An accurate finite element method for the numerical solution of isothermal and incompressible flow of viscous fluid

TL;DR

This paper introduces a new finite element method for viscous, incompressible fluid flow that avoids stabilization techniques, offering a robust and accurate solution with straightforward extension to multiphysics problems.

Contribution

A novel finite element approach that solves viscous fluid flow using only balance equations, eliminating the need for stabilization or splitting schemes.

Findings

Method achieves high accuracy compared to closed-form solutions.

Robustness demonstrated through benchmark problems.

Implementation using open-source packages.

Abstract

Despite its numerical challenges, finite element method is used to compute viscous fluid flow. A consensus on the cause of numerical problems has been reached; however, general algorithms---allowing a robust and accurate simulation for any process---are still missing. Either a very high computational cost is necessary for a direct numerical solution (DNS) or some limiting procedure is used by adding artificial dissipation to the system. These stabilization methods are useful; however, they are often applied relative to the element size such that a local monotonous convergence is challenging to acquire. We need a computational strategy for solving viscous fluid flow using solely the balance equations. In this work, we present a general procedure solving fluid mechanics problems without use of any stabilization or splitting schemes. Hence, its generalization to multiphysics applications is straightforward. We discuss emerging numerical problems and present the methodology rigorously. Implementation is achieved by using open-source packages and the accuracy as well as the robustness is demonstrated by comparing results to the closed-form solutions and also by solving well-known benchmarking problems.