83 research outputs found
On adaptive BDDC for the flow in heterogeneous porous media
We study a method based on Balancing Domain Decomposition by Constraints
(BDDC) for a numerical solution of a single-phase flow in heterogenous porous
media. The method solves for both flux and pressure variables. The fluxes are
resolved in three steps: the coarse solve is followed by subdomain solves and
last we look for a divergence-free flux correction and pressures using
conjugate gradients with the BDDC preconditioner. Our main contribution is an
application of the adaptive algorithm for selection of flux constraints.
Performance of the method is illustrated on the benchmark problem from the 10th
SPE Comparative Solution Project (SPE 10). Numerical experiments in both 2D and
3D demonstrate that the first two steps of the method exhibit some numerical
upscaling properties, and the adaptive preconditioner in the last step allows a
significant decrease in number of iterations of conjugate gradients at a small
additional cost.Comment: 21 pages, 7 figure
Dual-primal FETI algorithms for edge finite-element approximations in 3D
A family of dual-primal finite-element tearing and interconnecting methods for edge-element approximations in 3D is proposed and analysed. The key part of this work relies on the observation that for these finite-element spaces there is a strong coupling between degrees of freedom associated with subdomain edges and faces and a local change of basis is therefore necessary. The primal constraints are associated with subdomain edges. We propose three methods. They ensure a condition number that is independent of the number of substructures and possibly large jumps of one of the coefficients of the original problem, and only depends on the number of unknowns associated with a single substructure, as for the corresponding methods for continuous nodal elements. A polylogarithmic dependence is shown for two algorithms. Numerical results validating our theoretical bounds are give
Dual-Variable Schwarz Methods for Mixed Finite Elements
Schwarz methods for the mixed finite element discretization of second order elliptic problems are considered. By using an equivalence between mixed methods and conforming spaces first introduced in [13], it is shown that the condition number of the standard additive Schwarz method applied to the dual-variable system grows at worst like O(1+H/delta) in both two and three dimensions and for elements of any order. Here, H is the size of the subdomains, and delta is a measure of the overlap. Numerical results are presented that verify the bound
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