The region examined here is the 45-degree right triangle under Dirichlet Boundary conditions. Side lengths are 1, 1, and root(2). 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 distinct positive integers i,j the eigenvalue spectrum is the set {pi^2*(i^2+j^2)}. 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 1750
values were used here, without the pi^2 scalar.
Below is the Eigenvalue spectrum without the pi^2
scalar. These are the calculated and predicted eigenvalues, accompanied by their true values.
| Euclidean Right Isosceles Triangle | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Refinement | |||||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | Predicted | True Value | |
| 5.1358992 | 5.034796 | 5.0087952 | 5.0022078 | 5.0005527 | 5.0001382 | 5.0000346 | 0 | 5.0000001 | 5 |
| 10.573572 | 10.144838 | 10.036456 | 10.00914 | 10.002287 | 10.000572 | 10.000143 | 0 | 10 | 10 |
| 13.904243 | 13.228178 | 13.057381 | 13.014381 | 13.003598 | 13.0009 | 13.000225 | 0 | 13 | 13 |
| 18.678357 | 17.419488 | 17.105136 | 17.026324 | 17.006585 | 17.001647 | 17.000412 | 0 | 17 | 17 |
| 22.342576 | 20.580652 | 20.145131 | 20.03631 | 20.009081 | 20.002271 | 20.000568 | 0 | 20 | 20 |
| 28.514047 | 25.876076 | 25.219075 | 25.054814 | 25.013709 | 25.003428 | 25.000857 | 0 | 25 | 25 |
| 29.799264 | 26.947356 | 26.237073 | 26.059339 | 26.014843 | 26.003711 | 26.000928 | 0 | 26 | 26 |
| 33.582504 | 30.152658 | 29.289188 | 29.072444 | 29.018126 | 29.004533 | 29.001133 | 0 | 29 | 29 |
| 40.604561 | 35.648513 | 34.411454 | 34.102907 | 34.025736 | 34.006435 | 34.001609 | 0 | 34 | 34 |
| 44.931704 | 38.993491 | 37.498098 | 37.124625 | 37.031171 | 37.007794 | 37.001949 | 0 | 37.000001 | 37 |
| 49.09279 | 42.268451 | 40.564808 | 40.141148 | 40.03529 | 40.008823 | 40.002206 | 0 | 40 | 40 |
| 51.016044 | 43.443473 | 41.60449 | 41.150793 | 41.037683 | 41.00942 | 41.002355 | 0 | 41 | 41 |
| 56.927891 | 47.954726 | 45.732731 | 45.182898 | 45.045714 | 45.011428 | 45.002857 | 0 | 45 | 45 |
| 65.138301 | 53.785737 | 50.939495 | 50.234632 | 50.058659 | 50.014666 | 50.003667 | 0 | 50.000001 | 50 |
| 68.109596 | 55.954368 | 52.975215 | 52.24312 | 52.060749 | 52.015186 | 52.003797 | 0 | 52.000001 | 52 |
| 70.117751 | 57.245013 | 54.047213 | 53.261015 | 53.065215 | 53.016302 | 53.004075 | 0 | 52.999999 | 53 |
| 77.860813 | 62.983948 | 59.232691 | 58.307581 | 58.07688 | 58.01922 | 58.004805 | 0 | 58 | 58 |
| 80.916213 | 66.266507 | 62.310834 | 61.327472 | 61.081873 | 61.02047 | 61.005118 | 0 | 61.000001 | 61 |
| 88.137432 | 70.98335 | 66.479122 | 65.368893 | 65.092186 | 65.023045 | 65.005761 | 0 | 65 | 65 |
| 88.733669 | 71.180965 | 66.529753 | 65.381618 | 65.095375 | 65.023844 | 65.005961 | 0 | 65 | 65 |
| 93.738748 | 74.816942 | 69.687362 | 68.42093 | 68.105202 | 68.0263 | 68.006575 | 0 | 68 | 68 |
