The region examined here is the Equilateral triangle under Dirichlet Boundary conditions. Side lengths are 1. This triangle was one of our test cases to determine the accuracy of MATLAB's finite element solver and whether we could accurately assume that we could treat the remainder of the error as a portion of the geometric series. This is because this triangle is one of the regions for which the eigenvalue spectrum is known: for all positive integers i,j the eigenvalue spectrum is the set {(4/3*pi)^2*(i^2+j^2+ij)}. It confirmed both that the MATLAB solver was very accurate, and that our estimate of the remaining error resulted in even more accurate values. Together with our other known cases, this triangle also helped us to determine that the accuracy of the built in 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 1000
values were used here.
Here are the difference graphs for the eigenvalue
spectrum, using the true values since the spectrum is known. The first approximately 2000
values were used here, without the (4/3*pi)^2 scalar.
Below is the Eigenvalue spectrum without the (4/3*pi)^2
scalar. These are the calculated and predicted eigenvalues, accompanied by their true values.
| Euclidean Equilateral Triangle | ||||||||
|---|---|---|---|---|---|---|---|---|
| Refinements | ||||||||
| 0 | 1 | 2 | 3 | 4 | 5 | Predicted | True Value | |
| 3.087324873 | 3.02223838 | 3.005610505 | 3.001407519 | 3.000352288 | 3.000088104 | 3.000000525 | 3 | |
| 7.441032062 | 7.111702073 | 7.028132658 | 7.007054355 | 7.001765408 | 7.000441496 | 7.000002611 | 7 | |
| 7.498712147 | 7.124315766 | 7.031130049 | 7.00779124 | 7.001948688 | 7.000487247 | 7.00000277 | 7 | |
| 13.46061037 | 12.36125627 | 12.09015387 | 12.02254069 | 12.00563608 | 12.00140913 | 12.00000786 | 12 | |
| 14.6421936 | 13.40837398 | 13.1021276 | 13.02555376 | 13.00639105 | 13.001598 | 13.00000908 | 13 | |
| 14.70989939 | 13.42403728 | 13.10603615 | 13.026537 | 13.00663763 | 13.00165972 | 13.00000951 | 13 | |
| 22.38660778 | 19.84792755 | 19.21220494 | 19.0531165 | 19.01328668 | 19.00332236 | 19.00001912 | 19 | |
| 22.75809065 | 19.94248664 | 19.2349928 | 19.05873552 | 19.01468536 | 19.00367157 | 19.00002044 | 19 | |
| 25.32907871 | 22.05932218 | 21.26292664 | 21.06566334 | 21.01641483 | 21.00410384 | 21.00002267 | 21 | |
| 25.35512878 | 22.07051364 | 21.26635355 | 21.06655624 | 21.01664036 | 21.00416037 | 21.00002318 | 21 | |
| 33.90334675 | 28.77836686 | 27.44631251 | 27.11178626 | 27.02796686 | 27.00699343 | 27.0000406 | 27 | |
| 35.44905824 | 29.88745578 | 28.47005828 | 28.11743399 | 28.02935784 | 28.00733972 | 28.00004057 | 28 | |
| 36.10013125 | 30.01523843 | 28.49687353 | 28.12383095 | 28.03093953 | 28.00773414 | 28.00004141 | 28 | |
| 40.32830315 | 33.32134654 | 31.57462713 | 31.14343602 | 31.03585555 | 31.00896431 | 31.00004969 | 31 | |
| 40.66165429 | 33.4370836 | 31.60475757 | 31.15105186 | 31.03776625 | 31.0094425 | 31.000053 | 31 | |
| 49.04541475 | 40.16339167 | 37.78974444 | 37.19744655 | 37.04937401 | 37.01234508 | 37.00006976 | 37 | |
| 50.60515955 | 40.59688811 | 37.88904234 | 37.22158774 | 37.05536135 | 37.01383857 | 37.0000735 | 37 | |
| 53.68358556 | 42.8489338 | 39.94855042 | 39.23631901 | 39.05904055 | 39.01475851 | 39.00007873 | 39 | |
