University of Twente Student Theses


Towards an Efficient Multigrid Algorithm for Solving Pressure-Robust Discontinuous Galerkin Formulations of the Stokes Problem

Lange, T. de (2023) Towards an Efficient Multigrid Algorithm for Solving Pressure-Robust Discontinuous Galerkin Formulations of the Stokes Problem.

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Abstract:For the numerical simulation of incompressible turbulent flows using direct numerical simulation (DNS) and large eddy simulation (LES), solving the Stokes problem efficiently is an important component. In this context, high-order discretization methods such as the discontinuous Galerkin (DG) method receive increasing attention because of their high accuracy, low numerical dissipation and superior convergence properties. However, solving DG formulations of the Stokes problem efficiently is a significant challenge due to the structure and characteristics of the discrete system. In theory, a multigrid algorithm should be capable of solving the problem in an amount of work that is only proportional to the problem size, making it an attractive candidate as a solver for the DG discretization of the Stokes problem. In this thesis, a multigrid algorithm that can be applied to the DG discretization of the Poisson problem is developed, implemented and analysed. In addition, the DG discretization and a suitable smoother for the Stokes problem are derived, implemented and validated. These three topics form the building blocks for the development of an efficient multigrid algorithm for the DG formulation of the Stokes problem A multigrid algorithm based on polynomial and geometric coarsening was found to be very effective, it was capable of solving the Poisson problem in an amount cycles that was independent of the problem size. The implementation of the discretization of the Stokes problem could be validated on a Cartesian grid, while some unexpected results were found on curvilinear grids. Further research should point out if this is due to a implementation error or a fundamental problem in the discretization. Lastly, a distributive Gauss-Seidel smoother based on the least-squares commutator was found to be capable of smoothing the Stokes system effectively, provided that the order of the discretization was not higher than fourth order.
Item Type:Essay (Master)
Faculty:ET: Engineering Technology
Subject:52 mechanical engineering
Programme:Mechanical Engineering MSc (60439)
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