Euclidean Disc with Dirichlet Boundary Conditions

The region examined here is the Euclidean Disc of radius 1 with Dirichlet Boundary conditions. This was an important test case for us for the accuracy of MATLAB's FEM solver on curved regions, because the eigenvalue spectrum is known for this region: Eigenvalues correspond to the squares of the zeros of the Bessel Functions of the first kind, J_k(n). Eigenvalues corresponding to zeros of zero-eth Bessel Function, J_0(n), have multiplicity 1, and Eigenvalues corresponding to zeros of all other Bessel Functions, k>0, have multiplicity 2. This region also helped confirm that the accuracy of the MATLAB FEM solver begins to deteriorate after the first approximately 250 Eigenvalues.

Here are the counting graphs for the eigenvalue spectrum, using the true values since the spectrum is known. The first approximately 660 values were used here. The equation for the predicted counting function is N'(t) = 1/4*t - 1/2*sqrt(t) + 1/6.

Here are the difference graphs for the eigenvalue spectrum, using the true values since the spectrum is known. The first approximately 1250 values were used here.

Below is the Eigenvalue spectrum for the Euclidean Disc of radius 1 under Dirichlet Boundary conditions. The calculated and predicted eigenvalue spectrum is presented here, as well as their true values, and the Bessel zeroes that they correspond to. The columns k and n represent that the value corresponds to the nth zero of J_k.

Eigenvalues of Euclidean Disc, r = 1          
Refinements          
01234 PredictedTrue ValueBessel Zerokn
5.8557286845.8014509565.7877671825.784332615.783472733 5.7831861075.7831859632.40482555801
15.1033333214.7883961214.7086977614.6886632114.68364465 14.681971814.681970643.8317059711
15.1210885514.7929597514.709859514.6889557714.68371797 14.6819720414.681970643.8317059711
27.802560326.7330838826.4644793326.3971079326.38024155 26.3746194326.374616435.13562230221
27.8292012226.7390025126.4659399726.3974728726.38033282 26.3746194726.374616435.13562230221
32.3206079330.9336127130.5870951930.5002510230.47851233 30.4712660930.471262345.5200781102
44.0740227141.5462753640.9165922740.7590309640.71961061 40.7064704940.706465826.38016189631
44.1265181341.5620638240.9207401740.7600839640.7198751 40.7064721440.706465826.38016189631
53.9711588750.4025272549.5148660549.2926291749.23700676 49.2184659549.218456327.0155866712
54.1407377950.4537873449.5281652649.2959911549.2378501 49.2184697549.218456327.0155866712
64.3042897459.257088458.0014155957.6876000757.60911117 57.582948257.58294097.58834243541
64.4457276859.2821009258.0072126857.689021757.60946478 57.5829458157.58294097.58834243541
80.792617873.3282494671.4698389971.005077470.88878251 70.8500175470.849998928.4172441422
81.2805045773.4466212271.4986811271.0122292670.89056631 70.8500119970.849998928.4172441422
86.3766950177.7480968875.60145475.0656602774.93167944 74.8870191774.887006798.65372791303
89.0033199279.9233231677.683008277.1248982476.98542306 76.9389313376.938928338.77148381651
89.3839062279.9938891577.6991632177.1288328476.98639918 76.9389212976.938928338.77148381651
113.417523899.7862874396.4033425895.5591004895.34797158 95.2775952895.277572549.7610231332
113.961841599.925609896.4358364895.5670279195.34993913 95.2775762195.277572549.7610231332
118.8474099103.687908899.959881199.034335398.80327457 98.7262543298.726272489.93610952461
119.1692654103.748212499.9734828599.0376349798.80409239 98.7262448698.726272489.93610952461
124.9434073108.802176104.8192819103.8292174103.5818956 103.499455103.499453910.1734681413
126.0741975109.086611104.8881072103.84624103.5861381 103.4994375103.499453910.1734681413
