A Helmholtz--Hodge projection method using an iterative gauge computation to solve the 3D generalized Stokes problem
Résumé
The generalized Stokes problem (GSP) is broadly recognized as a keystone in implicit or semi-implicit discretizations of the Navier--Stokes equations (NSE), either incompressible or of low Mach number type, i.e., for which there exists a constraint on the vector field. The GSP is also known as the steady Darcy--Brinkman model for flows in porous media or in Hele--Shaw configurations. Up to now, only pressure- (preconditioned) Uzawa methods claimed to solve the three-dimensional (3D) GSP exactly (i.e., without introducing an error due to some time stepping). In the present article, we present another exact 3D solver that is developed in the particular context of the Helmholtz--Hodge projection in $(H^1)^d$. Instead of working iteratively with the pressure operator (i.e., the Schur complement), we are interested here in using an alternative scalar field, which performs the projection and is called the gauge. It turns out that iterative computation of the gauge presents efficient properties in terms of operator conditioning. To prove the relevance of the gauge method, we demonstrate several mathematical properties of the operator acting on the gauge, especially when preconditioned in an appropriate way. In particular, we prove that the gauge operator can be solved efficiently with any gradient method. The convergence properties of the continuous preconditioned operator are quantitatively estimated in the context of a semiperiodic two-dimensional (2D) domain. We then study the discrete gauge operator implemented in the framework of Chebyshev 2D-3D pseudospectral methods. It exhibits the same favorable properties as those found for the continuous operator. Finally, we perform numerical experiments that corroborate our analyses.