# Fast matrix-free evaluation schemes for generic finite element programming with deal.II

**Martin Kronbichler
**

Lehrstuhl für Numerische Mechanik

Technische Universität München

Munich, Germany

### Abstract:

My talk will present our work on fast matrix-free methods in the deal.II
finite element library. The implementation is based on sum factorization
techniques for quadrilateral and hexahedral elements, which combine low
operation counts with a memory-efficient data layout. This design makes
the matrix-free operator evaluation two to five times faster than sparse
matrix-vector products already on quadratic elements. Due to better
complexity, the gap increases as the element order increases. In
particular, matrix-free operator evaluation cost is almost constant per
degree of freedom for element degrees between two and ten. The
algorithms are designed for support of adaptive meshes with hanging
nodes. In terms of generic finite element design, we aim to use similar
implementations for continuous and discontinuous bases. For both
continuous and discontinuous finite element formulations, our methods
reach 25 to 70 percent of arithmetic peak of modern CPUs, which is more
than an order of magnitude higher than matrix-based methods and very
competitive in the context of PDE solvers. Despite the high level of
optimization, we aim for a generic programming interface where the weak
form in quadrature points, the access to source and destination vectors,
and the desired order of derivatives are specified in a transparent way.
The matrix-free kernels have been used in several complex application
scenarios in computational fluid dynamics and wave propagation,
including matrix-free multigrid methods, and run on up to 150,000
processor cores.