Here the representative cell Z is an semi-infinite domain. The cell solution u solves the Laplace problem on Z with Dirichlet boundary condition on the bottom, periodic boundary conditions on the lateral faces, and it stabilizes exponentially fast in the upper direction. The Robin constant C we are interested in is precisely the constant to which u tabilizes.

We use Q

The linear solver was a two-grid cycle, the coarse grid being
Q

The solution and the sequence of adaptively refined meshes can be seen in the following picture. Note that the domain has been extended in the last step.

The approximation of the Robin constant C

Cells | Unknowns | Matrix entries | C |
Error (est.) |
---|---|---|---|---|

2 | 36 | 784 | 9.3593619487d-01 | 6.65d-04 |

9 | 156 | 5182 | 9.4073588307d-01 | 1.68d-04 |

21 | 364 | 13146 | 9.4064954874d-01 | 2.41d-06 |

45 | 780 | 28922 | 9.4065210305d-01 | 4.67d-08 |

141 | 2404 | 89142 | 9.4065215007d-01 | 5.49d-10 |

393 | 6573 | 242335 | 9.4065215057d-01 | 7.15d-11 |

Nicolas Neuss, 2017-01-31