The BDDC method [2] is the most advanced method from the BDD family [5]. Polylogarithmic condition number estimates for BDDC were obtained in [6, 7] and a proof that eigenvalues of BDDC and FETI-DP are same except for an eigenvalue equal to one was given in [7]. For important insights, alternative formulations of BDDC, and simplified proofs of these results, see [1] and [4].
In the case of many substructures, solving the coarse problem exactly is becoming a bottleneck. Since the coarse problem in BDDC has the same form as the original problem, the BDDC method can be applied recursively to solve the coarse problem approximately, leading to a multilevel form of BDDC in a straightforward manner [2]. Polylogarithmic condition number bounds for three-level BDDC (BDDC with two coarse levels) were proved in [10, 9]. This contribution is concerned with condition number estimates of BDDC with an arbitrary number of levels.
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Mandel, J., Sousedík, B., Dohrmann, C.R. (2008). On Multilevel BDDC. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W. (eds) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75199-1_33
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