-379.
25. Han H, Wu X. A new mixed finite element formulation and the MAC method for the Stokes equations. SIAM Journal on Numerical Analysis 1998; 35(2):560-571.
26. Dubois F, Salaün M, Salmon S. First vorticity-velocity-pressure numerical scheme for the Stokes problem. Computer Methods in Applied Mechanics and Engineering 2003; 192:4877-4907.
27. Salaün M, Salmon S. Numerical stabilization of the Stokes problem in vorticity-velocity-pressure formulation. Computer Methods in Applied Mechanics and Engineering 2007; 196:1767-1786.
28. Kanschat G. Divergence-free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme. International Journal for Numerical Methods in Fluids 2007; 56:941-950.
29. Minev PD. Remarks on the links between low-order DG methods and some finite-difference schemes for the Stokes problem. International Journal for Numerical Methods in Fluids 2008; DOI: 10.1002/fld.1723.
30. Chou SH, Kwan DY. A covolume method based on rotated bilinears for the generalized Stokes problem. SIAM Journal on Numerical Analysis 1998; 35(2):494-507.
31. Wesseling P, Segal A, Kassels CGM. Computing flows on general three-dimensional nonsmooth staggered grids. Journal of Computational Physics 1999; 149:333-362.
32. Eymard R, Herbin R. A cell-centered finite volume scheme on general meshes for the Stokes equations in two space dimensions. Comptes Rendus de l'Academie des Sciences Paris, Series I 2003; 337:125-128.
33. Vanselow R. A cell-centered FVM for the two-dimensional Stokes equation. Zeitschrift für Angewandte Mathematik und Mechanik 2003; 83(11):779-789.
34. Blanc P, Eymard R, Herbin R. An error estimate for finite volume methods for the Stokes equations on equilateral triangular meshes. Numerical Methods for Partial Differential Equations 2004; 20(6):907-918.
35. Eymard R, Herbin R, Latché JC. Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3d meshes. SIAM Journal on Numerical Analysis 2007; 45(1):1-36.
36. Eymard R, Herbin R. A new colocated finite volume scheme for the incompressible Navier-Stokes equations on general non matching grids. Comptes Rendus de l'Academie des Sciences Paris, Series I 2007; 344:659-662.
37. Girault V, Raviart PA. Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer: Berlin, Heidelberg, 1987.
38. Nicolaides RA. Direct discretization of planar div-curl problems. SIAM Journal on Numerical Analysis 1992; 29:32-56.
39. Nicolaides RA. Analysis and convergence of the MAC scheme. I. The linear problem. SIAM Journal on Numerical Analysis 1992; 29(6):1579-1591.
40. Gatica GN. Analysis of a new augmented mixed finite element method for linear elasticity allowing RT0-P1-P0 approximations. Mathematical Modelling and Numerical Analysis 2006; 40:1-28.
41. Bercovier M, Engelman M. A finite element for the numerical simulation of viscous incompressible flows. Journal of Computational Physics 1979; 30:181-201.