A three-level BDDC algorithm for a saddle point problem |
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Authors: | Xuemin Tu |
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Institution: | (1) Institute of Mathematics, Academy of Sciences of the Czech Republic, CZ-11567 Prague 1, Czech Republic;(2) Faculty of Mechanical Engineering, Department of Mathematics, Czech Technical University, CZ-121 35 Praha 2, Czech Republic;(3) Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO 80217-3364, USA;(4) Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Praha 8, CZ-182 00, Czech Republic;(5) Faculty of Civil Engineering, Department of Mathemetics, Czech Technical University in Prague, Praha 8, CZ-166 29, Czech Republic;(6) Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO 80217-3364, USA |
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Abstract: | BDDC algorithms have previously been extended to the saddle point problems arising from mixed formulations of elliptic and
incompressible Stokes problems. In these two-level BDDC algorithms, all iterates are required to be in a benign space, a subspace
in which the preconditioned operators are positive definite. This requirement can lead to large coarse problems, which have
to be generated and factored by a direct solver at the beginning of the computation and they can ultimately become a bottleneck.
An additional level is introduced in this paper to solve the coarse problem approximately and to remove this difficulty. This
three-level BDDC algorithm keeps all iterates in the benign space and the conjugate gradient methods can therefore be used
to accelerate the convergence. This work is an extension of the three-level BDDC methods for standard finite element discretization
of elliptic problems and the same rate of convergence is obtained for the mixed formulation of the same problems. Estimate
of the condition number for this three-level BDDC methods is provided and numerical experiments are discussed. |
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