## Results from Rennes, computed with Mélina

The benchmark problems are presented in the page BENCHMAX.

Authors, affiliation

Authors
Martin COSTABEL, Monique DAUGE (numerical methods and computations)
Daniel MARTIN (main contributor to the code), Grégory VIAL (contributor to the code).

Affiliation
IRMAR (UMR 6625 CNRS-University of Rennes 1)

Coordinates
e-mail Monique.Dauge@univ-rennes1.fr
web http://perso.univ-rennes1.fr/monique.dauge

Method and code

The method is the weighted Regularization.

It consists in adding to the bilinear form \int_Domain curl u . curl v dx a part in \int_Domain div u div v d\mu with, instead the standard measure dx on the Domain, a measure of the form d\mu = \rho(x) dx where \rho(x) is a suitable weight (it tends to zero with a special rate as x tends to the singular support of the solution).

Reference 1 :
Martin COSTABEL, Monique DAUGE,
"Weighted Regularization of Maxwell Equations in Polyhedral Domains", To appear in Numer. Math..

Reference 2 :
Martin COSTABEL, Monique DAUGE, Daniel MARTIN, Grégory VIAL
"Weighted Regularization of Maxwell Equations -- Computations in Curvilinear Polygons".

Code Mélina :
Daniel MARTIN.

Technical data:
Computations in double precision arithmetics on a Digital Alpha station 600 (5/266) with system Digital UNIX V4.0.

Results

 Eigenvalues Choose a domain 2DomA 2DomB 2DomC 2DomD 3DomB

Maxwell eigenvalues in 2DomA (L-shape)

Computed by a Galerkin approximation with
a geometrical mesh of 60 elements refined near the corner (10 layers and ratio 4)
polynomials of degree 1 to 10.

###### Eigenvalues.
Q1, 154 DOF Q2, 546 DOF Q3, 1178 DOF Q4, 2050 DOF Q5, 3162 DOF
2.6291475959e+00
4.0900512769e+00
1.1825600354e+01
1.1835204552e+01
1.7018260513e+01
1.5620870140e+00
3.5406924568e+00
9.9336143576e+00
9.9337891692e+00
1.1544323096e+01
1.4881346700e+00
3.5343128808e+00
9.8707313298e+00
9.8707331407e+00
1.1392090765e+01
1.4779186835e+00
3.5340556671e+00
9.8696153506e+00
9.8696153588e+00
1.1389509981e+01
1.4760175366e+00
3.5340336955e+00
9.8696044691e+00
9.8696044698e+00
1.1389480655e+01
Q6, 4514 DOF Q7, 6106 DOF Q8, 7938 DOF Q9, 10010 DOF Q10, 12322 DOF
1.4756828209e+00
3.5340316022e+00
9.8696044014e+00
9.8696044016e+00
1.1389479504e+01
1.4756304380e+00
3.5340313917e+00
9.8696044011e+00
9.8696044014e+00
1.1389479409e+01
1.4756229725e+00
3.5340313700e+00
9.8696044011e+00
9.8696044017e+00
1.1389479399e+01
1.4756219774e+00
3.5340313679e+00
9.8696044011e+00
9.8696044019e+00
1.1389479398e+01
1.4756218450e+00
3.5340313683e+00
9.8696044011e+00
9.8696044013e+00
1.1389479398e+01

###### Relative errors with benchmark.
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10
7.82e-01
1.57e-01
1.98e-01
1.99e-01
4.94e-01
5.86e-02
1.88e-03
6.49e-03
6.50e-03
1.36e-02
8.48e-03
7.97e-05
1.14e-04
1.14e-04
2.29e-04
1.56e-03
6.88e-06
1.11e-06
1.11e-06
2.69e-06
2.68e-04
6.59e-07
6.89e-09
6.96e-09
1.10e-07
4.13e-05
6.66e-08
3.14e-11
5.01e-11
9.35e-09
5.84e-06
7.06e-09
1.66e-13
2.85e-11
9.63e-10
7.78e-07
9.00e-10
3.69e-13
6.07e-11
1.19e-10
1.04e-07
3.29e-10
3.69e-13
7.74e-11
2.03e-11
1.42e-08
4.32e-10
5.72e-13
2.55e-11
2.03e-11

Maxwell eigenvalues in 2DomB (Cracked domain)

Computed by a Galerkin approximation with
a geometrical mesh of 40 elements refined near the corner (10 layers and ratio 4)
polynomials of degree 1 to 10.

