  Adaptive finite elements combined with geometric multigrid solvers are one of
the most efficient numerical methods for problems such as the instationary
Navier-Stokes equations. Yet despite their efficiency, computations remain
expensive and the simulation of, for example, complex flow problems can take
many hours or days. GPUs provide an interesting avenue to speed up the
calculations due to their very large theoretical peak performance. However, the
large degree of parallelism and non-standard API make the use of GPUs in
scientific computing challenging. In this work, we develop a GPU acceleration
for the adaptive finite element library Gascoigne and study its effectiveness
for different systems of partial differential equations. Through the systematic
formulation of all computations as linear algebra operations, we can employ
GPU-accelerated linear algebra libraries, which simplifies the implementation
and ensures the maintainability of the code while achieving very efficient GPU
utilizations. Our results for a transport-diffusion equation, linear
elasticity, and the instationary Navier-Stokes equations show substantial
speedups of up to 20X compared to multi-core CPU implementations.
