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Abstract This work presents an overview of the most relevant results obtained by the authors regarding the numerical solution of the Biot’s consolidation problem by preconditioning techniques. The emphasis here is on the design of parameter-robust preconditioners for the efficient solution of the algebraic system of equations resulting after proper discretization of such poroelastic problems. The classical two- and three-field formulations of the problem are considered, and block preconditioners are presented for some of the discretization schemes that have been proposed by the authors for these formulations. These discretizations have been proved to be well-posed with respect to the physical and discretization parameters, what provides a framework to develop preconditioners that are robust with respect to such parameters as well. In particular, we construct both norm-equivalent (block diagonal) and field-of-value-equivalent (block triangular) preconditioners, which are proved to be parameter-robust. The theoretical results on this parameter-robustness are demonstrated by considering typical benchmark problems in the literature for Biot’s model. 

Abstract Discretization of flow in fractured porous media commonly lead to large systems of linear equations that require dedicated solvers. In this work, we develop an efficient linear solver and its practical implementation for mixed‐dimensional scalar elliptic problems. We design an effective preconditioner based on approximate block factorization and algebraic multigrid techniques. Numerical results on benchmarks with complex fracture structures demonstrate the effectiveness of the proposed linear solver and its robustness with respect to different physical and discretization parameters. 

We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that, with the proper choice of subspace splitting, we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regard to the number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely, the electric signaling in excitable tissue modeled by bidomain, the extracellular-membrane-intracellular (EMI) model, and reduced EMI equations. 

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. We introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for 𝐻(div) problems in ℝ2 establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.