Symmetric interior penalty discontinuous Galerkin discretizations and block preconditioning for heterogeneous Stokes flow. (English) Zbl 1516.65080
Summary: Provable stable arbitrary order symmetric interior penalty (SIP) discontinuous Galerkin discretizations of heterogeneous, incompressible Stokes flow utilizing \(Q^2_k\)-\(Q_{k-1}\) elements and hierarchical Legendre basis polynomials are developed and investigated. For solving the resulting linear system, a block preconditioned iterative method is proposed. The nested viscous problem is solved by a \(hp\)-multilevel preconditioned Krylov subspace method. For the \(p\)-coarsening, a two-level method utilizing element-block Jacobi preconditioned iterations as a smoother is employed. Piecewise bilinear (\(Q^2_1\)) and piecewise constant (\(Q^2_0\)) \(p\)-coarse spaces are considered. Finally, Galerkin \(h\)-coarsening is proposed and investigated for the two \(p\)-coarse spaces considered. Through a number of numerical experiments, we demonstrate that utilizing the \(Q^2_1\) coarse space results in the most robust \(hp\)-multigrid method for heterogeneous Stokes flow. Using this \(Q^2_1\) coarse space we observe that the convergence of the overall Stokes solver appears to be robust with respect to the jump in the viscosity and only mildly depending on the polynomial order \(k\). It is demonstrated and supported by theoretical results that the convergence of the SIP discretizations and the iterative methods rely on a sharp choice of the penalty parameter based on local values of the viscosity.
MSC:
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 65F08 | Preconditioners for iterative methods |
| 65F10 | Iterative numerical methods for linear systems |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 86A05 | Hydrology, hydrography, oceanography |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q86 | PDEs in connection with geophysics |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
Keywords:
heterogeneous Stokes flow; variable viscosity; incompressible flow; block preconditioners; DG; SIP; Galerkin multigrid; geodynamicsSoftware:
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