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In this paper, we introduce a multilevel algorithm for approximatingvariational formulations of symmetric saddle point systems. The algorithm isbased on availability of families of stable finite element pairs and on theavailability of fast and accurate solvers for it symmetric positive definitesystems. On each fixed level an efficient solver such as the gradient or theconjugate gradient algorithm for inverting a Schur complement is implemented.The level change criterion follows the cascade principle and requires that theiteration error be close to the expected discretization error. We prove newestimates that relate the iteration error and the residual for the constraintequation. The new estimates are the key ingredients in imposing an efficientlevel change criterion. The first iteration on each new level uses informationabout the best approximation of the discrete solution from the previous level.The theoretical results and experiments show that the algorithms achieveoptimal or close to optimal approximation rates by performing a non-increasingnumber of iterations on each level. Even though numerical results supportingthe efficiency of the algorithms are presented for the Stokes system, thealgorithms can be applied to a large class of boundary value problems,including first order systems that can be reformulated at the continuous levelas symmetric saddle point problems, such as the Maxwell equations.

Cascadic Multilevel Algorithms for Symmetric Saddle Point Systems