| 54.23587164 | 42.92509291 | 39.96557739 | 39.2404127 | 39.06005112 | 39.01501019 | 39.00007884 | 39 | |
| 60.62696263 | 47.63945553 | 44.13903758 | 43.28358441 | 43.0708408 | 43.01770792 | 43.00009403 | 43 | |
| 61.36066557 | 47.7522888 | 44.1726603 | 43.2925171 | 43.07311794 | 43.01828058 | 43.00010164 | 43 | |
| 68.33371122 | 53.84462489 | 49.4397259 | 48.35857792 | 48.08957191 | 48.02238921 | 48.00011773 | 48 | |
| 69.89383034 | 54.96741518 | 50.47045667 | 49.36598062 | 49.09140377 | 49.02284614 | 49.00011886 | 49 | |
| 71.70155161 | 55.31863434 | 50.54932395 | 49.3852152 | 49.09618705 | 49.0240407 | 49.00012373 | 49 | |
| 76.39025997 | 58.91301772 | 53.6880445 | 52.41927036 | 52.1046582 | 52.02615552 | 52.00013139 | 52 | |
| 78.31827646 | 59.19583213 | 53.75395137 | 52.43535045 | 52.10865167 | 52.02715219 | 52.0001346 | 52 | |
| 83.62308636 | 65.11023896 | 58.99666902 | 57.49691073 | 57.12411101 | 57.03102237 | 57.00016289 | 57 | |
| 85.04526238 | 65.23503323 | 59.01322747 | 57.50050734 | 57.12498272 | 57.03123859 | 57.00016182 | 57 | |
| 92.26703042 | 70.34973809 | 63.28141376 | 61.56636775 | 61.14135494 | 61.0353251 | 61.00017554 | 61 | |
| 95.77312028 | 71.18381438 | 63.47406522 | 61.61328032 | 61.15300505 | 61.03823272 | 61.00018497 | 61 | |
| 97.24226871 | 73.26227562 | 65.48270267 | 63.61473437 | 63.1533327 | 63.03831292 | 63.00018314 | 63 | |
| 100.6153145 | 74.05475162 | 65.66592722 | 63.66015707 | 63.16468695 | 63.04115275 | 63.00020039 | 63 | |
| 103.4830163 | 78.89891053 | 69.88514455 | 67.71447401 | 67.17820773 | 67.044528 | 67.00021233 | 67 | |
| 108.8522074 | 79.0699005 | 69.91633046 | 67.72246455 | 67.18024414 | 67.04504068 | 67.00021988 | 67 | |
| 112.066467 | 86.33799586 | 76.24886666 | 73.80536746 | 73.20092918 | 73.05020842 | 73.00024353 | 73 | |
| 119.8307861 | 86.88137151 | 76.39761365 | 73.84327083 | 73.21046799 | 73.05259831 | 73.00026351 | 73 | |
| 128.1099672 | 89.36319813 | 78.57377256 | 75.89336323 | 75.22336745 | 75.05584601 | 75.00031159 | 75 | |
| 134.2652105 | 91.08496118 | 79.60296954 | 76.88740449 | 76.22103305 | 76.05520882 | 76.00023703 | 76 | |
| 140.9783504 | 91.80980305 | 79.76281597 | 76.92493937 | 76.23026438 | 76.0575075 | 76.0002375 | 76 | |
| 145.3494965 | 94.77049997 | 82.85574755 | 79.95615476 | 79.23857126 | 79.05961593 | 79.0002911 | 79 | |
| 158.2375258 | 95.28485933 | 82.95226484 | 79.97815689 | 79.243957 | 79.06095657 | 79.00029077 | 79 | |
| 173.3558714 | 102.1444003 | 88.38793476 | 85.08454428 | 84.27035805 | 84.06754231 | 84.00030761 | 84 | |
| 174.5722997 | 103.0693549 | 88.5744491 | 85.12749704 | 84.28086225 | 84.07015362 | 84.00030237 | 84 | |
| 199.2947534 | 110.8208545 | 95.87624829 | 92.20975582 | 91.30187586 | 91.07543641 | 91.0003703 | 91 | |
| 216.5383097 | 112.1305898 | 96.15824557 | 92.27375354 | 91.31748239 | 91.07931465 | 91.00036055 | 91 | |
| 230.4667939 | 112.4020232 | 96.27418331 | 92.31039965 | 91.32716572 | 91.08177235 | 91.00042289 | 91 | |
| 241.4271102 | 112.9595955 | 96.36819266 | 92.3321541 | 91.33243911 | 91.08307547 | 91.00040984 | 91 | |
| 256.032981 | 115.0791249 | 98.42474676 | 94.338992 | 93.33356701 | 93.08331895 | 93.00036014 | 93 | |
| 283.783083 | 115.660956 | 98.52325026 | 94.36422946 | 93.34005319 | 93.0849545 | 93.00038768 | 93 | |