| 101.58783 | 80.621904 | 74.887567 | 73.470885 | 73.117682 | 73.029419 | 73.007355 | 0 | 73 | 73 |
| 107.3497 | 82.54575 | 76.084141 | 74.517999 | 74.129341 | 74.032327 | 74.008081 | 0 | 73.999999 | 74 |
| 114.29407 | 89.392835 | 82.317561 | 80.57754 | 80.1443 | 80.036072 | 80.009018 | 0 | 80 | 80 |
| 115.89916 | 91.642547 | 84.392409 | 82.59641 | 82.149015 | 82.03725 | 82.009312 | 0 | 81.999999 | 82 |
| 120.11806 | 95.247251 | 87.534651 | 85.63166 | 85.157816 | 85.03945 | 85.009862 | 0 | 84.999999 | 85 |
| 124.72669 | 95.917626 | 87.676777 | 85.665613 | 85.166206 | 85.041541 | 85.010385 | 0 | 85 | 85 |
| 130.92012 | 100.1335 | 91.739607 | 89.682615 | 89.170554 | 89.042635 | 89.010659 | 0 | 89 | 89 |
| 134.37186 | 102.06203 | 92.940273 | 90.730266 | 90.182295 | 90.045559 | 90.011389 | 0 | 89.999999 | 90 |
| 138.58517 | 109.88953 | 100.22392 | 97.80583 | 97.201484 | 97.050375 | 97.012594 | 0 | 97 | 97 |
| 150.71156 | 114.98941 | 103.66065 | 100.90939 | 100.22703 | 100.05674 | 100.01418 | 0 | 99.999993 | 100 |
| 153.68908 | 116.9352 | 104.85383 | 101.95431 | 101.23804 | 101.05948 | 101.01487 | 0 | 101 | 101 |
| 155.48578 | 120.338 | 108.01129 | 104.99796 | 104.24926 | 104.06231 | 104.01558 | 0 | 104 | 104 |
| 159.29678 | 122.16808 | 109.9689 | 106.98655 | 106.24631 | 106.06156 | 106.01539 | 0 | 106 | 106 |
| 167.7084 | 126.0886 | 113.21452 | 110.04896 | 109.26199 | 109.06549 | 109.01637 | 0 | 109 | 109 |
| 172.17847 | 132.4232 | 117.70891 | 114.16795 | 113.29149 | 113.07285 | 113.01821 | 0 | 113 | 113 |
| 178.13019 | 135.5824 | 120.83515 | 117.20383 | 116.3007 | 116.07516 | 116.01879 | 0 | 116 | 116 |
| 188.12143 | 136.85206 | 121.86248 | 118.20782 | 117.30152 | 117.07536 | 117.01884 | 0 | 117 | 117 |
| 192.87308 | 144.9315 | 127.554 | 123.37549 | 122.34314 | 122.08575 | 122.02143 | 0 | 121.99999 | 122 |
| 199.93722 | 147.68457 | 130.51041 | 126.36718 | 125.34122 | 125.08527 | 125.02132 | 0 | 125 | 125 |
| 211.52369 | 148.51437 | 130.74691 | 126.42554 | 125.35572 | 125.08889 | 125.02222 | 0 | 125 | 125 |
| 216.66199 | 154.39639 | 135.98453 | 131.4847 | 130.37046 | 130.09257 | 130.02314 | 0 | 130 | 130 |
| 223.17471 | 155.64837 | 136.22634 | 131.5432 | 130.38503 | 130.09621 | 130.02405 | 0 | 130 | 130 |
| 225.39797 | 164.06726 | 142.92756 | 137.71265 | 136.42704 | 136.1067 | 136.02667 | 0 | 135.99999 | 136 |
| 232.46235 | 164.80475 | 143.64862 | 138.65114 | 137.41215 | 137.103 | 137.02575 | 0 | 137 | 137 |
| 249.71866 | 175.49069 | 152.50298 | 146.86249 | 145.46486 | 145.11617 | 145.02904 | 0 | 145 | 145 |
| 263.72223 | 176.35795 | 152.66826 | 146.89815 | 145.47335 | 145.11827 | 145.02956 | 0 | 144.99999 | 145 |
| 268.8411 | 177.5773 | 153.8315 | 147.94234 | 146.48446 | 146.12105 | 146.03026 | 0 | 146 | 146 |
| 285.02198 | 180.73 | 155.81989 | 149.93297 | 148.48214 | 148.12047 | 148.03012 | 0 | 148 | 148 |
| 300.10932 | 181.85997 | 156.99581 | 150.97807 | 149.49322 | 149.12323 | 149.0308 | 0 | 148.99999 | 149 |
