Mesh Adaptation for Computational Fluid Dynamics, Volume 1

5Multiscale Adaptation for Steady Simulations

Multiscale adaptation has been previously defined for the interpolation of known analytic or continuous functions. The multiscale adaptation for PDE works on the same basis as an exact analytic function, with the difference that information is derived from numerical solutions. In a feature-based approach, we choose a special field, the sensor, generally identical to the solution when it is a scalar field. The derivatives of the sensor need to be computed. This is permitted by recovery methods. On the new mesh, we need a good initial solution for recomputing the numerical solution, and this is done by interpolation methods.

5.1. Introduction

While previous chapters address the multiscale adaptation to a known function, this chapter focuses on the application of mesh adaptation to numerical simulations. The relation between a numerical solution and the derivation of a better mesh is very frequently based on the transformation of the existing mesh, using information derived from the numerical solution already computed. We give in the notes (section 5.9) a short review of these approaches. In contrast, in the standpoint which is proposed by this chapter, the mesh is defined as an approximate of the best mesh for the exact solution.

Then two mains issues have to be addressed:

1. i) the solution u of the problem is not known, neither are its linear interpolate Π_hu and its Hessian H_u. We only know the numerical solution u_h and we want to ...