151.7713704129.794939124.2664047122.8875815122.5427722 122.4278357122.427796111.0647094942
152.207554129.8517587124.2992545122.897828122.5454673 122.4280138122.427796111.0647094942
154.542014130.715252124.8290168123.3849356123.0267398 122.9073413122.907600211.0863700271
155.4635912130.8241285124.8531578123.3920369123.0285982 122.907452122.907600211.0863700271
170.2158181143.8944933137.2306873135.5729287135.1587706 135.0207179135.020708911.6198411723
172.3405646144.411346137.3533154135.6030667135.1662674 135.0206677135.020708911.6198411723
178.3219791148.932132141.4966043139.6534832139.1935512 139.0402405139.040284411.7915344404
192.2210606160.3111027152.1568124150.1281735149.6216713 149.4528372149.452880912.2250922681
196.8539127161.5037323152.4403971150.1979525149.6390484 149.4527471149.452880912.2250922681
198.8363744163.7581775155.0879082152.9509891152.4185197 152.2410299152.241153512.338604252
201.2418068163.975601155.1263138152.9600251152.4207524 152.2409949152.241153512.338604252
224.7638245183.5727457172.9215533170.2760658169.6155797 169.3954176169.395449813.0152007233
226.7144215184.0200714173.0251043170.3013907169.6218743 169.3953688169.395449813.0152007233
239.8211192193.7292044181.5359854178.5221972177.7710084 177.5206121177.520766813.3236919414
240.7411513194.5238109181.7350745178.5699795177.7828292 177.5204458177.520766813.3236919414
244.1418669194.6508327182.3548715179.339453178.587737 178.337165178.337341213.3543004891
245.2712743194.97445182.4360307179.3581563178.5923227 178.3370448178.337341213.3543004891
251.6729061201.5979952188.8454846185.70972184.928869 184.6685853184.668807313.5892901762
255.529812202.085535188.9309681185.7289004184.9335271 184.6684027184.668807313.5892901762
287.3683625227.9529378211.8655609207.8905066206.8998192 206.5695901206.569810414.3725366743
288.8933606228.4731956211.9920196207.922192206.9077566 206.5696115206.569810414.3725366743
290.6406153231.4793095214.98139210.8967062209.8790446 209.5398241209.540120114.47550069101
299.7864949233.1062442215.3182396210.9769002209.8988496 209.5394994209.540120114.47550069101
307.4838137243.2263194225.0332787220.4552305219.3043582 218.9207341218.920189114.7959517824
309.8342585243.3911044225.3080344220.5441554219.3265745 218.9207142218.920189114.7959517824
312.8817952245.0812775225.7069743221.1448383220.0376361 219.6685687219.670006714.8212687372
326.0456386246.1675913225.8587261221.1882242220.0481499 219.6681252219.670006714.8212687372
335.663789250.8151615229.6595419224.5985899223.3479485 222.931068222.932303614.9309177105
348.561798273.1327982250.5005116244.9016039243.5075511 243.0428668243.043357115.58984788111
355.0741122274.2762794250.6911346244.9439023243.5178004 243.0424331243.043357115.58984788111
364.5461706277.4097538254.1163545248.3922646246.9691685 246.4948031246.495466115.7001740853
365.6697833277.9331401254.2102455248.4158775246.9751262 246.4948758246.495466115.7001740853
374.1159282289.929398265.2871807259.2212724257.7124866 257.209558257.21020116.0377741982
391.1379879292.1309706265.676128259.3112743257.7345986 257.2090401257.21020116.0377741982
401.1901171298.8712509271.9226791265.3685241263.7420432 263.1998829263.200854316.2234661634
404.8589003302.9094426272.7894454265.5740352263.792673 263.1988856263.200854316.2234661634
410.1013917311.3258984281.0016932273.6916248271.8829351 271.2800385271.281654316.4706300515
421.3234842312.8306709281.305416273.7632704271.9005868 271.2796923271.281654316.4706300515
424.5427056319.0239245288.7365127281.2944336279.4464293 278.8304278278.831550916.69824993121
440.5133305319.8569475288.8529085281.3196407279.4525128 278.8301368278.831550916.69824993121
