This session covers techniques for GPU-based Finite Element multigrid solvers. After presenting the necessary preliminaries, we outline the broader context in which the techniques can be applied. We introduce structured, block-structured and unstructured discretization grids for finite element, finite difference and finite volume techniques and discuss their computational and bandwidth requirements. Here, we also describe fine-grained parallelization techniques for the ``finite element assembly``, i.e., the construction of linear systems of equations from the discretization of a PDE on a mesh covering the computational domain.
The second part of our lecture covers another building block in finite element simulation software, namely iterative solvers for sparse linear systems. We briefly present the necessary building blocks from numerical linear algebra that are needed to implement solvers of multigrid and Krylov subspace type; and then focus on numerically powerful preconditioning and smoothing techniques and the general trade-off between recursive, inherently sequential numerically advantageous properties and scalable parallelization for fine-grained architectures such as GPUs. In addition, we present mixed precision methods as a generic performance improvement technique in the context of iterative solvers.
The third part of the lecture is then concerned with integrating the GPU-based components from the previous parts into large-scale PDE software that executes on heterogeneous clusters. We discuss the benefits and drawbacks of a ``minimally invasive`` integration vs. a full re-implementation in such clusters, technical details of an efficient management of heterogeneous resources (scheduling, overlapping communication with computation, etc.) and present case studies for applications from fluid dynamics, solid mechanics and geophysics.