Marriage of least-squares and Galerkin FEM to solve Inompressible flows.

What is it about?

It is about solving Navier-Stokes equations in fluid dynamics using a novel finite element method that combines the classic Galerkin method and the least-squares finite element methods. this method has been formulated in such a way that it avoids the negative aspects of the two methods and takes advantage of their positive aspects.

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Why is it important?

The least-squares finite element method (LSFEM), which is based on minimizing the l2 -norm of the residual, has many attractive advantages over Galerkin finite element method (GFEM). It is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations, unlike GFEM. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have prevented the application of LSFEM to large-scale practical problems, such as three-dimensional compressible viscous flows. A simple finite element method is proposed that combines the positive of the LSFEM and the Galerkin method and avoids their above-mentioned drawbacks.

Perspectives