| 318.4381 | 185.78751 | 161.16402 | 155.03537 | 153.50855 | 153.12713 | 153.03178 | 0 | 153 | 153 |
| 350.99151 | 193.54741 | 165.82909 | 159.17793 | 157.54269 | 157.13557 | 157.03389 | 0 | 157 | 157 |
| 376.39548 | 196.33929 | 168.93314 | 162.21595 | 160.55294 | 160.13818 | 160.03454 | 0 | 159.99999 | 160 |
| 0 | 204.87818 | 173.91212 | 166.44922 | 164.61056 | 164.15254 | 164.03813 | 0 | 163.99999 | 164 |
| 0 | 208.97921 | 178.91677 | 171.46184 | 169.61442 | 169.15355 | 169.03838 | 0 | 168.99999 | 169 |
| 0 | 211.88581 | 180.3621 | 172.5684 | 170.64071 | 170.1601 | 170.04002 | 0 | 169.99999 | 170 |
| 0 | 212.31392 | 180.50536 | 172.60723 | 170.65061 | 170.16258 | 170.04064 | 0 | 169.99999 | 170 |
| 0 | 217.11814 | 183.85841 | 175.68459 | 173.66927 | 173.16721 | 173.0418 | 0 | 173 | 173 |
| 0 | 222.15862 | 189.03768 | 180.75411 | 178.68835 | 178.17209 | 178.04302 | 0 | 178 | 178 |
| 0 | 228.25041 | 191.85516 | 182.9187 | 180.72666 | 180.18148 | 180.04536 | 0 | 179.99999 | 180 |
| 0 | 228.64042 | 192.80835 | 183.92151 | 181.72839 | 181.18198 | 181.04549 | 0 | 180.99999 | 181 |
| 0 | 231.95376 | 196.73357 | 187.92105 | 185.7295 | 185.18233 | 185.04558 | 0 | 185 | 185 |
| 0 | 234.61749 | 197.28893 | 188.04082 | 185.75823 | 185.18945 | 185.04736 | 0 | 185 | 185 |
| 0 | 244.25551 | 205.88027 | 196.19502 | 193.79713 | 193.19919 | 193.04979 | 0 | 192.99999 | 193 |
| 0 | 245.14945 | 206.9461 | 197.21991 | 194.80381 | 194.20089 | 194.05022 | 0 | 194 | 194 |
| 0 | 254.08123 | 211.12501 | 200.4923 | 197.87054 | 197.21748 | 197.05436 | 0 | 196.99999 | 197 |
| 0 | 257.50487 | 214.40051 | 203.56044 | 200.88756 | 200.22174 | 200.05543 | 0 | 199.99999 | 200 |
| 0 | 261.06516 | 216.78648 | 205.65096 | 202.90975 | 202.22726 | 202.0568 | 0 | 201.99998 | 202 |
| 0 | 262.31271 | 219.04549 | 208.48843 | 205.87076 | 205.21762 | 205.0544 | 0 | 204.99999 | 205 |
| 0 | 267.79921 | 220.67688 | 208.86469 | 205.96253 | 205.24041 | 205.06009 | 0 | 204.99998 | 205 |
| 0 | 271.03994 | 223.72677 | 211.88683 | 208.9687 | 208.24199 | 208.06049 | 0 | 207.99999 | 208 |
| 0 | 272.10647 | 227.25609 | 215.80988 | 212.9521 | 212.23801 | 212.0595 | 0 | 212 | 212 |
| 0 | 286.40128 | 234.9742 | 222.17621 | 219.03968 | 218.25966 | 218.0649 | 0 | 217.99998 | 218 |
| 0 | 288.14786 | 237.77419 | 225.16004 | 222.03787 | 221.25934 | 221.06483 | 0 | 220.99999 | 221 |
| 0 | 289.08029 | 238.57062 | 225.3567 | 222.08646 | 221.27145 | 221.06785 | 0 | 220.99998 | 221 |
| 0 | 298.84738 | 243.10644 | 229.4759 | 226.11579 | 225.27876 | 225.06968 | 0 | 224.99999 | 225 |
| 0 | 301.50163 | 245.00089 | 230.69439 | 227.16968 | 226.29219 | 226.07303 | 0 | 225.99998 | 226 |
| 0 | 305.58109 | 248.22242 | 233.73672 | 230.17927 | 229.29451 | 229.07361 | 0 | 228.99998 | 229 |
| 0 | 308.55264 | 250.97556 | 236.70062 | 233.1725 | 232.29297 | 232.07323 | 0 | 231.99998 | 232 |
