A Lagrange multiplier/fictitious domain method for the Dirichlet problem -- generalization to some flow problems
The authors discuss a family of fictitious domain methods (sometimes called domain imbedding methods) which are based on the explicit use of Lagrange multipliers defined on the actual boundary and associated with the genuine boundary conditions. This approach allows regular grids, and therefore fast specialized solvers for problems on space domains with complicated geometries can be used. The resulting saddle-point system can be solved by an Uzawa/conjugate gradient algorithm. The methods described in the article do not require a strong coupling between the actual boundary discretization and the grid used in the auxiliary domain. In the model, it is assumed that the domain under consideration is bounded in \(\mathbb{R}^d (d \geq 1)\), and its boundary is Lipschitz continuous. Then the solution of a nonlinear time-dependent problem related to the Ginzburg-Landau equations is considered. In order to solve the equivalent fictitios domain formulation, a time discretization by the Peaceman-Rachford scheme is used. Some numerical results for a two-dimensional test problem where the actual domain is the unit circle with a center at the origin and the auxiliary domain the square \((- 2, - 2) \times (- 2,2)\) are presented. The resulting methodology is applied to study the external incompressible viscous flows modeled by the Navier-Stokes equations with Neumann downstream boundary conditions. The problems with Dirichlet downstream boundary conditions have been already considered by the authors [Comput. Meth. Appl. Mech. Eng., 112, 133-148 (1994)]. After obtaining the equivalent fictitios domain formulation, a time discretization by an operator splitting method which allows to decouple the nonlinearity and the incompressibility in the problem is used. Finally, some numerical experiments for a test problem where the actual domain is a NACA0012 airfoil with zero degrees or five degrees (the Kármán vortex shedding occurs) angle of attack are presented.
scientific article; zbMATH DE number 780133
scientific article