BDDC for Mixed-Hybrid Formulation of Flow in Porous Media with Combined Mesh Dimensions

TL;DR

This paper extends the BDDC method to handle mixed-hybrid finite element discretizations of porous media flow with combined mesh dimensions, enabling efficient parallel solutions for complex geological models.

Contribution

The paper introduces a novel extension of BDDC for saddle-point problems with mixed mesh dimensions, including implementation and performance analysis.

Findings

The extended BDDC method effectively solves large-scale porous media flow problems.

Numerical experiments demonstrate the method's scalability and efficiency.

Parallel implementation shows good performance on real-world geological models.

Abstract

We extend the Balancing Domain Decomposition by Constraints (BDDC) method to flows in porous media discretised by mixed-hybrid finite elements with combined mesh dimensions. Such discretisations appear when major geological fractures are modelled by 1D or 2D elements inside three-dimensional domains. In this set-up, the global problem as well as the substructure problems have a symmetric saddle-point structure, containing a `penalty' block due to the combination of meshes. We show that the problem can be reduced by means of iterative substructuring to an interface problem, which is symmetric and positive definite. The interface problem can thus be solved by conjugate gradients with the BDDC method as a preconditioner. A parallel implementation of this algorithm is incorporated into an existing software package for subsurface flow simulations. We study the performance of the iterative solver on several academic and real-world problems. Numerical experiments illustrate its efficiency and scalability.