| 0 | 311.32929 | 252.37659 | 237.80988 | 234.19991 | 233.29981 | 233.07494 | 0 | 232.99998 | 233 |
| 0 | 314.38214 | 253.53939 | 238.76343 | 235.1837 | 234.29551 | 234.07385 | 0 | 233.99996 | 234 |
| 0 | 321.62612 | 261.19177 | 246.01219 | 242.2505 | 241.31247 | 241.07811 | 0 | 240.99999 | 241 |
| 0 | 331.34854 | 265.76549 | 249.36038 | 245.3349 | 244.33341 | 244.08334 | 0 | 243.99998 | 244 |
| 0 | 332.4453 | 266.33101 | 250.22986 | 246.30078 | 245.32479 | 245.08117 | 0 | 244.99996 | 245 |
| 0 | 334.16961 | 271.38505 | 255.30849 | 251.32438 | 250.33093 | 250.08272 | 0 | 249.99998 | 250 |
| 0 | 343.47808 | 273.25024 | 255.7133 | 251.42168 | 250.35501 | 250.08873 | 0 | 249.99997 | 250 |
| 0 | 349.82096 | 281.08199 | 262.96315 | 258.4867 | 257.37143 | 257.09284 | 0 | 256.99998 | 257 |
| 0 | 353.08472 | 284.32161 | 266.00844 | 261.49637 | 260.37374 | 260.09341 | 0 | 259.99997 | 260 |
| 0 | 354.66533 | 284.39757 | 266.1276 | 261.5277 | 260.38161 | 260.09538 | 0 | 259.99997 | 260 |
| 0 | 361.7486 | 285.51974 | 266.95755 | 262.48477 | 261.37098 | 261.09273 | 0 | 260.99998 | 261 |
| 0 | 364.89817 | 289.92671 | 271.11134 | 266.51961 | 265.3794 | 265.09482 | 0 | 264.99996 | 265 |
| 0 | 366.71421 | 290.89397 | 271.35569 | 266.58067 | 265.39466 | 265.09864 | 0 | 264.99997 | 265 |
| 0 | 369.05069 | 295.04802 | 275.42075 | 270.5986 | 269.39925 | 269.09979 | 0 | 268.99997 | 269 |
| 0 | 375.24213 | 297.13076 | 278.24337 | 273.56036 | 272.3901 | 272.09753 | 0 | 272.00001 | 272 |
| 0 | 379.80611 | 300.56732 | 280.5017 | 275.61473 | 274.40301 | 274.10071 | 0 | 273.99994 | 274 |
| 0 | 386.33974 | 304.70547 | 283.81053 | 278.69448 | 277.42311 | 277.10575 | 0 | 276.99996 | 277 |
| 0 | 389.44111 | 308.56967 | 287.78665 | 282.68973 | 281.42202 | 281.10548 | 0 | 280.99997 | 281 |
| 0 | 402.82105 | 319.26809 | 296.44323 | 290.85229 | 289.46256 | 289.11561 | 0 | 288.99996 | 289 |
| 0 | 0 | 320.60872 | 297.55535 | 291.88138 | 290.46987 | 290.11744 | 0 | 289.99996 | 290 |
| 0 | 0 | 321.11789 | 297.73953 | 291.92902 | 290.48184 | 290.12044 | 0 | 289.99997 | 290 |
| 0 | 0 | 322.21393 | 299.45142 | 293.85473 | 292.46319 | 292.11577 | 0 | 291.99996 | 292 |
| 0 | 0 | 324.48862 | 300.74053 | 294.92817 | 293.48164 | 293.12039 | 0 | 292.99997 | 293 |
| 0 | 0 | 328.30304 | 305.85684 | 297.97997 | 296.49443 | 296.12357 | 0 | 295.99995 | 296 |
| 0 | 0 | 330.0775 | 312.75831 | 299.95341 | 298.48769 | 298.12188 | 0 | 297.99994 | 298 |
| 0 | 0 | 336.36503 | 313.42452 | 306.93638 | 305.48407 | 305.12102 | 0 | 305 | 305 |
| 0 | 0 | 339.32052 | 314.65279 | 307.09556 | 305.52324 | 305.13077 | 0 | 304.99995 | 305 |
| 0 | 0 | 341.59917 | 321.85002 | 308.14329 | 306.53441 | 306.13351 | 0 | 305.99988 | 306 |
| 0 | 0 | 348.168 | 322.41238 | 315.20237 | 313.54994 | 313.13745 | 0 | 312.99995 | 313 |
| 0 | 0 | 348.89756 | 325.92095 | 316.09395 | 314.52294 | 314.1307 | 0 | 313.99995 | 314 |
