Solving Stokes flow problems with many blobs using GPU based preconditioned minimum residual method

Numerical solutions for solving Stokes flow problems with strongly variable viscosity are widely discussed in Geodynamics. Discrete Stokes system with coupled equations of momentum conservation and mass conservation, which is generated by finite element method, leads to a linear algebraic problem. This linear coefficient matrix is sparse, symmetric and indefinite so minimum residual method (MINRES) is suitable. Nowadays, using Graphical Processing Unit (GPU) for general-purpose massive computing shows its huge power in computational sciences, which also brings challenges for a lot of interdisciplinary sciences. We implemented a GPU based preconditioned MINRES solver using an open source library - CUSP, which has a good Sparse Matrix-Vector Multiplication (SpMV) solution for GPU. Taking advantage of modern GPU, gives a more than 5x speedup compared with pure CPU computing, for which the relative residual is about 10e-6 within 36000 iterations for a viscosity ratio of 10e4. Meanwhile a quad-tree finite element method with low-memory-usage is applied, which benefits the limitation of device memory and bandwidth of current GPU card. In addition, we try to utilize our GPU solver to model an early Earth which has many blobs sinking inside. Primitive incomplete mixing mantle with many blobs is involved with variable viscosity. The difference of rheological characters between lower and upper mantle may lead to the existence of high viscosity blobs which have primordial materials. A kind of whole mantle convection model with blobs of primordial mantle may explain this situation. By this simulation we can research the early state of solid planetary body including our Earth, especially the core formation processes.