## Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations.(English)Zbl 1224.65263

The paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (DG) finite element approximations for elliptic problems of second order. A nested hierarchy of meshes is generally assumed. This approach utilizes a general two-level scheme, where the finite element space for the DG method is decomposed into a subspace plus a correction which can be handled by a standard smoothing procedure. The authors consider three different auxiliary subspaces, namely, piecewise linear $$C^0$$-conforming functions, piecewise linear functions that are continuous at the centroids of the edges/faces (Crouzeix-Raviart finite elements) and piecewise constant functions over the finite elements. Finally, numerical experiments for a 3D model problem showing uniform convergence of the constructed methods are presented.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65F08 Preconditioners for iterative methods 35J25 Boundary value problems for second-order elliptic equations
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