| 0 | 0 | 353.02666 | 329.11006 | 319.22446 | 317.55578 | 317.13893 | 0 | 316.99998 | 317 |
| 0 | 0 | 357.04837 | 334.05769 | 322.26618 | 320.56584 | 320.14142 | 0 | 319.99995 | 320 |
| 0 | 0 | 361.7155 | 334.15646 | 327.2544 | 325.56296 | 325.1407 | 0 | 324.99995 | 325 |
| 0 | 0 | 362.31826 | 334.92509 | 327.27908 | 325.56921 | 325.14227 | 0 | 324.99996 | 325 |
| 0 | 0 | 365.10723 | 337.59206 | 327.46684 | 325.61575 | 325.15388 | 0 | 324.99992 | 325 |
| 0 | 0 | 367.12219 | 342.66683 | 330.38737 | 328.59618 | 328.149 | 0 | 327.99994 | 328 |
| 0 | 0 | 372.40132 | 347.10185 | 335.40485 | 333.60051 | 333.15008 | 0 | 332.99994 | 333 |
| 0 | 0 | 377.08453 | 347.93134 | 339.51935 | 337.62929 | 337.15729 | 0 | 336.99996 | 337 |
| 0 | 0 | 379.00859 | 349.93222 | 340.472 | 338.61749 | 338.15434 | 0 | 337.99996 | 338 |
| 0 | 0 | 380.92306 | 350.65074 | 342.46085 | 340.61386 | 340.15338 | 0 | 339.99989 | 340 |
| 0 | 0 | 383.36783 | 356.79005 | 342.64736 | 340.66086 | 340.16516 | 0 | 339.99993 | 340 |
| 0 | 0 | 389.80816 | 359.58465 | 348.6841 | 346.67019 | 346.1675 | 0 | 345.99994 | 346 |
| 0 | 0 | 392.20063 | 364.14654 | 351.63324 | 349.65754 | 349.16434 | 0 | 348.99994 | 349 |
| 0 | 0 | 398.60954 | 367.3959 | 355.76758 | 353.69069 | 353.1726 | 0 | 352.9999 | 353 |
| 0 | 0 | 0 | 371.26948 | 358.83081 | 356.70657 | 356.17657 | 0 | 355.9999 | 356 |
| 0 | 0 | 0 | 373.8787 | 362.80937 | 360.70192 | 360.17546 | 0 | 359.99997 | 360 |
| 0 | 0 | 0 | 376.84713 | 364.95314 | 362.73714 | 362.18421 | 0 | 361.9999 | 362 |
| 0 | 0 | 0 | 376.94842 | 367.94475 | 365.73513 | 365.18372 | 0 | 364.99992 | 365 |
| 0 | 0 | 0 | 381.07135 | 367.99156 | 365.74801 | 365.18701 | 0 | 365.00001 | 365 |
| 0 | 0 | 0 | 381.79748 | 371.99456 | 369.74741 | 369.18678 | 0 | 368.9999 | 369 |
| 0 | 0 | 0 | 382.31515 | 372.93382 | 370.73256 | 370.18309 | 0 | 369.99993 | 370 |
| 0 | 0 | 0 | 384.8589 | 373.06193 | 370.76428 | 370.19099 | 0 | 369.99989 | 370 |
| 0 | 0 | 0 | 389.2285 | 375.95946 | 373.73955 | 373.18487 | 0 | 372.99998 | 373 |
| 0 | 0 | 0 | 389.65682 | 380.03292 | 377.75678 | 377.18911 | 0 | 376.99989 | 377 |
| 0 | 0 | 0 | 398.87375 | 380.1341 | 377.78163 | 377.19529 | 0 | 376.99984 | 377 |
| 0 | 0 | 0 | 401.28359 | 389.21884 | 386.8049 | 386.20124 | 0 | 386.00002 | 386 |
| 0 | 0 | 0 | 0 | 391.29689 | 388.82279 | 388.20561 | 0 | 387.99988 | 388 |
| 0 | 0 | 0 | 0 | 392.3415 | 389.83297 | 389.20809 | 0 | 388.9998 | 389 |
| 0 | 0 | 0 | 0 | 397.5299 | 394.88091 | 394.22013 | 0 | 393.99987 | 394 |
| 0 | 0 | 0 | 0 | 400.49655 | 397.87321 | 397.21825 | 0 | 396.99993 | 397 |
| 0 | 0 | 0 | 0 | 403.65346 | 400.9125 | 400.22807 | 0 | 399.99993 | 400 |
| 0 | 0 | 0 | 0 | 404.64103 | 401.90866 | 401.22707 | 0 | 400.99987 | 401 |
| 0 | 0 | 0 | 0 | 0 | 404.91994 | 404.22993 | 0 | 403.99993 | 404 |
| 0 | 0 | 0 | 0 | 0 | 0 | 405.22582 | 0 | 0 | 405 |