We have meshed one half of the domain only and computed only the eigenvalues which correspond to eigenvectors which have their normal component zero across the extension [OA] of the crack line. We compare them with the Dirichlet-Neumann eigenvalues.

Q1, 110 DOF Q2, 378 DOF Q3, 806 DOF Q4, 1394 DOF Q5, 2142 DOF
5.6197268128e+00
6.0460333917e+00
2.0812992175e+01
2.5196477922e+01
5.0819971405e+01
1.9638031721e+00
4.0586806561e+00
1.1065368300e+01
1.2821800704e+01
2.2816741662e+01
1.2025333529e+00
4.0473831884e+00
1.0849160910e+01
1.2320663824e+01
2.1321672869e+01
1.0738251647e+00
4.0469599402e+00
1.0844900983e+01
1.2276487108e+01
2.1255164484e+01
1.0432879681e+00
4.0469283135e+00
1.0844856513e+01
1.2267572042e+01
2.1246462406e+01
Q6, 3050 DOF Q7, 4118 DOF Q8, 5346 DOF Q9, 6734 DOF Q10, 8282 DOF
1.0359481654e+00
4.0469255794e+00
1.0844854454e+01
1.2265461457e+01
2.1244623583e+01
1.0343951334e+00
4.0469253198e+00
1.0844854295e+01
1.2264993355e+01
2.1244199642e+01
1.0341226686e+00
4.0469252946e+00
1.0844854280e+01
1.2264908132e+01
2.1244116917e+01
1.0340832503e+00
4.0469252936e+00
1.0844854281e+01
1.2264896112e+01
2.1244105301e+01
1.0340769669e+00
4.0469252916e+00
1.0844854278e+01
1.2264894544e+01
2.1244104110e+01

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10
4.43e+00
4.94e-01
9.19e-01
1.05e+00
1.39e+00
8.99e-01
2.90e-03
2.03e-02
4.54e-02
7.40e-02
1.63e-01
1.13e-04
3.97e-04
4.55e-03
3.65e-03
3.84e-02
8.56e-06
4.31e-06
9.45e-04
5.20e-04
8.91e-03
7.47e-07
2.06e-07
2.18e-04
1.11e-04
1.81e-03
7.12e-08
1.62e-08
4.61e-05
2.43e-05
3.11e-04
7.01e-09
1.53e-09
7.95e-06
4.34e-06
4.71e-05
7.88e-10
1.61e-10
1.00e-06
4.45e-07
8.94e-06
5.41e-10
2.32e-10
2.15e-08
1.01e-07
2.86e-06
5.41e-11
1.29e-11
1.06e-07
1.58e-07

Maxwell eigenvalues in 2DomC (Part of a ring)

Computed by a Galerkin approximation with
a geometrical mesh of 1 element and a non-isoparametric approximation of domain of degree 9,
polynomials of degree 1 to 10.

###### Eigenvalues.
Q1, no result Q2, 18 DOF Q3, 32 DOF Q4, 50 DOF Q5, 72 DOF
2.6344459378e+00
1.0160844380e+01
1.7041534794e+01
2.0553358265e+01
2.4955084142e+01
2.6117490382e+00
8.9237779958e+00
1.0104936793e+01
1.2986378028e+01
3.1825222286e+01
2.5605781775e+00
9.9920394519e+00
1.0458942799e+01
1.2797273763e+01
1.9889340438e+01
2.5593128769e+00
9.8187903689e+00
9.9916221794e+00
1.2792389947e+01
2.1669789467e+01
Q6, 98 DOF Q7, 128 DOF Q8, 162 DOF Q9, 200 DOF Q10, 242 DOF
2.5592059949e+00
9.8666091572e+00
9.9915164004e+00
1.2792025643e+01
2.0831936125e+01
2.5592025895e+00
9.8600660313e+00
9.9915161343e+00
1.2792016500e+01
2.1256479431e+01
2.5592025340e+00
9.8601862885e+00
9.9915161134e+00
1.2792016322e+01
2.1176676927e+01
2.5592025325e+00
9.8601672624e+00
9.9915161133e+00
1.2792016319e+01
2.1183974647e+01
2.5592025325e+00
9.8601671077e+00
9.9915161133e+00
1.2792016319e+01
2.1183262393e+01

