# On adaptive BDDC for the flow in heterogeneous porous media

## TL;DR

This paper develops an adaptive BDDC method for simulating single-phase flow in heterogeneous porous media, improving computational efficiency through adaptive constraint selection and demonstrating effectiveness on benchmark problems.

## Contribution

The paper introduces an adaptive algorithm for selecting flux constraints within BDDC, enhancing solver efficiency for flow in heterogeneous porous media.

## Key findings

- Adaptive BDDC reduces iteration counts significantly.
- Numerical upscaling properties observed in the first two steps.
- Method performs well on 2D and 3D benchmark problems.

## 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.