A fixed-stress type splitting method for nonlinear poroelasticity

TL;DR

The paper addresses nonlinear poroelasticity with a stress dependent permeability $K(div u)$ by extending the fixed-stress splitting to the Biot-type system and proving linear convergence for sufficiently small time steps. The method stabilizes the flow equation with a diagonal term controlled by $L$ and alternates between updating pressure and displacement, yielding contraction bounds in energy norms that are robust to parameter values when $K$ satisfies positivity, boundedness and Lipschitz continuity. The main theoretical contribution is a contraction result with explicit bounds, including a practical choice $L=1/(c_K^2+\lambda)$, where $c_K=1/\sqrt{d}$, and a limiting contraction factor independent of the Lamé and Biot coefficients as $\tau \to 0$. Numerical experiments on linear and nonlinear forms of $K$ on an L-shaped domain confirm the theory, showing stability and efficiency across mesh sizes, time steps and parameter regimes, with clear guidance on the stabilization parameter for robust convergence.

Abstract

In this paper we consider a nonlinear poroelasticity model that describes the quasi-static mechanical behaviour of a fluid-saturated porous medium whose permeability depends on the divergence of the displacement. Such nonlinear models are typically used to study biological structures like tissues, organs, cartilage and bones, which are known for a nonlinear dependence of their permeability/hydraulic conductivity on solid dilation. We formulate (extend to the present situation) one of the most popular splitting schemes, namely the fixed-stress split method for the iterative solution of the coupled problem. The method is proven to converge linearly for sufficiently small time steps under standard assumptions. The error contraction factor then is strictly less than one, independent of the Lamé parameters, Biot and storage coefficients if the hydraulic conductivity is a strictly positive, bounded and Lipschitz-continuous function.

Figures (7)

• Figure 1: Left - triangulation of $\Omega$, $h=1/16$, right - $\boldsymbol u$ for model (o), where $h=1/32$, $\tau=0.01$, $S=10^{-4}$, $K_0=10^{-6}$ and $\lambda=10^2$.
• Figure 2: Left - $p$ for model (o), right - $p$ for model (i). For both $h=1/32$, $\tau=0.01$, $S=10^{-4}$, $K_0=10^{-6}$ and $\lambda=10^2$, additionally for the non-linear model we have set $K_1=10^{-1}$.
• Figure 3: Number of fixed-stress iterations for the linear model where we have set $\tau = 0.01$, $S=10^{-4}$ and $h\in\{1/16,1/32,1/64,1/128\}$. Blue range of colors - $\lambda=10^1$, violet range of colors - $\lambda=10^2$ (single dotted line for $h=1/16$), red range of colors - $\lambda = 10^3$.
• Figure 4: Number of fixed-stress iterations for $K= K_0 + K_1 (\text{div$\,$} \boldsymbol u)^2$, $S=10^{-4}$ where we have set $\tau = 0.01$, $S=10^{-4}$ and $\lambda=10^2$. Red line - $K_0=10^{-8}$, gray line - $K_0=10^{-6}$, yellow line - $K_0 = 10^{-4}$, blue line - $K_0=10^{-2}$. The blue-green line shows the results for a different stopping criterion, namely a residual reduction by factor $10^{6}$, which are identical for the four different choices of $K_0$.
• Figure 5: Number of fixed-stress iterations for $K= K_0 + K_1 (\text{div$\,$} \boldsymbol u)^2$. Red line - $\tau=0.0001$, orange line - $\tau = 0.0005$, black line - $\tau=0.001$, violet line - $\tau=0.005$, blue line - $\tau=0.01$, grey line - $\tau=0.05$, green line - $\tau=0.1$. Missing points means that the algorithm does not converge for the corresponding parameter values.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 4
• Theorem 3
• proof