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

TL;DR

This paper introduces and validates a stabilized finite element method for flow problems from 4D flow MRI, enabling accurate velocity and pressure estimation while avoiding invasive procedures.

Contribution

It develops a stabilized finite element approach for equal-order velocity-pressure approximation in MRI flow analysis, with proven stability and error estimates.

Findings

The method provides stable and accurate solutions for linearized and nonlinear models.

Numerical experiments confirm optimal error estimates and practical effectiveness.

The approach offers a non-invasive alternative for pressure reconstruction from MRI data.

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.