Physics-oriented solvers for multicompartmental poromechanics
Final Report Abstract
Classical Biot theory provides the foundation for a fully dynamic poroelasticity model describing the propagation of elastic waves in fluid-saturated media. Multiple network poroelastic theory (MPET) takes into account that the elastic matrix (solid) can be permeated by one or several superimposed interacting single fluid networks of possibly highly diverse characteristics. Biological multicompartmental poroelasticity models can be used to embed more specific medical models, e.g., to describe the water transport in the cerebral environment. This project has focused on physics-oriented solvers for quasi-static and dynamic MPET models. Solvers are understood here to be combined discretization and iterative solution methods. By physics-oriented, we mean that formulations are used which adequately describe the physics of the studied problem, discretizations are chosen which preserve its differential algebraic structure and correctly reflect physical laws such as energy conservation/dissipation, conservation of mass and momentum, and, finally, iterative solution methods are carefully adjusted to harmonize with the discretization process–resulting in high-performance solvers. The main outcomes of the project have been the development and convergence analysis of new hybridized hybrid mixed higher-order methods for the quasistatic MPET equations, the design of new parameter-robust preconditioners for the resulting discrete problems, and generalizations of these discretizations to Biot-Brinkman and dynamic Biot problems. For the latter problem class, variational space-time discrtetizations have been proposed, whose convergence analysis in space and time has been carried out in detail which has led to optimal-order error estimates in space and time. Moreover, a useful abstract framework has been developed for the stability analysis of perturbed saddle-point problems that frequently arise in the well-posed analysis, construction of norm-equivalent preconditioners, and parameter-robust near-best-approximation results of the related variational formulations.
Publications
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Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models. Numerical Linear Algebra with Applications, 26(4).
Hong, Qingguo; Kraus, Johannes; Lymbery, Maria & Philo, Fadi
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Uniformly well-posed hybridized discontinuous Galerkin/hybrid mixed discretizations for Biot’s consolidation model. Computer Methods in Applied Mechanics and Engineering, 384, 113991.
Kraus, Johannes; Lederer, Philip L.; Lymbery, Maria & Schöberl, Joachim
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A new practical framework for the stability analysis of perturbed saddle-point problems and applications. Mathematics of Computation, 92(340), 607-634.
Hong, Qingguo; Kraus, Johannes; Lymbery, Maria & Philo, Fadi
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C1-conforming variational discretization of the biharmonic wave equation. Computers & Mathematics with Applications, 119, 208-219.
Bause, Markus; Lymbery, Maria & Osthues, Kevin
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Robust Approximation of Generalized Biot-Brinkman Problems. Journal of Scientific Computing, 93(3).
Hong, Qingguo; Kraus, Johannes; Kuchta, Miroslav; Lymbery, Maria; Mardal, Kent-André & Rognes, Marie E.
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Hybridized Discontinuous Galerkin/Hybrid Mixed Methods for a Multiple Network Poroelasticity Model with Application in Biomechanics. SIAM Journal on Scientific Computing, 45(6), B802-B827.
Kraus, Johannes; Lederer, Philip L.; Lymbery, Maria; Osthues, Kevin & Schöberl, Joachim
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A fixed-stress type splitting method for nonlinear poroelasticity.
Kraus, Johannes; Kumar, Kundan; Lymberly, Maria & Radu, Florin Adrian
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Analysis of a family of time-continuous strongly conservative space-time finite element methods for the dynamic Biot model
Kraus, Johannes; Lymbery, Maria; Osthues, Kevin & Philo, Fadi
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Monolithic two-level Schwarz preconditioner for Biot’s consolidation model in two space dimensions
Meggendorfer, Stefan; Kanschat, Guido & Kraus, Johannes
