The benchmark problems are presented in the page BENCHMAX.
Authors, affiliation
Authors
Philipp
FRAUENFELDER, Kersten SCHMIDT,
Christoph SCHWAB.
Affiliation
ETH Zürich, Seminar for Applied Mathematics
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 s \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). s is a scaling parameter not affecting physical Eigenmodes.
Reference 1:
Martin COSTABEL, Monique DAUGE,
"Weighted Regularization of Maxwell Equations in Polyhedral
Domains. A rehabilitation of nodal finite elements",
Numer. Math., 93(2): 239277, 2002.
Reference 2:
Philipp FRAUENFELDER, Christian LAGE,
"ConceptsAn ObjectOriented Software Package for Partial
Differential Equations", M2AN Math. Model. Numer. Anal. 36(5): 937951, 2002.
Reference 3:
Philipp FRAUENFELDER,
"hpFinite Element Methods for
Anisotropically, Locally Refined Meshes in Three Dimensions", Doctoral
Dissertation, in preparation. (To be found
online.)
Reference 4:
Code Concepts
Reference 5:
Mark AINSWORTH and Joe COYLE,
"Hierarchic hpedge element families for Maxwell's
equations on hybrid quadrilateral/triangular meshes",
Comput. Methods Appl. Mech. Engrg. 190(4950): 67096733, 2001.
Reference 6:
Paul David LEDGER,
"An hpadaptive finite element procedure for
electromagnetic scattering problems", Doctoral Dissertation,
University of Wales, Nov. 2001.
Technical data:
Computations in double precision arithmetics on a SunFire880
running Solaris 2.8. Eigensolver: ARPACK together
with Umfpack as
direct sparse linear solver.
Results
Maxwell eigenvalues in 2DomA (Lshape), (20031112)
Computed by a Galerkin approximation with a geometric mesh with increasing number of layers n and linearly distributed polynomial degree (between 2 in the smallest two layers and n+1 in the largest layer).
In the above plot (15770 dofs), red are spurious Eigenvalues, green are physical Eigenvalues, horizontal blue lines are 'exact' values from Benchmax. Below are the numeric values of the first three physical Eigenvalues for different number of layers n. We chose s = 9.
N = 92 n = 2  N = 254 n = 3  N = 524 n = 4  N = 920 n = 5  N = 1460 n = 6 

2.86573959840516  2.13750910748911  1.78898572721029  1.61889362116687  1.53736021852868 
3.65505284848033  3.57396281907217  3.54665671778786  3.53757610395472  3.53486006898055 
9.94384679647975  9.87085428640141  9.8696194331942  9.86960450926821  9.86960440165983 
N = 2162 n = 7  N = 3044 n = 8  N = 4124 n = 9  N = 5420 n = 10  N = 6950 n = 11 
1.50117163414723  1.48594500302172  1.47976090879924  1.47727860896356  1.47628579229304 
3.5341913131655  3.53405875446339  3.5340358390327  3.53403208704629  3.53403148248334 
9.86960440109168  9.86960440108937  9.86960440108936  9.86960440108925  9.86960440108923 
N = 8732 n = 12  N = 10784 n = 13  N = 13124 n = 14  N = 15770 n = 15 

1.47588720508015  1.47572764092995  1.47566392437459  1.47563855438825  
3.53403138526762  3.53403136972463  3.53403136725283  3.53403136686121  
9.86960440108941  9.86960440108869  9.86960440103989  9.86960440106789 
The convergence histories below for the 1st, 2nd and 3rd Eigenvalue
show three different curves each: for the values s = 9, s = 6 and s = 2.
Maxwell eigenvalues in 2DomB (Cracked domain), (20031112)
Computed by a Galerkin approximation with a geometric mesh with increasing number of layers n and linearly distributed polynomial degree (between 2 in the smallest two layers and n+1 in the largest layer).
We have meshed the whole domain despite of its symmetry.
In the above plot (21028 dofs), red are spurious Eigenvalues, green are physical Eigenvalues, horizontal blue lines are 'exact' values from Benchmax. Below are the numeric values of the first three physical Eigenvalues for different number of layers n. We chose s = 9.
N = 28 n = 1  N = 124 n = 2  N = 340 n = 3  N = 700 n = 4  N = 1228 n = 5 

