A Lagrange Multiplier Based Fictitious Domain Method And It's Applications

A Lagrange multiplier based fictitious domain method for the Dirichlet problem of a class of linear elliptic operators is discussed. We construct an equivalent variational formulation for the Dirichlet problem via introducing Lagrange multipliers, and the resulting saddle point problem is solved by the conjugate gradient algorithm. In this fictitious domain method, structured meshes independent of domain boundary can be used. So, some fast solvers based on regular meshes should be applied to solve the deduced linear system in the finite dimensional spaces. The emphasis is then put on the application to the incompressible Navier-Stokes equations. We construct an variational formulation via introducing distributed Lagrange multipliers to relax the constraint of moving rigid bodies. The methodology for the Navier-Stokes equations takes advantage of time discretization byàla Marchuk-Yanenko operator splitting in order to treat separately advection, body-imbedding and incompressibility. The quasi-Stokes problem can be solved by the conjugate gradient algorithm.For the particularity of this approach, the calculation of correlative integral on fictitious domain is described in detail. The cyclic reduction procedure for linear system is deduced integrally.Finally, we present the numerical results of a linear elliptic problem and flows around a fixed disk, and the comparison with body-fitted mesh results or experimental data. Numerical results for unsteady two-dimensional flows around a disk with known movement are also presented.