Optimal $L^2$-norm error estimate of multiphysics finite element method for poroelasticity model and simulating brain edema

TL;DR

This work addresses an optimal $L^2$-norm error estimate for a multiphysics finite element method (MFEM) applied to the Biot poroelasticity model and demonstrates brain edema simulations using the reformulated system. A reformulation introduces variables $q=div\mathbf{u}$, $\eta=c_0 p + \alpha q$, $\xi=\alpha p - \lambda q$, and expresses $p$ and $q$ through $p = \kappa_1\xi + \kappa_2\eta$, $q = \kappa_1\eta - \kappa_3\xi$ with the constants $\kappa_1,\kappa_2,\kappa_3$ defined from $\alpha,\lambda,$ and $c_0$. The MFEM algorithm (Algorithm 2.1) solves for $(\mathbf{u}_h^{n+1},\xi_h^{n+1},\eta_h^{n+1})$ in a first step and updates $(p_h^{n+1},q_h^{n+1})$ in a second, with coupling controlled by $\theta$; the analysis uses projections and an auxiliary problem to establish error bounds. The main results show that the displacement error in $L^2$-norm attains $O(h^3)$ and the pressure error attains $O(h^2)$, with a time-step requirement $\Delta t = O(h^2)$ for the decoupled case ($\theta=0$) and no such restriction for the coupled case ($\theta=1$). Numerical tests on manufactured solutions validate the predicted rates, and brain edema simulations reveal that the permeability $K$ most strongly influences intracranial pressure and tissue deformation, while Young's modulus $E$ and Poisson ratio $\nu$ mainly affect deformation magnitude and edema development speed, respectively. Overall, the work provides a rigorous error framework for MFEM in poroelasticity and yields mechanistic insights into parameter effects on brain edema, with implications for modeling and therapy.

Abstract

In this paper, we derive an optimal $L^2$-norm error estimate of the multiphysics finite element method for the poroelasticity model by introducing an auxiliary problem. We show some numerical tests to verify the theoretical result and apply the multiphysics finite element method to simulate the brain edema which caused by abnormal accumulation of cerebrospinal fluid in injured areas. And we investigate the effects of the key physical parameters on brain edema and observed that the permeability $K$ has the biggest influence on intracranial pressure and tissue deformation, Young's modulus $E$ and Poisson ratio $ν$ have little effect on the maximum value of intracranial pressure, but have great effect on the tissue deformation and the developing speed of brain edema.

Figures (13)

• Figure 3.1: (a) surface plot of the computed pressure $p$ and (b) arrow plot of the computed displacement $\mathbf{u}$ at $T$.
• Figure 3.2: Surface plot of the computed displacement (a) $u_1$ and (b) $u_2$ at the terminal time $T$, respectively.
• Figure 3.3: The solution $p$ at the terminal time $T$: (a) the exact solution, (b) the numerical solution.
• Figure 3.4: Surface plots of the computed displacement (a) $u_1$ and (b) $u_2$ at the terminal time $T$, respectively.
• Figure 3.5: Arrow plot of the computed displacement $\mathbf{u}$ at $T$.

Theorems & Definitions (8)

• Lemma 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Remark 2.1
• Theorem 2.4
• Theorem 2.5
• proof