Multilevel discretization of symmetric saddle point systems without the discrete LBB condition

Using an inexact Uzawa algorithm at the continuous level, we study the convergence of multilevel algorithms for solving saddle-point problems. The discrete stability Ladyshenskaya–Babušca–Brezzi (LBB) condition does not have to be satisfied. The algorithms are based on the existence of a multilevel sequence of nested approximation spaces for the constrained variable. The main idea is to maintain an accurate representation of the residual associated with the main equation at each step of the inexact Uzawa algorithm at the continuous level. The residual representation is approximated by a Galerkin projection. Whenever a sufficient condition for the accuracy of the representation fails to be satisfied, the representation of the residual is projected on the next (larger) space available in the prescribed multilevel sequence. Numerical results supporting the efficiency of the algorithms are presented for the Stokes equations and a div – curl system.