Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations
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2018
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Abstract
Multigrid methods belong to the best-known methods for solving linear systems
arising from the discretization of elliptic partial differential equations. The
main attraction of multigrid methods is that they have an asymptotically meshindependent
convergence behavior. Multigrid with Vanka (or local multilevel
pressure Schur complement method) as smoother have been frequently used for
the construction of very effcient coupled monolithic solvers for the solution of
the stationary incompressible Navier-Stokes equations in 2D and 3D. However,
due to its innate Gauß-Seidel/Jacobi character, Vanka has a strong influence
of the underlying mesh, and therefore, coupled multigrid solvers with Vanka
smoothing very frequently face convergence issues on meshes with high aspect
ratios. Moreover, even on very nice regular grids, these solvers may fail when
the anisotropies are introduced from the differential operator.
In this thesis, we develop a new class of robust and efficient monolithic finite
element multilevel Krylov subspace methods (MLKM) for the solution of the
stationary incompressible Navier-Stokes equations as an alternative to the coupled
multigrid-based solvers. Different from multigrid, the MLKM utilizes a
Krylov method as the basis in the error reduction process. The solver is based
on the multilevel projection-based method of Erlangga and Nabben, which accelerates
the convergence of the Krylov subspace methods by shifting the small
eigenvalues of the system matrix, responsible for the slow convergence of the
Krylov iteration, to the largest eigenvalue.
Before embarking on the Navier-Stokes equations, we first test our implementation
of the MLKM solver by solving scalar model problems, namely the
convection-diffusion problem and the anisotropic diffusion problem. We validate
the method by solving several standard benchmark problems. Next, we
present the numerical results for the solution of the incompressible Navier-Stokes
equations in two dimensions. The results show that the MLKM solvers produce
asymptotically mesh-size independent, as well as Reynolds number independent
convergence rates, for a moderate range of Reynolds numbers. Moreover, numerical
simulations also show that the coupled MLKM solvers can handle (both
mesh and operator based) anisotropies better than the coupled multigrid solvers.
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Keywords
Monolithic multilevel methods, Krylov subspaces, GMRES, FEM, Navier-Stokes equations, Saddle point problems