Very weak estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions. (English) Zbl 1281.76052

Milišić, Vuk

Methods Appl. Anal. 16, No. 2, 157-186 (2009).

Summary: This work is a continuation of [E. Bonnetier, D. Bresch and the author, Blood flow modelling in stented arteries: new convergence results of first order boundary layers and wall-laws for a rough Neumann-Laplace problem (submitted), see
http://arxiv-web3.library.cornell.edu/abs/0809.2454v2];
it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation of a flat edge on one side and natural homogeneous Neumann boundary conditions are applied on the inlet/outlet of the domain. To prevent oscillations on the Neumann-like boundaries, we introduce a microscopic vertical corrector defined in a rough quarter-plane. In the paper cited above we studied a priori estimates in this setting; here we fully develop very weak estimates à la J. Nečas [Les méthodes directes en théorie des équations elliptiques. Paris: Masson et Cie; Prague: Academia (1967; Zbl 1225.35003); for a review see the English translation, Berlin: Springer (2012; Zbl 1246.35005)] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [Bonnetier et al. (op. cit.)]. We validate these results numerically, proving first order results for boundary layer approximation including the vertical correctors and a little less for the averaged wall-law introduced in the literature [W. Jäger, the author and N. Neuss, SIAM J. Sci. Comput. 22, No.6, 2006–2028 (2001; Zbl 0980.35124); A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations. Oxford: Clarendon Press (1999; Zbl 0931.65118)].

MSC:

76Z05	Physiological flows
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs

Keywords:

wall-laws; rough boundary; Laplace equation; multi-scale modelling; boundary layers; error estimates; natural boundary conditions; vertical boundary correctors

Citations:

Zbl 1225.35003; Zbl 1246.35005; Zbl 0980.35124; Zbl 0931.65118

Software:

FreeFem++