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Register a new userRobust block preconditioners for poroelasticity
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Added by
Qingguo Hong
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper we study the linear systems arising from discretized poroelasticity problems. We formulate one block preconditioner for the two-filed Biot model and several preconditioners for the classical three-filed Biot model under the unified relationship framework between well-posedness and preconditioners. By the unified theory, we show all the considered preconditioners are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.
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In this paper, two types of Schur complement based preconditioners are studied for twofold and block tridiagonal saddle point problems. One is based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We discuss different preconditioners incorporating the exact Schur complements. It is shown that some of them will lead to positive stable preconditioned systems. Our theoretical analysis is instructive for devising various exact and inexact preconditioners, as well as iterative solvers for many twofold and block tridiagonal saddle point problems.
Linear poroelasticity models have a number of important applications in biology and geophysics. In particular, Biots consolidation model is a well-known model that describes the coupled interaction between the linear response of a porous elastic medium and a diffusive fluid flow within it, assuming small deformations. Although deterministic linear poroelasticity models and finite element methods for solving them numerically have been well studied, there is little work to date on robust algorithms for solving poroelasticity models with uncertain inputs and for performing uncertainty quantification (UQ). The Biot model has a number of important physical parameters and inputs whose precise values are often uncertain in real world scenarios. In this work, we introduce and analyse the well-posedness of a new five-field model with uncertain and spatially varying Youngs modulus and hydraulic conductivity field. By working with a properly weighted norm, we establish that the weak solution is stable with respect to variations in key physical parameters, including the Poisson ratio. We then introduce a novel locking-free stochastic Galerkin mixed finite element method that is robust in the incompressible limit. Armed with the `right norm, we construct a parameter-robust preconditioner for the associated discrete systems. Our new method facilitates forward UQ, allowing efficient calculation of statistical quantities of interest and is provably robust with respect to variations in the Poisson ratio, the Biot--Willis constant and the storage coefficient, as well as the discretization parameters.
We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge-Kutta methods. Recasting the semi-discrete solution as the minimizer of a properly defined energy functional, the proof of convergence uses its alternating minimization. The scheme is closely related to the undrained split for the quasi-static Biot system.
Using the framework of operator or Cald{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s in [0,2]$ discretized with continuous finite (or boundary) elements. The cost of the preconditioner is the cost of the application an elliptic opposite order operator discretized with discontinuous or continuous finite elements on the same mesh, plus minor cost of linear complexity. Herewith the construction of a so-called dual mesh is avoided.
Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, these preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and non-affine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare truncation preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones.
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