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A Note on the Global Extraction Element‐by‐Element Method

In the engineering literature, the so‐called Global Extraction Element‐by‐Element (GE‐EBE) Methods have been successfully used for solving real‐life mechanical problems, especially, in 3D [2], although the EBE preconditioners C haven't any asymptotically preconditioning effects on the stiffness matrix K, i. e. the spectral condition number k(C‐1K) behaves like O(h‐2), say, for elasticity problems. However, the method has quite good scaling properties. From a theoretical point of view, the GE‐EBE preconditioner is nothing else than an Additive Schwarz Method (ASM) preconditioner generated by an overlapping splitting of the finite element space into subspaces spanned by the element ansatz functions. The authors propose to use the GE‐EBE methods and their patch counterpart (GE‐PBP) either as smoothers, especially, in their multiplicative version, or as preconditioners in their multilevel additive version. In the multilevel version, such preconditioners are asymptotically optimal. The GE‐PBP smoother is especially suited for the use in Algebraic Multi‐Grid (AMG) Methods based on matrix graph coarsening strategies. The numerical results presented confirm the good smoothing properties of GE‐PBP smoothers as well as the preconditioning properties of multilevel additive GE‐PBP preconditioners. Finally, the authors discuss parallelization aspects.