Project:

Hybrid
Coupling

Researcher(s) | : | Prof. Dr. R. Hiptmair |

: | Prof. Dr. Ana Alonso-Rodriguez, Universita Degli Studi Di Trento, Ital, | |

: | Prof. Dr. Alberto Valli, Universita Degli Studi Di Trento, Italy | |

Funding | : | no external funding |

Duration | : | ongoing project |

**Description.** The general principle
underlying hybrid coupling can best
be explained in the case the scalar model problem

in on | (1) |

for some . This way to write the boundary value problem is called the

in on | (2) |

This is sometimes referred to as the

Let us assume that is partitioned into two connected Lipschitz subdomains (``primal'') and (``dual'') such that . Let be their common interface, with the unit normal on pointing into . On we resort to the primal variational formulation obtained from integration by parts applied to (1): seek such that

On the mixed variational formulation will be used, which emerges from casting the first equation of (2) into weak form and retaining the second strongly: seek , such that

(remember that is the unit

Both problems can be linked by the *transmission conditions*
on :

These can be used to express the interface terms in both (3) and (4) through quantities from the other subdomain. Subsequently merging the variational formulations we arrive at the final coupled problem:

Let us denote by the bilinear form on corresponding to the left-upper block of (6). The crucial observation is that itself features a

because the interface contributions cancel. This makes it possible to show that satisfies an inf-sup condition on . Beware, that this is not straightforward from (7), because the one-dimensional

It is important to note that the coupling of the two subdomains in (7) is
purely variational, because none of the transmission conditions (5) shows
up in the definition of the spaces. This makes (7) attractive, if
unrelated (``*non-matching*'') finite element meshes are to be
used on
and
.

**Hybrid coupling for eddy current problems**. The eddy current
equations
in frequency domain

give rise to another example of a second order boudary value problem. The hybrid coupling idea outlined above can also be applied to this system, though the theoretical analysis becomes more involved [AHV04,ARHV04]. The reason is that is encountered in non-conducting regions.

**Hybrid domain decomposition on non-matching grids**. We want to
resort to
the hybrid coupling idea to deal with non-matching grids arising from a
partitioning of into subdomains
, across whose boundaries
the finite element meshes need not match
see Fig. 1.

The idea is to introduce narrow stripes of width along the interfaces, see Fig. 2. In these stripes the problem is cast into primal form, whereas in the remainder of the domain we use the mixed form. Letting width of the stripes tend to zero, we hope to obtain a mixed formulation on the non-matching meshes.