443.6477519331.6826766299.6053251291.7366841289.7808798 289.128945289.129883317.0038196763
451.3896165332.6689741299.7263221291.7604435289.7864477 289.1284491289.129883317.0038196763
456.2021342341.7393768308.110205299.9540301297.9322037 297.2582615297.259680317.2412203892
466.2610351344.5359609308.6626417300.0828636297.9638046 297.2574516297.259680317.2412203892
477.7370767359.700964322.4784906313.3443538311.0767402 310.320869310.322259917.6159660544
482.948712362.9477139322.9894573313.4570027311.104018 310.3196898310.322259917.6159660544
489.4455875368.8519033329.6768002320.0675852317.683951 316.8894063316.891093517.80143515131
497.6192735370.3214208329.8954069320.1146751317.6952682 316.8887992316.891093517.80143515131
500.4419476376.5589309335.7842141325.8379058323.3742978 322.5530952322.555116317.9598194925
516.1424022379.4871668336.5641782326.0358387323.423891 322.5532417322.555116317.9598194925
525.3953228384.8612028340.6426184330.0453708327.4315353 326.5602568326.563352918.0710639706
539.3283449391.247893348.3386398337.8931005335.2990296 334.4343393334.435685918.2875828373
539.7063355393.0769606348.6334565337.9521951335.3128463 334.4330634334.435685918.2875828373
544.484096400.3540731354.3798393343.3956323340.6906174 339.7889458339.792582818.43346367102
546.8947449400.7111675354.4581188343.4205999340.6972443 339.7894591339.792582818.43346367102
557.5926919423.6178843373.525225361.2575083358.2197944 357.2072231357.209922618.89999795141
561.5213565425.00151373.7273608361.3020812358.2306056 357.2067804357.209922618.89999795141
577.8592821425.8818525376.3365245364.2431242361.2433843 360.2434709360.245481918.9801338854
581.3393316428.6516852376.8841741364.3668061361.273422 360.2422939360.245481918.9801338854
591.7390157449.5808468394.6184897381.17331377.8358642 376.7233823376.725399419.4094152335
596.371605450.5902477395.5706214381.4279201377.8999941 376.7240188376.725399419.4094152335
603.6058909456.3709275400.0688429386.7804961383.4797082 382.3794456382.37989519.5545364383
617.9047032459.6413123401.2171605387.0551097383.5466731 382.3771942382.37989519.5545364383
621.8732677462.3456517403.3777838389.36511385.9261704 384.7798572384.781905119.6158585116
634.0171952464.8275041403.9690635389.4905443385.9562511 384.7781533384.781905119.6158585116
642.028839466.041334404.3151712389.6017951385.9827731 384.7764324384.786157619.6159669112
660.3617824467.8133516404.6368381389.673637386.0004917 384.7761099384.786157619.6159669112
687.8861959482.9606044420.200252404.8382297401.0395796 399.7733628399.777256219.99443063151
690.2915459485.2179802420.5821086404.9234649401.0602519 399.7725143399.777256219.99443063151
699.9531842498.6490515434.0309759418.1730291414.2418792 412.931496412.934474320.3207892164
747.4193418500.4859458434.2827635418.2235581414.2537614 412.9304959412.934474320.3207892164
749.8046124524.0085233454.9801791437.9796494433.6670125 432.2294668432.220206520.78990636122
751.4725122526.2471449455.1920453438.011314433.6794418 432.2354845432.220206520.78990636122
769.9986495528.9251226456.438895438.7276079434.3751767 432.9243663432.933237720.8070477993
790.0152312534.5061039457.2179033438.8457366434.3994316 432.9173299432.933237720.8070477993
823.1134404535.1903777458.3029157439.7579265435.2477581 433.7443686433.761136420.8269329645
833.1792434536.2240423458.7083853439.8995298435.2859233 433.7480545433.761136420.8269329645
868.9018227545.0308824469.4380569450.7422537446.1193041 444.5783208444.583386621.08514611161
879.1389687550.3382385470.8626483451.0648737446.1981092 444.5758543444.583386621.08514611161
907.725313553.7735685471.3512408452.2086507447.4931261 445.9212845445.927564521.1169970526
