On adaptive BDDC for the flow in heterogeneous porous media

TL;DR

This work extends adaptive BDDC to mixed RT0 discretizations of single-phase Darcy flow in highly heterogeneous porous media, introducing an adaptive enrichment of the flux coarse space via local generalized eigenvalue problems. The method solves the flux in three stages with a CG solve preconditioned by a two-level BDDC, where the coarse space includes flux averages over faces and subdomain pressure averages; adaptation adds constraints until a target condition number is achieved, yielding a bound of the form $\kappa \le \tilde{\omega} N_F^2$. Numerical results on the SPE 10 benchmark in both 2D and 3D demonstrate noticeable upscaling in the initial steps and substantial reductions in CG iterations with adaptive flux constraints, outperforming multiscale MFEM-inspired constraints. Overall, the adaptive BDDC framework provides a robust, scalable solver for heterogeneous reservoir flow, with partitioning quality remaining a key practical consideration for efficiency.

Abstract

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.

Figures (7)

• Figure 6.1: Substructuring and the base $10$ logarithm of the permeability $k$ for the problem with jumps aligned with the substructure interfaces (left panel) and the largest $300$ eigenvalues of the BDDC preconditioned operator for this problem (right panel).
• Figure 6.2: Substructuring and the base $10$ logarithm of the permeability $k$ in layer 1 (left panel) and layer 20 (right panel) of the SPE 10 problem. Left panel also illustrates irregular partitioning A, and the right panel illustrates irregular partitioning B.
• Figure 6.3: Substructuring and the base $10$ logarithm of the permeability $k$ in layer 60 (left panel) and layer 85 (right panel) of the SPE 10 problem. Left panel illustrates irregular partitioning A, and the right panel illustrates irregular partitioning B.
• Figure 6.4: The base $10$ logarithm of the permeability $k$ in the subdomains $1$ and $2$ from the layer 85 of the SPE 10 problem (Fig. \ref{['fig:layers_60_85']}, right panel), and the first $12$ eigenvalues of the corresponding local generalized eigenvalue problem (\ref{['eq:eigenvalue-problem-local']}). Here $\lambda_{i}=0$ for $i=11$ and $12$.
• Figure 6.5: The largest $300$ eigenvalues of the BDDC preconditioned operator for the layer $85$ of the SPE 10 problem (Fig. \ref{['fig:layers_60_85']}, right panel) without adaptivity ($\tau=\infty$) and for the adaptive BDDC with the target condition number $\tau=100$.

Theorems & Definitions (4)

• Remark 3.3
• Theorem 3.5
• Lemma 4.1: Mandel-2012-ABTSousedik-2013-AMB
• Remark 4.3