Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method
Faisal Fairag
TL;DR
The paper develops and analyzes a two-level finite-element method for the stream-function formulation of the incompressible Navier–Stokes equations, enabling a nonlinear solve on a coarse mesh followed by a linear solve on a fine mesh to approximate high-Re flows efficiently. It proves a superlinear relation between coarse- and fine-mesh errors and provides an error bound $|\psi-\psi^h|_2 \le C\{ \inf_{w^h \in X^h} |\psi-w^h|_2 + |\ln h|^{1/2} |\psi-\psi^H|_1 \}$, along with practical mesh-scaling rules $h = O(H^{3/2} |\ln H|^{1/4})$ for common $C^1$ elements. The method is implemented with conforming $C^1$ elements (Argyris, Clough–Tocher, Bogner–Fox–Schmit, Bicubic Spline) and demonstrated on three numerical problems, showing substantial computational savings while maintaining accuracy comparable to a traditional one-level approach. The results indicate the approach is robust for increasing $Re$ and effective for classic benchmark flows such as the driven cavity, highlighting its practical potential for efficient high-Re incompressible flow simulations in the stream-function framework.
Abstract
We consider two-level finite element discretization methods for the stream function formulation of the Navier-Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh, then solving a linear system on the fine mesh. The basic result states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the two-level method can be implemented to approximate efficiently solutions to the Navier-Stokes equations. Two fluid flow calculations are considered to test problems which have a known solution and the driven cavity problem. Stream function contours are displayed showing the main features of the flow.