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Semi-explicit time discretization for linear thermo-poroelasticity

R. Altmann, R. Maier, J. Schmeck

Abstract

Within this paper, we introduce partially and fully decoupled time stepping schemes for linear thermo-poroelasticity. This means that the mechanics, heat, and flow equations can be solved sequentially. We provide sufficient conditions on the material parameters, which can be checked a priori, guaranteeing first-order convergence of the introduced schemes. Hence, the proposed methods have the same order as the implicit Euler scheme but are computationally more efficient due to the decoupling of the system equations. Numerical examples validate the proven convergence results and analyze the sharpness of the mentioned parameter condition. Further, we compare the schemes with other decoupling schemes from the literature.

Semi-explicit time discretization for linear thermo-poroelasticity

Abstract

Within this paper, we introduce partially and fully decoupled time stepping schemes for linear thermo-poroelasticity. This means that the mechanics, heat, and flow equations can be solved sequentially. We provide sufficient conditions on the material parameters, which can be checked a priori, guaranteeing first-order convergence of the introduced schemes. Hence, the proposed methods have the same order as the implicit Euler scheme but are computationally more efficient due to the decoupling of the system equations. Numerical examples validate the proven convergence results and analyze the sharpness of the mentioned parameter condition. Further, we compare the schemes with other decoupling schemes from the literature.
Paper Structure (25 sections, 11 theorems, 99 equations, 5 figures)

This paper contains 25 sections, 11 theorems, 99 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Thermo--poroelasticity
  3. Classical poroelasticity
  4. Linear thermo--poroelasticity
  5. Implicit time discretization
  6. An Abstract Convergence Result
  7. Half-Decoupled Schemes
  8. HF--M scheme from BruABNR20
  9. Semi-explicit discretization (M--HF)
  10. A semi-explicit discretization with inner iteration (M--HF)
  11. Fully Decoupled Schemes
  12. M--F--H scheme from KolV17
  13. H--F--M scheme from BruABNR20
  14. F--H--M scheme from BruABNR20
  15. Fully decoupled semi-explicit discretization (M--F--H)
  16. ...and 10 more sections

Key Result

Theorem 2.3

Let Assumption ass:operatorsI hold and consider initial data $p(0)=p^0\in {\mathcal{H}_{\mathcal{Q} }}$ together with right-hand sides $f\in H^1(0,T;\mathcal{V}^*)$, $g\in L^2(0,T;\mathcal{Q} ^*)$. Then there exists a unique (weak) solution pair

Figures (5)

  • Figure 6.1: Convergence history for the half-decoupled schemes for the geothermal model problem from Section \ref{['sec:num:geothermal']} with fixed mesh size $h=0.125$. The dash-dotted line shows order $1$.
  • Figure 6.2: Convergence history for the fully decoupled schemes for the geothermal model problem from Section \ref{['sec:num:geothermal']} with fixed mesh size $h=0.125$. The dash-dotted line shows order $1$.
  • Figure 6.3: Relative errors at the final time point $t=T=1$ for different spatial mesh sizes, namely $h=0.0625$ (dash-dotted), $h=0.0312$ (solid), $h=0.0156$ (dashed), and $0.0078$ (dotted). For the spatial discretization, we use $P1$--$P1$--$P1$ (left) and $P2$--$P1$--$P1$ finite elements (right).
  • Figure 6.4: Illustration of the sharpness of the weak coupling condition guaranteeing convergence for the half decoupled scheme \ref{['eq:halfDecoupled:semiExpl']}.
  • Figure 6.5: Illustration of the sharpness of the weak coupling condition guaranteeing convergence for the fully decoupled scheme \ref{['eq:fullyDecoupled:semiExpl']}.

Theorems & Definitions (29)

  • Remark 2.2
  • Theorem 2.3: weak solution, poroelasticity
  • proof
  • Remark 2.4
  • Remark 2.6
  • Lemma 2.8: properties of $\mathbb{B}$, $\mathbb{C}$, and $\mathbb{D}$
  • proof
  • Theorem 2.9: weak solution, thermo--poroelasticity
  • proof
  • Theorem 2.10: well-posedness of the implicit Euler scheme
  • ...and 19 more