Details

Autor(en) / Beteiligte

• Manteuffel, Thomas A.
• Mccormick, Stephen F.
• Röhrle, Oliver

Titel

Projection Multilevel Methods for Quasilinear Elliptic Partial Differential Equations: Theoretical Results

Ist Teil von

• SIAM journal on numerical analysis, 2006-01, Vol.44 (1), p.139-152

Ort / Verlag

Philadelphia: Society for Industrial and Applied Mathematics

Erscheinungsjahr

2006

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• In a companion paper [T. A. Manteuffel et al., SIAM J. Numer. Anal, 44 (2006), pp. 120-138], we propose a new multilevel solver for two-dimensional elliptic systems of partial differential equations with nonlinearity of type u∂v. The approach is based on a multilevel projection method (PML) [S. F. McCormick, Multilevel Projection Methods for Partial Differential Equations, SIAM, Philadelphia, 1992] applied to a first-order system least-squares functional that allows us to treat the nonlinearity directly. While the companion paper focuses on computation, here we concentrate on developing a theoretical framework that confirms optimal two-level convergence. To do so, we choose a first-order formulation of the Navier-Stokes equations as a basis of our theory. We establish continuity and coercivity bounds for the linearized Navier-Stokes equations and the full nonquadratic least-squares functional, as well as existence and uniqueness of a functional minimizer. This leads to the immediate result that one cycle of the two-level PML method reduces the functional norm by a factor that is uniformly less than 1.