Hybrid high-order methods: overview, implementation and latest developments
The Hybrid High-order method, a member of the Discontinuous Skeletal methods family, presents several attractive features as the dimension-independent mathematical construction, the possibility to use arbitrary polynomial orders and the support for meshes with polytopal cells with matching and non-matching interfaces. Positioning unknowns at mesh faces is also a natural way to express locally the fundamental balance properties satisfied by the boundary-value problem at hand.
Implementations of DiSk methods should conserve their mathematical peculiarities: a single piece of code should be able to work with arbitrary space dimension and cell shapes. It is infrequent to find this approach in software: a major reason is that programming languages commonly used by the scientific computing community are not readily amenable to an implementation which is general and efficient at the same time. With DiSk++ , a C++ template library, we efficiently replicated the mathematical flexibility of DiSk methods in software, obtaining a ``write once, run on any kind of mesh'' numerical framework.
In the first part of this talk, Dr Cicuttin presents the HHO method, and then the software library supporting it will be discussed. In the second half of the talk, two variants of HHO will be introduced: the Unfitted HHO method, allowing to use meshes not conforming to interfaces, and the Multiscale HHO method, capable to handle efficiently problems with highly oscillatory material parameters.