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
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.