Robust Preconditioners for Perturbed Saddle-Point Problems and Conservative Discretizations of Biot's Equations Utilizing Total Pressure (original) (raw)
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Numerical Linear Algebra With Applications, 2019
The parameters in the governing system of partial differential equations of multicompartmental poroelastic models typically vary over several orders of magnitude making its stable discretization and efficient solution a challenging task. In this paper, inspired by the approach recently presented by Hong and Kraus [Parameter-robust stability of classical three-field formulation of Biot's consolidation model, ETNA (to appear)] for the Biot model, we prove the uniform stability, and design stable disretizations and parameter-robust preconditioners for flux-based formulations of multiple-network poroelastic systems. Novel parameter-matrix-dependent norms that provide the key for establishing uniform inf-sup stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ, but also with respect to all the other model parameters such as permeability coefficients Ki, storage coefficients cp i , network transfer coefficients βij , i, j = 1, • • • , n, the scale of the networks n and the time step size τ. Moreover, strongly mass conservative discretizations that meet the required conditions for parameterrobust stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm-equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.
Weakly Imposed Symmetry and Robust Preconditioners for Biot’s Consolidation Model
Computational Methods in Applied Mathematics
We discuss the construction of robust preconditioners for finite element approximations of Biot’s consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger–Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.
Preconditioning Mixed Finite Element Saddle‐point Elliptic Problems
Numerical Linear Algebra with Applications, 1996
We consider saddle-point problems that typically arise from the mixed finite element discretization of second order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form R Ω a ?1 + 1 r r ( is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditionerfor the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.
An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations
International Journal for Numerical Methods in Engineering, 2002
Finite element simulations of very large-scale soil-structure interaction problems (e.g. excavations, tunnelling, pile-rafts, etc.) typically involve the solution of a very large, ill-conditioned, and indeÿnite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very ine cient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely e ective for solving very large-scale Biot's ÿnite element equations using the symmetric quasi-minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an 'element-by-element' framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The e ectiveness of the GJ preconditioner over a wide range of soil sti ness and permeability was demonstrated numerically using a simple three-dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coe cient matrix.
Preconditioning saddle-point problems arising from mixed nite element discretization of elliptic pro
1992
We consider saddle-point problems that typically arise from the mixed finite element discretization of second order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form R Ω a ?1 + 1 r r ( is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditionerfor the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.
A Preconditioned Scheme for Nonsymmetric Saddle-Point Problems
High-Performance Scientific Computing, 2012
In this paper, we present an effective preconditioning technique for solving nonsymmetric saddle-point problems. In particular, we consider those saddlepoint problems that arise in the numerical simulation of particulate flows-flow of solid particles in incompressible fluids, using mixed finite element discretization of the Navier-Stokes equations. These indefinite linear systems are solved using a preconditioned Krylov subspace method with an indefinite preconditioner. This creates an inner-outer iteration, in which the inner iteration is handled via a preconditioned Richardson scheme. We provide an analysis of our approach that relates the convergence properties of the inner to the outer iterations. Also "optimal" approaches are proposed for the implicit construction of the Richardson's iteration preconditioner. The analysis is validated by numerical experiments that demonstrate the robustness of our scheme, its lack of sensitivity to changes in the fluid-particle system, and its "scalability".
International Journal for Numerical and Analytical Methods in Geomechanics, 2010
Large-scale simulations of flow in deformable porous media require efficient iterative methods for solving the involved systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in geological applications, such as basin evolution at regional scales. The success of iterative methods for this type of problems depends strongly on finding effective preconditioners. This paper investigates how the block-structured matrix system arising from single-phase flow in elastic porous media should be preconditioned, in particular for highly discontinuous permeability and significant jumps in elastic properties. The most promising preconditioner combines algebraic multigrid with a Schur complement-based exact block decomposition. The paper compares numerous block preconditioners with the aim of providing guidelines on how to formulate efficient preconditioners.
A major computational issue in the Finite Element (FE) integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system. Because of ill-conditioning, the iterative solution, which is recommended in large size 3D settings, requires the computation of a suitable preconditioner to guarantee convergence. In this paper the coupled system is solved by a Krylov subspace method preconditioned by a Relaxed Mixed Constraint Preconditioner (RMCP) which is a generalization based on a parameter ω of the Mixed Constraint Preconditioner (MCP) developed in . Choice of optimal ω is driven by the spectral distribution of suitable symmetric positive definite (SPD) matrices. Numerical tests performed on realistic 3D problems reveal that RMCP accelerates Krylov subspace solvers by a factor up to three with respect to MCP.
Performance Analysis of Block AMG Preconditioning of Poroelasticity Equations
Lecture Notes in Computer Science, 2015
The goal of this study is to develop, analyze, and implement efficient numerical algorithms for equations of linear poroelasticity, a macroscopically diphasic description of coupled flow and mechanics. We suppose that the solid phase is governed by the linearized constitutive relationship of Hooke's law. Assuming in addition a quasi-steady regime of the fluid structure interaction, the media is described by the Biot's system of equations for the unknown displacements and pressure (u, p). A mixed Finite Element Method (FEM) is applied for discretization. Linear conforming elements are used for the displacements. Following the approach of Arnold-Brezzi, non-conforming FEM approximation is applied for the pressure where bubble terms are added to guarantee a local mass conservation. Block-diagonal preconditioners are used for iterative solution of the arising saddle-point linear algebraic system. The BiCGStab and GMRES are the basic iterative schemes, while algebraic multigrid (AMG) is utilized for approximation of the diagonal blocks. The HYPRE implementations of BiCGStab, GMRES and AMG (BoomerAMG, [6]) are used in the presented numerical tests. The aim of the performance analysis is to improve both: (i) the convergence rate of the solvers measured by the iteration counts, and (ii) the CPU time to solve the problem. The reported results demonstrate some advantages of GMRES for the considered real-life, large-scale, and strongly heterogeneous test problems. Significant improvement is observed due to tuning of the BoomerAMG settings.