hp-Multigrid as Smoother algorithm for the solution of higher order accurate space-time discontinuous Galerkin discretizations
Funded by: Directorate General of Higher Education, Ministry of National Education, government of Indonesia
PhD: Mochamad Tito Julianto
Supervisor: Jaap van der Vegt / Onno Bokhove
Space-time discontinuous Galerkin finite element methods are a class of discontinuous Galerkin (DG) methods in which space and time are simultaneously discretized using basis functions which are discontinuous, both in space and time. The resulting discretization is implicit in time and fully conservative. However, the efficient solution of the algebraic system resulting from an implicit time discretization for higher order accurate DG discretizations is non trivial, in particularly for steady state solutions of advection dominated flows. Standard multigrid and Krylov subspace methods are suboptimal for such problems since to mathematical properties of the space-time DG are discretizations are quite different from to lower order finite volume and finite difference discretizations. Previous research on the theoretical performance of hp-Multigrid as Smoother algorithm (hp-MGS) showed that the algorithm has excellent convergence rates for several test cases on regular and Shiskin hexahedral meshes.
In this research, we further investigate the hp-MGS algorithm in handling high order accurate space-time DG discretizations of advection-diffusion and Euler equations. Several aspects of the algorithm are considered, including: optimal semi-implicit Runge-Kutha smoothers on more general mesh geometries, mesh refinement techniques, and software implementation issues using hpGEM. We will also consider domain-decomposition and parallelization issues of the algorithm.
M.T. Julianto, V.R. Ambati, A.R. Thornton, S. Rhebergen, W.E.H. Sollie, T.Weinhart, O. Bokhove, J.J.W. van der Vegt, “Discontinuous Galerkin implementation with hpGEM”, European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2010), June 14th - 17th, 2010. Lisbon, Portugal.
V.R. Ambati & M.T. Julianto, "hpGEM -- A Versatile Package for Discontinuous Galerkin Methods," 34th Woudschoten Conference of the Dutch-Flemish Numerical Analysis Communities, 7th-9th October 2009, Woudschoten Conference Centre, Zeist, The Netherlands.