###### Relative errors with benchmark.
Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10
2.94e-02
3.05e-02
7.06e-01
6.07e-01
1.78e-01
2.05e-02
9.50e-02
1.14e-02
1.52e-02
5.02e-01
5.38e-04
1.34e-02
4.68e-02
4.11e-04
6.11e-02
4.31e-05
4.20e-03
1.06e-05
2.92e-05
2.30e-02
1.35e-06
6.53e-04
2.87e-08
7.29e-07
1.66e-02
2.23e-08
1.02e-05
2.10e-09
1.42e-08
3.45e-03
5.80e-10
1.95e-06
4.10e-12
2.53e-10
3.12e-04
4.31e-12
1.78e-08
1.42e-13
2.41e-12
3.22e-05
1.60e-13
2.16e-09
8.41e-14
1.07e-13
1.45e-06

Maxwell eigenvalues in 2DomD (curved L-shape)

Computed by a Galerkin approximation with
a geometrical mesh of 60 elements refined near the corner (10 layers and ratio 4) and a non-isoparametric approximation of domain of degree 6,
polynomials of degree 1 to 10.

###### Eigenvalues.
Q1, 154 DOF Q2, 546 DOF Q3, 1178 DOF Q4, 2050 DOF Q5, 3162 DOF
3.9877520483e+00
4.4957260805e+00
1.2174818969e+01
1.4051770583e+01
2.3129636801e+01
1.9929114871e+00
3.5033970437e+00
1.0126485236e+01
1.0227718561e+01
1.2640143428e+01
1.8416881305e+00
3.4913221097e+00
1.0066604393e+01
1.0114400455e+01
1.2446993014e+01
1.8225706506e+00
3.4906373136e+00
1.0065611590e+01
1.0111985363e+01
1.2436618720e+01
1.8192421358e+00
3.4905833159e+00
1.0065601591e+01
1.0111897180e+01
1.2435711700e+01
Q6, 4514 DOF Q7, 6106 DOF Q8, 7938 DOF Q9, 10010 DOF Q10, 12322 DOF
1.8186722768e+00
3.4905770638e+00
1.0065601506e+01
1.0111887817e+01
1.2435564303e+01
1.8185853001e+00
3.4905763309e+00
1.0065601501e+01
1.0111886450e+01
1.2435541082e+01
1.8185730493e+00
3.4905762468e+00
1.0065601500e+01
1.0111886259e+01
1.2435537749e+01
1.8185714013e+00
3.4905762357e+00
1.0065601500e+01
1.0111886234e+01
1.2435537302e+01
1.8185711806e+00
3.4905762330e+00
1.0065601501e+01
1.0111886230e+01
1.2435537204e+01

###### Relative errors with benchmark.
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10
1.19e+00
2.88e-01
2.10e-01
3.90e-01
8.60e-01
9.59e-02
3.67e-03
6.05e-03
1.15e-02
1.65e-02
1.27e-02
2.14e-04
9.96e-05
2.49e-04
9.21e-04
2.20e-03
1.75e-05
1.00e-06
9.80e-06
8.70e-05
3.69e-04
2.03e-06
9.05e-09
1.08e-06
1.40e-05
5.56e-05
2.38e-07
5.12e-10
1.57e-07
2.18e-06
7.78e-06
2.81e-08
7.69e-11
2.17e-08
3.08e-07
1.04e-06
4.03e-09
1.03e-11
2.83e-09
4.02e-08
1.37e-07
8.40e-10
1.13e-11
3.15e-10
4.31e-09
1.56e-08
6.43e-11
9.08e-11
7.12e-11
3.60e-09

Maxwell eigenvalues in 3DomB (Fichera corner)

We have used a tensor product mesh by 7 x 27 parallelepipeds, (strongly) refined towards the edges of the domain, and tensor polynomials of degree 4. Number of DOF 41691. The variation of a scalar coefficient in front of the divergence weighted integral allows us to estimate our results as follows.

###### 8 first eigenvalues.
Eigenvalues Number of hopefully reliable digits Guess for the next digit - conjectured eigenvalue
3.31380523357e+00
5.88634994607e+00
5.88634994619e+00
1.06945143272e+01
1.06945143276e+01
1.07005804029e+01
1.23345495948e+01
1.23345495949e+01
1
3
3
4
4
2
3
3
3.2???e+00
5.88??e+00
5.88??e+00
1.0694e+01
1.0694e+01
1.07??e+01
1.232?e+01
1.232?e+01

The first eigenvalue is simple and the corresponding eigenvector has the most singular part possible (the Fichera exponent is visible at the corner).

The eigenvalues 2 and 3 are double.

The eigenvalues 4 and 5 are double and the corresponding eigenvectors are "smooth" (they are correctly approximated without weight).

The eigenvalue 6 has a non-smooth eigenvector. Does it coincide with eigenvalues 4 and 5 ? (Case of a triple eigenvalue?)

The eigenvalues 7 and 8 are double.