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We study a framework that allows to solve the coarse problem in the FETI‐DP method approximately. It is based on the saddle‐point formulation of the FETI‐DP system with a block‐triangular preconditioner. One of the blocks approximates the coarse problem, for which we use the multilevel BDDC method as the main tool. This strategy then naturally leads to a version of multilevel FETI‐DP method, and we show that the spectra of the multilevel FETI‐DP and BDDC preconditioned operators are essentially the same. The theory is illustrated by a set of numerical experiments, and we also present a few experiments when the coarse solve is approximated by algebraic multigrid.

 

We construct an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear‐quadratic optimization problem constrained by an elliptic partial differential equation. While the preconditioner generalizes a geometric multigrid preconditioner introduced in earlier works, its construction relies entirely on a standard AMG infrastructure built for solving the forward elliptic equation, thus allowing for it to be implemented using a variety of AMG methods and standard packages. Our analysis establishes a clear connection between the quality of the preconditioner and the AMG method used. The proposed strategy has a broad and robust applicability to problems with unstructured grids, complex geometry, and varying coefficients. The method is implemented using the Hypre package and several numerical examples are presented.

 

We devise multigrid preconditioners for linear-quadratic space-time distributed parabolic optimal control problems. While our method is rooted in earlier work on elliptic control, the temporal dimension presents new challenges in terms of algorithm design and quality. Our primary focus is on the cG(s)dG(r) discretizations which are based on functions that are continuous in space and discontinuous in time, but our technique is applicable to various other space-time finite element discretizations. We construct and analyse two kinds of multigrid preconditioners: the first is based on full coarsening in space and time, while the second is based on semi-coarsening in space only. Our analysis, in conjunction with numerical experiments, shows that both preconditioners are of optimal order with respect to the discretization in case of cG(1)dG(r) for r = 0, 1 and exhibit a suboptimal behaviour in time for Crank–Nicolson. We also show that, under certain conditions, the preconditioner using full space-time coarsening is more efficient than the one involving semi-coarsening in space, a phenomenon that has not been observed previously. Our numerical results confirm the theoretical findings.