Adaptive Mesh Refinement for Error Estimation in Finite Element Methods

In this article, we discuss the challenges of modeling complex fluid dynamics in engineering applications, particularly when dealing with both 0D (flat domain) and 2D (surface domain) models. We propose a novel approach to estimate the error between these two models using an "a posteriori" method, which means we use information from the solution of the problem to estimate the error. This allows us to adaptively refine the mesh in certain areas where the error is higher, resulting in more accurate solutions.
Our approach builds upon existing techniques for model coupling, such as those used in the literature by Miglio et al. (2005). However, we introduce a new strategy for adaptive mesh refinement using the "adaptmesh" function in FreeFEM, a popular software library for numerical simulations. This approach enables us to efficiently and effectively refine only the regions of the mesh where needed, without affecting the rest of the mesh.
To demonstrate the effectiveness of our method, we perform numerical experiments on several challenging benchmark problems, including the Couette flow problem and the incompressible Navier-Stokes equation with a nonlinear source term. Our results show that our adaptive mesh refinement technique can significantly improve the accuracy of the solution compared to a fixed mesh, while also reducing computational time.
In summary, this article presents a novel approach for estimating the error between 0D and 2D models in fluid dynamics simulations, allowing for adaptive mesh refinement and improved accuracy. By leveraging the "adaptmesh" function in FreeFEM, we can efficiently and effectively refine only the regions of the mesh where needed, resulting in more accurate solutions with reduced computational time.