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Unconditionally stable second order convergent partitioned methods for multiple-network poroelasticity

Jeonghun J. Lee

TL;DR

This work tackles quasi-static MPET equations modeling multiple fluid networks in poroelastic media. It introduces two partitioned time-stepping schemes that decouple elasticity from diffusion by solving a Lamé subproblem and a parabolic network system in either order, avoiding iterative coupling. The authors prove unconditional stability and second- or third-order convergence in time for the elasticity-then-diffusion and diffusion-then-elasticity schemes, respectively, with an error analysis that remains robust as $\lambda$ grows large and storage vanishes. Numerical experiments on manufactured solutions corroborate the theory and reveal occasional superconvergence, highlighting the practical efficiency and reliability of the proposed methods in near-incompressible regimes.

Abstract

In this paper, we consider partitioned numerical methods for quasi-static multiple-network poroelasticity (MPET) equations, generalizations of the Biot model in poroelasticity for multiple pore networks. Two partitioned numerical methods are presented for the equations which split time discretization into solving two subequations, a Lame equation and a system of heat equations, alternatively. In contrast to the iterative coupling methods which require multiple iterations at each time step, our numerical methods solve these smaller equations only once at each time step. We prove their unconditional stability and high order convergence in time with a novel error analysis. A number of numerical results are presented to illustrate good performances of these partitioned methods.

Unconditionally stable second order convergent partitioned methods for multiple-network poroelasticity

TL;DR

This work tackles quasi-static MPET equations modeling multiple fluid networks in poroelastic media. It introduces two partitioned time-stepping schemes that decouple elasticity from diffusion by solving a Lamé subproblem and a parabolic network system in either order, avoiding iterative coupling. The authors prove unconditional stability and second- or third-order convergence in time for the elasticity-then-diffusion and diffusion-then-elasticity schemes, respectively, with an error analysis that remains robust as grows large and storage vanishes. Numerical experiments on manufactured solutions corroborate the theory and reveal occasional superconvergence, highlighting the practical efficiency and reliability of the proposed methods in near-incompressible regimes.

Abstract

In this paper, we consider partitioned numerical methods for quasi-static multiple-network poroelasticity (MPET) equations, generalizations of the Biot model in poroelasticity for multiple pore networks. Two partitioned numerical methods are presented for the equations which split time discretization into solving two subequations, a Lame equation and a system of heat equations, alternatively. In contrast to the iterative coupling methods which require multiple iterations at each time step, our numerical methods solve these smaller equations only once at each time step. We prove their unconditional stability and high order convergence in time with a novel error analysis. A number of numerical results are presented to illustrate good performances of these partitioned methods.

Paper Structure

This paper contains 17 sections, 3 theorems, 87 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Sobolev spaces
  4. Spaces involving time
  5. Finite element spaces
  6. Parameter values
  7. Boundary conditions
  8. Initial conditions
  9. The formulation with total pressure
  10. Governing equations introducing the total pressure
  11. Variational formulation
  12. Partitioned numerical methods with error analysis
  13. Preliminaries
  14. The first partitioned method (elasticity-then-diffusion)
  15. The second partitioned algorithm (diffusion-then-elasticity)
  16. ...and 2 more sections

Key Result

Theorem 1

For given compatible initial data $(\boldsymbol{u}^0, p_{t}^0, \boldsymbol{p}^0)$ and given $\boldsymbol{f}$ and $\{ g_i \}_{i=1}^N$, suppose that $(\boldsymbol{u}, p_{t}, \boldsymbol{p})$ is the solution of eq:mpet:tp. For numerical initial data $(\boldsymbol{u}_h^0, p_{t,h}^0, \boldsymbol{p}_h^0)$ and the implicit constants in these inequalities depend on the norms of the exact solutions and are

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 1
  • proof