0  0  0  1.58777378390929  1.32620845736318 
2.48596169911994  2.4686647564103  2.46740590431364  2.46740111987976  2.46740110030937 
4.20889371256103  4.09240705319858  4.05977414811014  4.0507501786559  4.04794410873117 
N = 1948 n = 6  N = 2884 n = 7  N = 4060 n = 8  N = 5500 n = 9  N = 7228 n = 10 
1.1837069859702  1.10935774375835  1.07158738859862  1.05275810922445  1.04339680492742 
2.46740110027244  2.46740110027235  2.46740110027234  2.46740110027227  2.46740110027231 
4.04714649402212  4.04696249807443  4.04693055206994  4.04692598281576  4.04692538017452 
N = 9268 n = 11  N = 11644 n = 12  N = 14380 n = 13  N = 17500 n = 14  N = 21028 n = 15 
1.03873979945416  1.03640903456255  1.03524249059564  1.03465847420399  1.03436626682797 
2.46740110027233  2.46740110027229  2.46740110027132  2.46740110026939  2.46740110027262 
4.04692530272451  4.04692529283525  4.0469252915798  4.0469252914196  4.04692529140291 
The convergence histories below for the 1st, 2nd and 3rd Eigenvalue
show three different curves each: for the values s = 9, s = 6 and s = 2.
Maxwell eigenvalues in 2DomE (Square for transmission problems, ε_{1} = 0.5), (20040408, 20040505, 20040511)
Computed by a Galerkin approximation with a geometric mesh with increasing number of layers n and linearly distributed polynomial degree (between 2 in the smallest two layers and n+1 in the largest layer). We have used three different methods:
Plain Weighted Regularization  Weighted Regularization with Node Doubling  Edge Elements  

Mesh  
Eigenvalues  


In the above plots (20766 and 7680 dofs, click for larger version), red are spurious Eigenvalues, green are physical Eigenvalues, horizontal blue lines are 'exact' values from Benchmax. Below are the numeric values of the first three physical Eigenvalues for different number of layers n. We chose s = 10.  



Relative errors with benchmarkThe convergence histories below for the 1st, 2nd and 3rd Eigenvalue show three different curves each: for the values s = 10, s = 6 and s = 2. In the case of edge elements, no s is used. 





The descent for the weighted regularization with node doubling (pictures in the middle) is steeper because it uses only a vertex mesh as opposed to edge meshes in the other two cases. 
Maxwell eigenvalues in 2DomE (Square for transmission problems, ε_{1} = 1e2), (20040527)
Computed by a Galerkin approximation with a geometric mesh with increasing number of layers n and linearly distributed polynomial degree (between 2 in the smallest two layers and n+1 in the largest layer). We have used three different methods:
The same mesh was used for all three methods.
In the above plot (7680 dofs), red are spurious Eigenvalues, green are physical Eigenvalues, horizontal blue lines are 'exact' values from Benchmax. Below are the numeric values of the first three physical Eigenvalues for different number of layers n. We chose s = 2.
Weighted Regularization with Node Doubling  Edge Elements  Equivalent Magnetic Problem  




The convergence histories below for the 1st, 2nd and 3rd Eigenvalue
The edge elements and the equivalent problem show exponential convergence for the first three Eigenvalues in the above plots. The weighted regularisation only does so for the third Eigenvalue. The convergence histories for the first and second Eigenvalue deteriorate at a level of the relative error of 10^{6}. This does not depend on s, the chosen Eigenvalue solver (or its parameters) or the formulation of the constraints at the origin.
The third plot above shows the convergence history for the third Eigenvalue: It is approximated poorly. The reason is that the third Eigenfunction has a strong unbounded singularity at the origin. This can also be seen in the eigenvaluess plot: The Eigenvalues computed with the weighted regularisation are above 20 for s >= 2 and even above 25 for s >= 16the true value is expected to be close to 15.
Maxwell eigenvalues in 2DomE (Square for transmission problems, ε_{1} = 1e8), (20040512)
Computed by a Galerkin approximation with a geometric mesh with increasing number of layers n and linearly distributed polynomial degree (between 2 in the smallest two layers and n+1 in the largest layer). We have used three different methods:
The same mesh was used for all three methods.
In the above plot (7680 dofs), red are spurious Eigenvalues, green are physical Eigenvalues, horizontal blue lines are 'exact' values from Benchmax. Below are the numeric values of the first three physical Eigenvalues for different number of layers n. We chose s = 6.
Weighted Regularization with Node Doubling  Edge Elements  Equivalent Magnetic Problem  