| 0 | 120.4334197 | 102.7683155 | 98.42873578 | 97.35634629 | 97.0890375 | 97.00042294 | 97 | |
| 0 | 120.804059 | 102.8344399 | 98.44165601 | 97.35932876 | 97.08976758 | 97.00040634 | 97 | |
| 0 | 130.2466343 | 109.6836272 | 104.6521512 | 103.4118259 | 103.1028837 | 103.0004674 | 103 | |
| 0 | 130.4901967 | 109.7265536 | 104.6619049 | 103.4142358 | 103.1034859 | 103.0004703 | 103 | |
| 0 | 135.3180569 | 115.0648241 | 109.7837034 | 108.4472108 | 108.1118883 | 108.0007268 | 108 | |
| 0 | 139.1765825 | 116.4272692 | 110.8270677 | 109.4548013 | 109.1135818 | 109.0004653 | 109 | |
| 0 | 140.2597192 | 116.6978026 | 110.8938358 | 109.4713606 | 109.1177116 | 109.0004747 | 109 | |
| 0 | 141.9001539 | 118.5551012 | 112.8674085 | 111.4654816 | 111.11629 | 111.0005308 | 111 | |
| 0 | 142.7431298 | 118.72026 | 112.9027625 | 111.4740405 | 111.1184134 | 111.0005208 | 111 | |
| 0 | 144.4719961 | 119.9605199 | 113.9579431 | 112.4873517 | 112.1217063 | 112.0004926 | 112 | |
| 0 | 145.1355663 | 120.0293554 | 113.9711357 | 112.4903701 | 112.1224439 | 112.0004741 | 112 | |
| 0 | 151.8895934 | 125.5021555 | 119.0942789 | 117.5215502 | 117.1302645 | 117.0005508 | 117 | |
| 0 | 152.9301405 | 125.7574034 | 119.1560599 | 117.5368768 | 117.1340888 | 117.0005621 | 117 | |
| 0 | 162.6317675 | 133.7063987 | 126.3908745 | 124.5953046 | 124.1486772 | 124.0006173 | 124 | |
| 0 | 163.8358155 | 133.8255744 | 126.4304405 | 124.6057993 | 124.1513382 | 124.0006815 | 124 | |
| 0 | 166.3336098 | 136.7555823 | 129.4162226 | 127.6026768 | 127.1505917 | 127.0007226 | 127 | |
| 0 | 167.7285429 | 137.3671892 | 129.5793474 | 127.6440896 | 127.1609813 | 127.0008279 | 127 | |
| 0 | 170.7284919 | 139.3917425 | 131.5672288 | 129.6396163 | 129.1597671 | 129.0006941 | 129 | |
| 0 | 173.3176566 | 139.8373055 | 131.6408305 | 129.6555976 | 129.1636175 | 129.000523 | 129 | |
| 0 | 176.0774574 | 143.533017 | 135.5877932 | 133.6441469 | 133.1608673 | 133.000657 | 133 | |
| 0 | 176.7055505 | 143.9462946 | 135.7050903 | 133.6741205 | 133.1684012 | 133.000752 | 133 | |
| 0 | 178.9242596 | 144.3020923 | 135.7754733 | 133.6905127 | 133.172421 | 133.0006704 | 133 | |
| 0 | 180.5078399 | 144.6056282 | 135.8403604 | 133.7060536 | 133.1762647 | 133.0006363 | 133 | |
| 0 | 187.5458253 | 151.0451331 | 141.9627438 | 139.7374388 | 139.1841623 | 139.0007476 | 139 | |
| 0 | 188.5577493 | 151.3671586 | 142.0454639 | 139.7582371 | 139.1893702 | 139.0007872 | 139 | |
| 0 | 198.3709701 | 160.0100513 | 150.2646859 | 147.8164638 | 147.2040841 | 147.0010763 | 147 | |
| 0 | 198.7355662 | 160.2670315 | 150.3038341 | 147.8282734 | 147.207321 | 147.0014713 | 147 | |
| 0 | 200.8569123 | 161.0568501 | 150.4843976 | 147.8685674 | 147.2169901 | 147.0009881 | 147 | |
| 0 | 203.3193883 | 161.6733113 | 151.3206517 | 148.823599 | 148.2054906 | 148.0005838 | 148 | |
| 0 | 204.5389683 | 162.3048253 | 151.4982295 | 148.8659326 | 148.2158903 | 148.0003972 | 148 | |
| 0 | 208.4660443 | 165.3824909 | 154.5092841 | 151.8723787 | 151.217805 | 151.0008097 | 151 | |
| 0 | 210.4767725 | 165.8857288 | 154.6457477 | 151.9062222 | 151.2262316 | 151.0008104 | 151 | |
| 0 | 216.3472456 | 171.3923588 | 159.7823071 | 156.941016 | 156.2349738 | 156.0009164 | 156 | |
| 0 | 216.6088785 | 171.5730253 | 159.8696181 | 156.9637672 | 156.2406855 | 156.0009794 | 156 | |
