A stabilized finite element method for a flow problem arising from 4D flow magnetic resonance imaging

TL;DR

The paper addresses reconstructing and assessing the quality of 4D Flow MRI velocity data by formulating a stabilized finite element method for a flow problem derived from the incompressible Navier–Stokes equations. By decomposing the velocity into MRI data $\boldsymbol{u}_m$ and a noise component $\boldsymbol{w}$, it yields a linearized Oseen-type model, which is discretized with equal-order velocity–pressure spaces stabilized to ensure stability and convergence. The authors prove well-posedness and optimal-order error estimates for the linearized problem and demonstrate, through numerical experiments, accurate pressure reconstruction and robust data-perturbation performance on realistic geometries. The work provides a practically efficient framework for MRI data quality assessment and pressure estimation in 4D Flow MRI applications, with potential extensions to time-dependent problems and pressure-robust discretizations.

Abstract

In this work we propose, {analyze}, and validate a stabilized finite element method for a flow problem arising from the assessment of {4D Flow Magnetic Resonance Imaging quality}. Starting from the Navier-Stokes equation and splitting its velocity as the MRI-observed one (considered a datum) plus an ``observation error'', a modified Navier-Stokes problem is derived. This procedure allows us to estimate the quality of the measured velocity fields, while also providing an alternative approach to pressure reconstruction, thereby avoiding invasive procedures. Since equal-order approximations have become a popular choice for problems linked to pressure recovery from MRI images, we design a stabilized finite element method allowing equal-order interpolations for velocity and pressure. In the linearized version of the resulting model, we prove stability and (optimal order) error estimates and test the method with a variety of numerical experiments testing both the linearized case and the more realistic nonlinear one.

Figures (12)

• Figure 1: Kovasznay flow for the model problem with $\boldsymbol{a}_h = 0.9\boldsymbol{I}_h({\boldsymbol w})$. Convergence histories for linear elements ($k=1$) (left), quadratic elements ($k=2$) (center) and cubic elements (right). From top to bottom the values of $\mu$ are $1, 10^{-1}, 10^{-2},$ and $10^{-3}$.
• Figure 2: Kovasznay flow for the perturbed Navier--Stokes problem with $\boldsymbol{a}_h = {\boldsymbol w}_h$. Convergence histories for linear elements ($k=1$) (left), quadratic elements ($k=2$) (center) and cubic elements (right). From top to bottom the values of $\mu$ are $1, 10^{-1}, 10^{-2},$ and $10^{-3}$.
• Figure 3: Pressure reconstructed on a bent computational domain. Top: Quadrilateral mesh (left) and map of the distribution of the random velocity field ${\boldsymbol u}$ (right). Bottom: Triangular criss-cross mesh (left) and the interpolated velocity ${\boldsymbol u}_m\in \boldsymbol{H}_h$, with $k=1$ (right).
• Figure 4: Pressure reconstructed on a bent computational domain. Top: Exact solution ${\boldsymbol w}$ (left) and computed solution ${\boldsymbol w}_h$ (right). Bottom: Exact solution $p$ (left) and computed solution $p_h$ (right). Here, we use linear elements ($k=1$) on a mesh formed by 2.772 triangles.
• Figure 5: Domain computational and boundary conditions.

Theorems & Definitions (16)

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