912.6615998555.6295215472.85486452.5841526447.5866788 445.9208541445.927564521.1169970526
980.3388307559.1251057476.6864821456.5427479451.5806674 449.9266405449.933528521.2116366307
995.1716567577.8335494495.2468862475.0253483470.0202049 468.3518238468.356297721.6415410274
1047.162086580.5052608495.7117501475.1234664470.0435998 468.3503109468.356297721.6415410274
1071.25342598.0848486510.9287855489.2643284483.8719544 482.0744963482.076653621.95624407132
1116.914752602.1923786511.5158174489.3857126483.9007622 482.0724454482.076653621.95624407132
1153.213143609.3325641515.9060728493.4098419487.8971453 486.0595798486.069563722.04698536103
1236.318872611.9487076516.1865007493.4773863487.9140907 486.0596589486.069563722.04698536103
1249.572424617.5917952522.8749806499.3320092493.5406697 491.6102232491.619517622.17249462171
0618.82822523.097625499.3887504493.5548613 491.6102316491.619517622.17249462171
0621.4195407524.3916231501.2437845495.5291415 493.6242606493.630632222.217799955
0622.7618153525.0306187501.3786777495.5609731 493.6217382493.630632222.217799955
0641.630629542.6946423518.1025555512.0069077 509.9750251509.979675922.5827295936
0644.7266561544.2406032518.5249218512.1141101 509.9771728509.979675922.5827295936
0656.7274936552.9805261526.6696487520.1762269 518.0117529518.02144122.7600843817
0657.987328553.10503526.6853147520.1795935 518.0110197518.02144122.7600843817
0666.3977934560.2228411534.7957195528.5530843 526.4722059526.480970122.9451731384
0669.2442408561.1948596535.0518298528.6170512 526.472125526.480970122.9451731384
0677.1968959569.9251567543.1817821536.5461994 534.3343386534.339208623.11577835142
0683.2922971570.6158577543.3043307536.5738468 534.3303521534.339208623.11577835142
0692.3444287578.3329441550.1886543543.2024904 540.8737691540.877633923.25677609181
0694.8825408578.4720044550.258103543.2213211 540.8757272540.877633923.25677609181
0697.7651256579.5615833550.928773544.0378312 541.7408506541.765366723.27585373113
0700.5998019579.9596727551.072743544.0754089 541.7429642541.765366723.27585373113
0711.6639219594.6823513565.8080484558.6736487 556.2955155556.30337923.5860844465
0712.5872535595.3257416565.9702957558.7140031 556.2952389556.30337923.5860844465
0739.8800358618.5880494587.3280844579.5160296 576.9120114576.913298924.0190195246
0745.5815921619.8675136587.6640458579.6017291 576.9142903576.913298924.0190195246
0755.6928756629.1459148597.6581592589.8696253 587.2734474587.281194724.2338852694
0761.2261936629.8375074597.7803939589.9007962 587.2742636587.281194724.2338852694
0764.2731387632.5442599599.7786725591.682337 588.9835584588.993099124.26918003152
0766.8812591632.8493358599.8351369591.6949389 588.9815395588.993099124.26918003152
0769.6777747633.2932216599.9135671591.7445717 589.0215733589.038351724.2701123127
0774.0672163636.1719909600.6472624591.9248965 589.0174412589.038351724.2701123127
0776.6854439638.0884994603.6050508595.1516538 592.3338548592.350394724.33824962191
0778.4195683638.1648555603.6320894595.1590595 592.3347163592.350394724.33824962191
0783.8442153639.9157201604.5135577595.8944873 593.0214638593.042869724.3524715308
0789.6184068646.0176553611.2762019602.8031268 599.9787684599.999393324.49488504123
0793.0322895646.6258842611.4044929602.8338074 599.9769122599.999393324.49488504123
0807.4760204668.7323822633.4228642624.6634857 621.7436929621.750628824.9349278975
0820.6803751670.5456475633.7659851624.7425394 621.7347242621.750628824.9349278975
0844.4868762697.972476658.8593169649.2248358 646.013342646.024855225.41701901162
0850.0505025698.069248658.9867187649.2567356 646.013408646.024855225.41701901162
0853.8167263698.4768685659.5354329649.3923838 646.0113674646.031047225.41714081201
