Maria Lymbery

Qinggou Hong, Johannes Kraus, Maria Lymbery et al.

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 $K_i$, storage coefficients $c_{p_i}$, network transfer coefficients $β_{ij}, i,j=1,\cdots,n$, the scale of the networks $n$ and the time step size $τ$. Moreover, strongly mass conservative discretizations that meet the required conditions for parameter-robust 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.

Johannes Kraus, Raytcho Lazarov, Maria Lymbery et al.

In this paper we propose and analyze a preconditioner for a system arising from a finite element approximation of second order elliptic problems describing processes in highly het- erogeneous media. Our approach uses the technique of multilevel methods and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus (see [8]). The main results are the design and a theoretical and numerical justification of an iterative method for such problems that is robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient (related to the permeability/conductivity).

Qingguo Hong, Johannes Kraus, Maria Lymbery et al.

We consider flux-based multiple-porosity/multiple-permeability poroelasticity systems describing multiple-network flow and deformation in a poro-elastic medium, sometimes also referred to as MPET models. The focus of the paper is on the convergence analysis of the fixed-stress split iteration, a commonly used coupling technique for the flow and mechanics equations in poromechanics. We formulate the fixed-stress split method in the present context and prove its linear convergence. The contraction rate of this fixed-point iteration does not depend on any of the physical parameters appearing in the model. The theory is confirmed by numerical results which further demonstrate the advantage of the fixed-stress split scheme over a fully implicit method relying on norm-equivalent preconditioning.

Maria Lymbery

This research explores the application of the auxiliary space multigrid method (ASMG) that is based on additive Schur complement approximation (ASCA) to graph Laplacian matrices arising from general graphs. A major predicament when considering algebraic multigrid (AMG) methods on such graphs is the choice of a general coarsening strategy which has to be both cheap and effective. Such a strategy has been incorporated in the presented approach which in addition has several advantages. First, it is purely algebraic in its construction which makes the algorithm easy to implement. Furthermore, the approach requires no limitation on the graph's structure and itself can be adjusted to the particular problem. Last but not least, its computational complexity can be easily analysed. A demonstrative set of numerical experiments is presented.