21. GPU Implementation of Multigrid Solver for Stokes Equation with Strongly Variable Viscosity

Solving Stokes flow problem is commonplace for numerical modeling of geodynamic processes, because the lithosphere and mantle can be always regarded as incompressible flow for long geological time scales. For Stokes flow, the Reynold Number is effectively zero so that one can ignore the advective transport of momentum equation thus resulting in the slowly creeping flow. Because of the ill-conditioned matrix due to the saddle points problem that coupling mass and momentum partial differential equations together, it is still extremely to efficiently solve this elliptic PDE system, especially with the strongly variable coefficients due to rheological structure of the earth. However, since NVIDIA issued the CUDA programming framework in 2007, scientists can use commodity CPU-GPU system to do such geodynamic simulation efficiently with the advantage of CPU and GPU respectively. In this paper, we try to implement a GPU solver for Stokes Equations with variable viscosity based on CUDA using geometric multigrid methods on the staggered grids. For 2D version, we used a mixture of Jacobi and Gauss-Seidel iteration with conservative finite difference as the smoother. For 3D version, we called the GPU smoother which is rewritten with the Red-Black Gauss-Seidel updating method to avoid the problem of disordered threads with Matlab 2010b.