0856.632316698.7662168659.562385649.3978191 646.0096304646.031047225.41714081201
0863.8038045700.8760479659.7736179649.9854816 646.7227695646.702251225.4303411556
0864.6144651701.0318738659.9462863650.0225303 646.7146116646.702251225.4303411556
0868.5358629703.6955963663.488878653.8948655 650.6968614650.732067625.50945055104
0871.3936426704.5252766663.7628061653.9608696 650.6935574650.732067625.50945055104
0880.1768629716.8687186674.5217636664.1756685 660.7269701660.75232325.70510305133
0889.8735085717.2514228674.577422664.1871009 660.7236605660.75232325.70510305133
0891.9495761718.7826038676.7412455666.400178 662.9531555662.968088425.748166737
0894.9918286721.5445161677.5280454666.6010219 662.9586808662.968088425.748166737
0903.7026932730.1726641685.5396966674.6145223 670.9727975671.000227625.9036720918
0909.9313825731.3832458685.7448066674.6614984 670.9670623671.000227625.9036720918
0924.1886632747.637119704.11745693.4743295 689.9266227689.945551426.2668146485
0929.5075984750.1525011704.802229693.6460226 689.9272872689.945551426.2668146485
0946.5087256765.4617951717.6892844705.8455648 701.8976583701.913353426.49364742211
0949.99583766.0968756717.8071431705.8716605 701.8931663701.913353426.49364742211
0956.5854883768.564307720.9001412709.2737585 705.3982976705.422133526.55978414172
0958.7665083769.5038415721.0395158709.3031346 705.3910075705.422133526.55978414172
0964.9715853777.2978616731.9794207720.6289227 716.8454234716.810800126.77332255114
0967.1356928778.8415677732.3445849720.7115071 716.8338145716.810800126.77332255114
0981.2312979785.432362735.5227816723.3338328 719.2708498719.320552426.8201519866
0988.0137391787.3293507735.9794592723.4432116 719.2644623719.320552426.8201519866
0995.2523375791.7795871740.5612211728.1149354 723.9661735724.006505226.90736898143
01004.542869792.7765099740.723439728.1522663 723.9618754724.006505226.90736898143
01007.949637809.5789667757.1717637744.1286616 739.7809608739.790375327.1990877747
01013.819007811.0533037757.540972744.2239234 739.7849072739.790375327.1990877747
01027.652847823.6990819769.585297756.2910201 751.8595945751.887853827.4205735528
01042.441055828.5065001770.9111093756.6219003 751.8588307751.887853827.4205735528
01044.842309830.3909531774.5523972760.5313191 755.8576264755.891394827.4934791309
01047.06482833.4020907778.4144392764.6157529 760.0161908759.991530427.56794389221
01049.11133835.1708056778.553258764.6294491 759.9881794759.991530427.56794389221
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0002774.0559232609.509572  2524.59288150.24532696912
0002774.9032282610.766395  2557.14130250.56818468713
0002775.9381452617.780243  2557.14130250.56818468713
0002779.0815162618.05277  2558.70778650.58367114198
0002780.1004022619.388898  2558.70778650.58367114198
0002794.3706612619.81921  2566.54026450.661033314
0002798.2191612634.847715  2566.54026450.661033314
0002799.9073712636.146861  2568.283750.678236951410
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0002808.2787592645.252759  2581.36803650.8071652514
0002813.1604952645.568979  2585.09888350.8438677441
0002814.3280722648.640777  2585.09888350.8438677441
0002815.3926682648.714238  2593.92601250.93059996392
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00002781.057596  2705.7933952.017241281510
00002781.363661  2722.92591252.1816626237
00002784.170202  2722.92591252.1816626237
00002784.363574  2724.28108952.19464617295
00002792.909343  2724.28108952.19464617295
00002794.455447  2724.81720552.19978166363
00002795.070968  2724.81720552.19978166363
00002795.421095  2733.141352.2794539614
00002813.239283  2733.141352.2794539614

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