Multilevel Preconditioning in H (div) and Applications to a Posteriori Error Estimates for Discontinuous Galerkin Approximations

An optimal order algebraic multilevel iterative (AMLI) method for solving systems of linear algebraic equations arising from the finite element discretization of certain boundary value problems, that have their weak formulation in the space H (div), is presented. The algorithm is used for the solution of the discrete minimization problem which arises in the functional-type a posteriori error estimates for the discontinuous Galerkin (DG) approximation of elliptic problems. The method is theoretically analyzed and supporting numerical examples are presented. By comparing the computing time for the proposed solver and the AMLI solver for the DG problem it is shown that the guaranteed and sharp error bounds can be computed with a reasonable cost.