Abstract
We consider L-scheme and Newton-based solvers for Biot model under large deformation. The mechanical deformation follows the Saint Venant-Kirchoff constitutive law. Furthermore, the fluid compressibility is assumed to be non-linear. A Lagrangian frame of reference is used to keep track of the deformation. We perform an implicit discretization in time (backward Euler) and propose two linearization schemes for solving the non-linear problems appearing within each time step: Newton’s method and L-scheme. Each linearization scheme is also presented in a monolithic and a splitting version, extending the undrained split methods to non-linear problems. The convergence of the solvers, here presented, is shown analytically for cases under small deformation and numerically for examples under large deformation. Illustrative numerical examples are presented to confirm the applicability of the schemes, in particular, for large deformation.
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Acknowledgments
The work was conducted as part of the Ph.D. thesis of Borregales Reverón M.A. at the University of Bergen. Nordbotten J.M. was funded in part by NRC grant number 250223. The research of Radu F.A. was partially supported by the VISTA project number AdaSim #6367. Kumar K. would like to acknowledge financial support from the project 811716 LAB2FIELD funded by the NRC. Open Access funding is provided by SINTEF AS.
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Appendix A: Convergence proof of the alternate Newton method
Appendix A: Convergence proof of the alternate Newton method
The following result provides the linear convergence of the alternate Newton method in Eqs. 28 and 29 for τ sufficiently small. In order to prove convergence , the following lemmas will be used.
Lemma 2
Let \(\left \lbrace x_{{k}} \right \rbrace _{{k}\geq 0}\) be a sequence of real positive number satisfying:
where a, b ≥ 0. Assuming that
holds, then the sequence \(\left \lbrace x_{{k}} \right \rbrace _{{k}\geq 0}\) converges to zero.
The result can be shown by induction; see p. 52 in [45] for more details.
Theorem 3
Assuming (A1)–(A4) and \(L_{s}\geq \frac {\alpha ^{2}}{\alpha _{\mathfrak {b}}}\), the alternate Newton splitting method in Eqs. 28 and 29 converges linearly if is small enough.τ
Proof
By subtracting problems Eqs. 28 and 29 and 16, taking as test functions \({\mathbf e}_{\textbf {q}}^{n,i}\), \(e_{p}^{n,i}\), and \({\mathbf e}_{\mathbf u}^{n,i}\), and rearranging some elements to the right-hand side we obtain:
The mechanics equation then gives:
By using similar steps as in Theorem 1, we obtain the following:
Next, by using the inverse inequality \(||\cdot ||_{L^{4}(\varOmega )}\leq C h^{-d/4}||\cdot ||\) [16], and by using the following formula \((x-y,x) = \displaystyle \frac {||x||^{2}}{2} +\frac {||x-y||^{2}}{2}-\frac { ||y||^{2}}{2}\), by choosing \(x = \boldsymbol \nabla \cdot {{\mathbf e}_{\mathbf u}}^{n,i}\) and \(y = \boldsymbol \nabla \cdot {\mathbf e}_{\mathbf u}^{n,i-1},\) we obtain from Eq. 34:
Finally, by reorganizing Eq. 35, using (A2) and choosing \(\gamma _{1}=\alpha _{\mathfrak {c}}\), we obtain the following inequality:
In a similar way, we obtain the following expression from Eq. 20:
By using Young’s inequality \((a,b)\leq \displaystyle \frac {||a||^{2}}{2\gamma }+\frac {\gamma ||b||^{2}}{2}\), for γ > 0 and choosing \(b = e_{p}^{n,i}\) and \(a = \boldsymbol \nabla \cdot \delta {\mathbf e}_{\mathbf u}^{n,i} \), we bound the coupling term (for γ2 > 0):
Then by using Eq. 39 and choosing \(\gamma _{2} = \frac {\alpha _{\mathfrak {b}}}{2}\), we obtain from Eq. 38:
Since \(L_{s}\geq \frac {\alpha ^{2}}{\alpha _{\mathfrak {b}}}\), we obtain:
By using \(||{\boldsymbol \nabla \cdot {\mathbf e}_{\mathbf u}^{n,0}}||\leq C\tau \), \(||{ {e}_{p}^{n,0}}||\leq C \tau \) which can be proven and the estimate in Lemma 2, the convergence is ensured if \(\tau = O (h^{\frac {d}{2}})\). □
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Borregales Reverón, M.A., Kumar, K., Nordbotten, J.M. et al. Iterative solvers for Biot model under small and large deformations. Comput Geosci 25, 687–699 (2021). https://doi.org/10.1007/s10596-020-09983-0
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DOI: https://doi.org/10.1007/s10596-020-09983-0