Spectral discretizations of the Darcy's equations with non standard boundary conditions

Contenido principal del artículo

Bernard Jean-Marie


This paper is devoted to the approximation of a
nonstandard Darcy problem, which modelizes the flow in porous media, by
spectral methods: the pressure is assigned on a part of the boundary.
We propose two variational formulations, as well as three spectral
discretizations. The second discretization improves the approximation of the
divergence-free condition, but the error estimate on the pressure is not
optimal, while the third one leads to optimal error estimate with a
divergence-free discrete solution, which is important for some
applications. Next, their numerical analysis is performed in detail
and we present some numerical experiments which confirm the interest
of the third discretization.


La descarga de datos todavía no está disponible.


Cargando métricas ...

Detalles del artículo

Cómo citar
Jean-Marie, B. (2018). Spectral discretizations of the Darcy’s equations with non standard boundary conditions. ACI Avances En Ciencias E Ingenierías, 10(1). https://doi.org/10.18272/aci.v10i1.824


Achdou, Y., Bernardi, C. & Coquel, F. “A priori a posteriori analysis of finite volume discretizations of Darcy’s equations”, Numer. Math. 96 (2003), 17-42.

Azaïez, M., Bernardi, C. & Grundmann, M. “Méthodes spectrales pour les équations du milieu poreux”, East West Journal of Numerical Analysis, No. 2, 1994.

Babuˇska, I. “The finite element method with lagrangian multipliers”, Numer. Math. 20, pp. 179-192 (1973).

Bègue, C., Conca, C., Murat, F. & Pironneau, O. “Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression”, Nonlinear Partial Differen- tial Equations and their Applications, Coll`ege de France Seminar, vol. IX (1988).

Bernard, J.M. “Spectral discretizations of the Stokes equations with non standard boundary conditions”, Journal of Scientific Computing, 20(3), 355-377 (2004).

Bernardi, C., Canuto, C. & Maday, Y. “Spectral approximations of the Stokes equa- tions with boundary conditions on the pressure”, SIAM J. Numer. Anal. 28, No 2 (1991), p. 333-362.

Bernardi,C.&Maday,Y.Approximations spectrales de problèmesauxlimitesellip- tiques, Springer-Verlag, Paris, 1992.

Bernardi, C. & Maday, Y. Spectral Methods, Handbook of Numerical Analysis, Vol. V, Paris (1997).

Brezzi, F. “On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers”, R.A.I.R.O., Anal. Mumer. R2, PP. 129-151 (1974).

Chorin, A.J. “ Numerical solution of the Navier-Stokes equations”, Math. Comput. 22 (1968), 745-762.

Girault, V. & Raviart, P. A. Finite Element Methods for Navier-Stokes Equations,Springer-Verlag, Berlin (1986).

Grisvard, P. “Elliptic Problems in Nonsmooth Domains”, Pitman Monographs and Studies in Mathematics, 24, Boston (1985).

Neˇcas, J. les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris (1967).

Quarteroni, A. & Valli, A. Domain decomposition methods for partial differential equations, Oxford science publication (1999).

Temam, R. “Une méthode d’approximation de la solution des équations de Navier- Stokes”, Bull. Soc. Math. France 98 (1968), 115-152.