The convergence histories below for the 1st, 2nd and 3rd Eigenvalue
All three methods show exponential convergence for the first three Eigenvalues in the above plots with the following two exceptions:
Maxwell eigenvalues in 3DomA (thick L), (20031124)
Computed by a Galerkin approximation with a geometric mesh with increasing number of layers n and linearly distributed polynomial degree (between 1 in the smallest two layers and n in the largest layer). We used two different geometric grading parameters: σ = 0.5 and σ = 0.15.
The mesh with σ = 0.5 on the left has hanging nodes and is automatically generated by refining geometrically towards the singular reentrant edge. The mesh with σ = 0.15 on the right is generated by and handwritten mesh generator and is regular (no hanging nodes).
In the above plot (18726 dofs), red are spurious Eigenvalues, green are physical Eigenvalues, blue circles are undecided, horizontal blue lines are 'exact' values from Benchmax. Below are the numeric values of the first three physical Eigenvalues for different number of layers n. We chose s = 9.
N = 35 n = 2  N = 262 n = 3  N = 849 n = 4  N = 2045 n = 5  N = 4079 n = 6 

11.7025800433922  9.93636033576106  9.703684287269  9.66199382740543  9.64832141646694 
14.4482798347581  12.9453012604953  11.8771170013488  11.4414228545519  11.2979362655014 
14.5879505606373  13.7176438766842  13.461396040052  13.4208624111407  13.4086701905748 
N = 7273 n = 7  N = 12003 n = 8  N = 18726 n = 9  N = 27953 n = 10  N = 40276 n = 11 
9.64311380519266  9.64106702614984  9.640256678357  9.63993527914001  9.63980775012728 
11.2627948488678  11.2237331775842  11.4993572250657  11.3653194525692  11.3484229353869 
13.4052388728482  13.4041732668983  13.4037883302838  13.4036686125225  13.4036417544865 
N = 56341 n = 12  
9.63975714199179  
11.3448098681505  
13.4036367812583 
N = 330 n = 2  N = 1007 n = 3  N = 2196 n = 4  N = 4096 n = 5  N = 6951 n = 6 

9.66154237518914  9.64236934625773  9.63992566703802  9.63974748821976  9.63972630864826 
11.6830348052992  11.3114906688245  11.4089339115341  11.3474108890186  11.3451931889982 
13.5830928990752  13.4180982414251  13.4071878529517  13.4038647251574  13.4036551326599 
N = 11041 n = 7  N = 16700 n = 8  N = 24298 n = 9  
9.63972420070038  9.63972390048556  9.63972385325434  
11.3446042886796  11.3458307118951  11.345288444485  
13.4036373807275  13.4036359650574  13.4036350967918 
The convergence histories below for the 1st, 2nd and 3rd Eigenvalue (top to bottom) show three different curves each for different values of s. The left and right plots show σ = 0.5 and σ = 0.15 respectively with the same scales! It is clearly visible that the mesh with σ = 0.15 beats the mesh with σ = 0.5. Note: the convergence histories for the different values of s are identical for the 1st Eigenvalue.
σ = 0.5  σ = 0.15 



Maxwell eigenvalues in 3DomB (Fichera), (20031127)
Computed by a Galerkin approximation with a geometric mesh with increasing number of layers n and linearly distributed polynomial degree (between 1 in the smallest two layers and n in the largest layer).
In the above plot (45450 dofs), red are spurious Eigenvalues, green are physical Eigenvalues, blue circles are undecided, horizontal blue lines are extrapolated values from the computations.
1st EV  2nd EV  3rd EV 

3.197  5.878  10.6955 
Below are the numeric values of the first three physical Eigenvalues for different number of layers n. We chose s = 9.
N = 102 n = 2  N = 990 n = 3  N = 3882 n = 4  N = 10551 n = 5  N = 23325 n = 6 

7.1844011718377  5.51951380126179  4.39519257419226  3.82778107840267  3.52738328207976 
7.18440117183771  6.24240156443094  6.03298285023165  5.94356763192768  5.90513973333147 
13.1062851040138  10.9763212873508  10.7529569081045  10.7138214384632  10.7010661314235 
45450 n = 7  N = 81285 n = 8  
3.37237419820498  3.29778479761133  
5.8899355428419  5.88424721054921  
10.6963740435117  10.6945254229041 
The convergence histories below for the 1st, 2nd and 3rd Eigenvalue
show three different curves each: for the values s = 9, s = 6 and s = 2.