Trygve Bærland

When solving a multi-physics problem one often decomposes a monolithic system into simpler, frequently single-physics, subproblems. A comprehensive solution strategy may commonly be attempted, then, by means of combining strategies devised for the constituent subproblems. When decomposing the monolithic problem, however, it may be that requiring a particular scaling for one subproblem enforces an undesired scaling on another. In this manuscript we consider the H(div)-based mixed formulation of the Darcy problem as a single-physics subproblem; the hydraulic conductivity, K, is considered intrinsic and not subject to any rescaling. Preconditioners for such porous media flow problems in mixed form are frequently based on H(div) preconditioners rather than the pressure Schur complement. We show that when the hydraulic conductivity, K, is small the pressure Schur complement can also be utilised for H(div)-based preconditioners. The proposed approach employs an operator preconditioning framework to establish a robust, K- and h-uniform block preconditioner. The mapping properties of the continuous operator are a key component in applying the theoretical framework point of view. As such, a main challenge addressed here is establishing a K-uniform inf-sup condition with respect to appropriately weighted Hilbert intersection- and sum-spaces.

Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would therefore be useful to have a preconditioner for the fractional Laplacian. In this work, we develop an additive multigrid preconditioner for the fractional Laplacian with positive fractionality and show a uniform bound on the condition number. For the case of negative fractionality, we reuse the preconditioner developed for the positive fractionality and left-right multiply a regular Laplacian with a preconditioner with positive fractionality to obtain the desired negative fractionality. Implementational issues are outlined in detail as the differences between the discrete operators and their corresponding matrices must be addressed when realizing these algorithms in code. We finish with some numerical experiments verifying the theoretical findings.

Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive- and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would then be useful to have a preconditioner for the fractional Laplacian. In this work, we develop an additive multigrid preconditioner for the fractional Laplacian with positive fractionality, and show a uniform bound on the condition number. For the case of negative fractionality, we re-use the preconditioner developed for the positive fractionality and left-right multiply a regular Laplacian with a preconditioner with positive fractionality to obtain the desired negative fractionality. Implementational issues are outlined in details as the differences between the discrete operators and their corresponding matrices must be addressed when realizing these algorithms in code. We finish with some numerical experiments verifying the theoretical findings.

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.

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.