European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS7: Robust discretizations for coupled elliptic/parabolic equations (Part 1)
Chair: Florin Radu
25 mins
Strongly conservative discretization and parameter-robust Uzawa-type methods for multiple-network poroelasticity models
Johannes Kraus, Qingguo Hong, Maria Lymbery, Fadi Philo
Abstract: In this talk we consider a multiple network poroelasticity model that enforces explicitly mass conservation on the continuous level. We propose its strongly conservative and parameter-robust stable discretization using $H$(div)-conforming finite elemet spaces for the displacement as well as for the flux fields. We devise a framework of augmenting and splitting the resulting three-by-three block system, which exhibits a double saddle point structure. The related block Gauss-Seidel methods perform the iterative coupling of the physical quantities of interest and converge uniformly at a rate independent of any of the physical and discretization parameters. Further we present numerical tests comparing different variants of these Uzawa-type methods that can be derived within this framework including the fixed-stress iterative method. This is joint work with Qingguo Hong (The Pennsylvania State University, USA), Maria Lymbery (University of Duisburg-Essen, Germany), Fadi Philo (University of Duisburg-Essen, Germany), Mary Wheeler (The University of Texas at Austin, USA).
25 mins
Efficient solvers for stabilized three-field mixed formulations of the poroelasticity problem
Massimiliano Ferronato, Matteo Frigo, Nicola Castelletto, Joshua White
Abstract: We focus on a three-field (displacement-velocity-pressure) stabilized mixed method for poroelasticity based on low-order elements. Precisely, piecewise trilinear (Q1), lowest order Raviart-Thomas (RT0), and piecewise constant (P0) spaces are used for the approximation of displacement, Darcy’s velocity, and fluid pore pressure, respectively. The attractive features of this choice are local (element-wise) mass-conservation and robustness with respect to highly heterogeneous hydromechanical properties, such as high-contrast permeability fields typically encountered in real-world geoscience applications. It is well-known that the selected discrete spaces do not intrinsically satisfy the LBB condition in the undrained/incompressible limit. Hence, suitable stabilization techniques are required to obtain reliable numerical solutions. In this work, we first propose a local stabilization based on the macro-element approach often applied in the context of Stokes problem. Then, we concentrate on the efficient solution of the non-symmetric large-size algebraic systems obtained by the application of the stabilized formulation. A class of block preconditioners for accelerating the iterative convergence by Krylov subspace methods are developed based on sparse Schur-complement approximations and hybridization techniques. Robustness and efficiency of the proposed approach is demonstrated in both theoretical benchmarks and real-world applications.
25 mins
The fixed-stress splitting scheme for Biot's equations as a modified Richardson iteration: Implications for optimal convergence
Erlend Storvik, Jakub Wiktor Both, Jan Martin Nordbotten, Florin Adrian Radu
Abstract: The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanic subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arizing from discretizing the equations in the regime of impermeable porous media in order to obtain {\it a priori} knowledge of the optimal choice of this tuning/stabilization parameter.
25 mins
Polynomial Chaos and Hybrid High-Order methods for poroelasticity with random coefficients
Michele Botti, Daniele A. Di Pietro, Olivier Le Maître, Pierre Sochala
Abstract: In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a fi nite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We carry out an in-depth study of the dependence between the physical parameters, leading to an uncertainty model where the storage coefficient is expressed in terms of the other coefficients. We then discuss the approximation of the problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. The method is robust with respect to rough variations of the permeability and delivers inf-sup stability of the hydro-mechanical coupling. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.