Block BDDC/FETI-DP Preconditioners for Three-Field mixed finite element Discretizations of Biot's consolidation model

TL;DR

This work addresses the numerical solution of Biot's consolidation model using a three-field mixed finite element formulation. It develops a block dual-primal BDDC/FETI-DP preconditioner that reduces the coupled problem to a symmetric positive definite interface system on $(\xi_\Gamma, p_\Gamma, \lambda_\Delta)$ and solves it with PCG, supported by specialized restriction/scaling operators and interface norms. Theoretical results establish condition-number bounds that scale polylogarithmically with the subdomain-to-mesh size ratio and are robust to model parameters, while numerical experiments confirm scalability, quasi-optimality, and robustness under almost incompressible limits and coefficient jumps. The approach offers an efficient, scalable framework for large Biot-type problems in domain-decomposed settings with strong practical impact for poroelastic simulations.

Abstract

In this paper, we construct and analyze a block dual-primal preconditioner for Biot's consolidation model approximated by three-field mixed finite elements based on a displacement, pressure, and total pressure formulation. The domain is decomposed into nonoverlapping subdomains, and the continuity of the displacement component across the subdomain interface is enforced by introducing a Lagrange multiplier. After eliminating all displacement variables and the independent subdomain interior components of pressure and total pressure, the problem is reduced to a symmetric positive definite linear system for the subdomain interface pressure, total pressure, and the Lagrange multiplier. This reduced system is solved by a preconditioned conjugate gradient method, with a block dual-primal preconditioner using a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for both the interface total pressure block and the interface pressure blocks, as well as a Finite Element Tearing and Interconnecting-Dual Primal (FETI-DP) preconditioner for the Lagrange multiplier block. By analyzing the conditioning of the preconditioned subsystem associated with the interface pressure and total pressure components, we obtain a condition number bound of the preconditioned system, which is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and mesh sizes, and robust with respect to the parameters of the model. Extensive numerical experiments confirm the theoretical result of the proposed algorithm.

Figures (1)

• Figure 1: 2D tests: least squares fits of $\mathrm{eig}_{\max}$ from Tables \ref{['2D:continuous']}-\ref{['2D:discontinuous']} with $C_1 + C_2(1 + \log H/h)^2$ plots. Left: continuous total pressure. Right: discontinuous total pressure.

Theorems & Definitions (19)

• Lemma 3.1
• Lemma 3.2
• Proof 1
• Lemma 3.3
• Lemma 4.1
• Lemma 4.2
• Proof 2
• Lemma 5.1
• Proof 3
• Lemma 5.2