| 0 | 219.315101 | 172.3219701 | 160.7261027 | 157.9246768 | 157.230764 | 157.0007275 | 157 | |
| 0 | 221.3910437 | 172.5824165 | 160.7814225 | 157.9396196 | 157.2345624 | 157.0008315 | 157 | |
| 0 | 227.2344703 | 179.4998778 | 167.0589341 | 164.0101058 | 163.2522389 | 163.0010012 | 163 | |
| 0 | 0 | 179.7798171 | 167.1248901 | 164.0263366 | 163.2562844 | 163.0010072 | 163 | |
| 0 | 0 | 186.1197733 | 173.2518181 | 170.0603665 | 169.264924 | 169.0012298 | 169 | |
| 0 | 0 | 187.4558634 | 173.6022542 | 170.1491141 | 169.2871696 | 169.0014295 | 169 | |
| 0 | 0 | 188.9331424 | 175.4846713 | 172.1196704 | 171.2795739 | 171.0010766 | 171 | |
| 0 | 0 | 189.7276412 | 175.5295756 | 172.1263024 | 171.2815287 | 171.0014809 | 171 | |
| 0 | 0 | 190.8332343 | 176.5876429 | 173.1370095 | 172.2836161 | 172.0007108 | 172 | |
| 0 | 0 | 191.2507508 | 176.6863395 | 173.1612106 | 172.2896131 | 172.0006731 | 172 | |
| 0 | 0 | 194.6546137 | 179.8079026 | 176.1943662 | 175.2981091 | 175.0009942 | 175 | |
| 0 | 0 | 194.92672 | 179.8518544 | 176.204104 | 175.3004837 | 175.0009279 | 175 | |
| 0 | 0 | 201.6690681 | 186.0863928 | 182.2652486 | 181.3159096 | 181.0011977 | 181 | |
| 0 | 0 | 202.3078388 | 186.2203225 | 182.2968823 | 181.3237022 | 181.0010868 | 181 | |
| 0 | 0 | 203.6628695 | 188.0607998 | 184.2587562 | 183.3143063 | 183.0012151 | 183 | |
| 0 | 0 | 204.0023398 | 188.1240993 | 184.2724123 | 183.3175766 | 183.0010425 | 183 | |
| 0 | 0 | 210.7958828 | 194.3677364 | 190.3363472 | 189.3337463 | 189.0013778 | 189 | |
| 0 | 0 | 211.9904576 | 194.6572734 | 190.4066486 | 189.3511696 | 189.0012717 | 189 | |
| 0 | 0 | 214.8986728 | 197.6925276 | 193.4296965 | 192.3580447 | 192.0027854 | 192 | |
| 0 | 0 | 215.8514241 | 198.6567597 | 194.4083537 | 193.3517004 | 193.0014132 | 193 | |
| 0 | 0 | 217.5725132 | 198.9974933 | 194.4790027 | 193.368278 | 193.0000657 | 193 | |
| 0 | 0 | 220.1920208 | 201.9053144 | 197.4682183 | 196.3665679 | 196.0013638 | 196 | |
| 0 | 0 | 220.4372757 | 201.9975535 | 197.4915107 | 196.3723937 | 196.0013993 | 196 | |
| 0 | 0 | 224.3430687 | 205.1961496 | 200.5371902 | 199.3835138 | 199.0010628 | 199 | |
| 0 | 0 | 224.6705329 | 205.2625422 | 200.5538643 | 199.3877286 | 199.0011473 | 199 | |
| 0 | 0 | 227.0783709 | 207.3398471 | 202.5716554 | 201.3920736 | 201.0010348 | 201 | |
| 0 | 0 | 227.4843841 | 207.4402721 | 202.5970261 | 201.3984397 | 201.0011007 | 201 | |
| 0 | 0 | 0 | 214.7229069 | 209.6705503 | 208.4169919 | 208.0014293 | 208 | |
| 0 | 0 | 0 | 214.8678473 | 209.7084719 | 208.4265655 | 208.0016054 | 208 | |
| 0 | 0 | 0 | 217.7015987 | 212.6669977 | 211.4162253 | 211.0015864 | 211 | |
| 0 | 0 | 0 | 217.9008456 | 212.7114132 | 211.4270199 | 211.0012354 | 211 | |
| 0 | 0 | 0 | 223.9191226 | 218.7249492 | 217.4310157 | 217.0020685 | 217 | |
| 0 | 0 | 0 | 224.0909219 | 218.7614339 | 217.4396499 | 217.0014702 | 217 | |
| 0 | 0 | 0 | 224.4318465 | 218.8507549 | 217.4622074 | 217.0018951 | 217 | |
| 0 | 0 | 0 | 224.6758478 | 218.9147456 | 217.4783877 | 217.002226 | 217 | |
| 0 | 0 | 0 | 226.4955058 | 220.8627015 | 219.4644686 | 219.0009455 | 219 | |
| 0 | 0 | 0 | 226.6052945 | 220.8707296 | 219.4662058 | 219.0005973 | 219 | |
| 0 | 0 | 0 | 0 | 224.910037 | 223.4769657 | 223.0018935 | 223 | |
| 0 | 0 | 0 | 0 | 224.9448567 | 223.4851851 | 223.0012947 | 223 |