On the Use of Anisotropic Triangles with Mixed Finite Elements: Application to an “Immersed” Approach for Incompressible Flow Problems

In this chapter, we discuss the use of some common mixed finite elements in the context of a locally anisotropic remeshing strategy, close in philosophy to "immersed" approaches for interface problems. A characteristic of the present method is the presence of highly flat triangles. Such a distinctive feature may imply stability issues for mixed elements with incompressible flow problems. First, we present a review of the literature dealing with interface problems and we illustrate these results with a simple 1D framework alongside of numerical tests. Second, we present the locally anisotropic remeshing approach for interface problems in 2D with a focus on the incompressible Stokes problem. We then present numerical tests to show stability issues of common mixed elements, as well as possible stable ones. We also deal with conditioning issues. Finally, we illustrate the results with two applications, including the fluid-structure interaction of a rotational rigid bar.