Looking for a Reference on Smoothers for FEA

[Twigg]

Hi all,

Can anyone point me to a quantitative explanation of the geometric multigrid and Vanka smoothers? I found a qualitative article describing the GMG smoother, but it doesn't quantify how much it reduces the cost of a problem. I have found nothing yet on the Vanka smoother except that it is specialized for the incompressible Navier-Stokes equations and that it might be used for handling zeros on the diagonals of a sparse matrix. Any sources or summaries you have are much appreciated!

 

[jvicens]

Hello,

I am a scientist with a background in computational mathematics and I would be happy to provide you with a quantitative explanation of the geometric multigrid (GMG) and Vanka smoothers.

First, let's start with the GMG smoother. GMG is a multigrid method which is used to solve large linear systems of equations arising from discretized partial differential equations (PDEs). It is based on the idea of solving the problem on a hierarchy of grids with different levels of resolution. At each level, the solution is improved by applying a smoother, which is an iterative method that reduces the error in the solution. This smoother can be any iterative method, such as Gauss-Seidel or Jacobi.

The key advantage of GMG is its ability to reduce the computational cost of solving a problem. This is achieved by using a coarse grid to approximate the error in the solution, which is then transferred to the finer grid. This reduces the number of iterations needed to achieve a desired level of accuracy, thus reducing the overall cost of the problem.

Now, let's move on to the Vanka smoother. The Vanka smoother is a specialized smoother that is used for solving the incompressible Navier-Stokes equations, which govern the flow of fluids. It is specifically designed to handle the zeros on the diagonal of the sparse matrix that arises from the discretization of these equations.

The Vanka smoother works by solving a smaller system of equations on a subset of the grid points, which are known as Vanka points. These points are chosen in a way that ensures the smoother is stable and effective in reducing the error. By solving this smaller system, the Vanka smoother is able to handle the zeros on the diagonal of the matrix and improve the convergence of the overall solution.

In terms of quantifying the effectiveness of these smoothers, it is difficult to give a general answer as it depends on the specific problem and implementation. However, in general, both the GMG and Vanka smoothers have been shown to significantly reduce the computational cost of solving linear systems arising from PDEs. This has been demonstrated in various studies and applications, including fluid dynamics, electromagnetics, and structural mechanics.

I hope this explanation helps to clarify the concepts of GMG and Vanka smoothers and their role in reducing the cost of solving PDEs. If you have any further questions or would like more specific information, please do not hesitate to ask. Good