FEM2D_LEASTSQUARES FORTRAN90 version Seek U(x,y), the solution of the least squares equation: Minimize L2 norm of U(x,y) - W(x,y), for W(x,y) given, with U(x,y) a piecewise linear function in the interior, and matching W(x,y) on the boundary. Reformulate this in terms of a finite element problem: ( U(x,y) - W(x,y), V(x,y) ) = 0 inside the region, U(x,y) = W(x,y) on the boundary The region is a rectangle, defined by: 0.00000 = XL<= X <= XR = 1.00000 0.00000 = YB<= Y <= YT = 1.00000 The finite element method is used, with piecewise linear basis functions on 3 node triangular elements. The corner nodes of the triangles are generated by an underlying grid whose dimensions are NX = 17 NY = 17 Number of nodes = 289 Number of elements = 512 K I J X Y U(x,y) W(x,y) 1 1 1 0.00 0.00 0.00000 0.00000 2 2 1 0.06 0.00 0.625000E-01 0.625000E-01 3 3 1 0.13 0.00 0.125000 0.125000 4 4 1 0.19 0.00 0.187500 0.187500 5 5 1 0.25 0.00 0.250000 0.250000 6 6 1 0.31 0.00 0.312500 0.312500 7 7 1 0.38 0.00 0.375000 0.375000 8 8 1 0.44 0.00 0.437500 0.437500 9 9 1 0.50 0.00 0.500000 0.500000 10 10 1 0.56 0.00 0.562500 0.562500 11 11 1 0.63 0.00 0.625000 0.625000 12 12 1 0.69 0.00 0.687500 0.687500 13 13 1 0.75 0.00 0.750000 0.750000 14 14 1 0.81 0.00 0.812500 0.812500 15 15 1 0.88 0.00 0.875000 0.875000 16 16 1 0.94 0.00 0.937500 0.937500 17 17 1 1.00 0.00 1.00000 1.00000 18 1 2 0.00 0.06 0.00000 0.00000 19 2 2 0.06 0.06 0.105790 0.100560 20 3 2 0.13 0.06 0.203545 0.199658 21 4 2 0.19 0.06 0.299951 0.295886 22 5 2 0.25 0.06 0.391603 0.387950 23 6 2 0.31 0.06 0.477924 0.474712 24 7 2 0.38 0.06 0.557860 0.555240 25 8 2 0.44 0.06 0.630775 0.628842 26 9 2 0.50 0.06 0.696262 0.695090 27 10 2 0.56 0.06 0.754206 0.753842 28 11 2 0.63 0.06 0.804784 0.805240 29 12 2 0.69 0.06 0.848452 0.849712 30 13 2 0.75 0.06 0.885935 0.887950 31 14 2 0.81 0.06 0.918190 0.920886 32 15 2 0.88 0.06 0.946453 0.949658 33 16 2 0.94 0.06 0.971294 0.975560 34 17 2 1.00 0.06 1.00000 1.00000 35 1 3 0.00 0.13 0.00000 0.00000 36 2 3 0.06 0.13 0.141045 0.137158 37 3 3 0.13 0.13 0.274772 0.271447 38 4 3 0.19 0.13 0.403832 0.400108 39 5 3 0.25 0.13 0.524307 0.520598 40 6 3 0.31 0.13 0.634306 0.630690 41 7 3 0.38 0.13 0.731923 0.728553 42 8 3 0.44 0.13 0.815828 0.812830 43 9 3 0.50 0.13 0.885192 0.882683 44 10 3 0.56 0.13 0.939754 0.937830 45 11 3 0.63 0.13 0.979819 0.978553 46 12 3 0.69 0.13 1.00625 1.00569 47 13 3 0.75 0.13 1.02043 1.02060 48 14 3 0.81 0.13 1.02417 1.02511 49 15 3 0.88 0.13 1.02009 1.02145 50 16 3 0.94 0.13 1.00895 1.01216 51 17 3 1.00 0.13 1.00000 1.00000 52 1 4 0.00 0.19 0.00000 0.00000 53 2 4 0.06 0.19 0.174951 0.170886 54 3 4 0.13 0.19 0.341332 0.337608 55 4 4 0.19 0.19 0.500453 0.496158 56 5 4 0.25 0.19 0.647291 0.642847 57 6 4 0.31 0.19 0.778923 0.774440 58 7 4 0.38 0.19 0.892616 0.888280 59 8 4 0.44 0.19 0.986420 0.982395 60 9 4 0.50 0.19 1.05913 1.05557 61 10 4 0.56 0.19 1.11035 1.10740 62 11 4 0.63 0.19 1.14052 1.13828 63 12 4 0.69 0.19 1.15087 1.14944 64 13 4 0.75 0.19 1.14344 1.14285 65 14 4 0.81 0.19 1.12079 1.12116 66 15 4 0.88 0.19 1.08667 1.08761 67 16 4 0.94 0.19 1.04319 1.04589 68 17 4 1.00 0.19 1.00000 1.00000 69 1 5 0.00 0.25 0.00000 0.00000 70 2 5 0.06 0.25 0.204103 0.200450 71 3 5 0.13 0.25 0.399307 0.395598 72 4 5 0.19 0.25 0.584791 0.580347 73 5 5 0.25 0.25 0.754818 0.750000 74 6 5 0.31 0.25 0.905487 0.900438 75 7 5 0.38 0.25 1.03336 1.02828 76 8 5 0.44 0.25 1.13593 1.13102 77 9 5 0.50 0.25 1.21167 1.20711 78 10 5 0.56 0.25 1.26005 1.25602 79 11 5 0.63 0.25 1.28163 1.27828 80 12 5 0.69 0.25 1.27797 1.27544 81 13 5 0.75 0.25 1.25164 1.25000 82 14 5 0.81 0.25 1.20594 1.20535 83 15 5 0.88 0.25 1.14543 1.14560 84 16 5 0.94 0.25 1.07344 1.07545 85 17 5 1.00 0.25 1.00000 1.00000 86 1 6 0.00 0.31 0.00000 0.00000 87 2 6 0.06 0.31 0.227924 0.224712 88 3 6 0.13 0.31 0.446806 0.443190 89 4 6 0.19 0.31 0.653923 0.649440 90 5 6 0.25 0.31 0.842987 0.837938 91 6 6 0.31 0.31 1.00929 1.00384 92 7 6 0.38 0.31 1.14881 1.14318 93 8 6 0.44 0.31 1.25859 1.25299 94 9 6 0.50 0.31 1.33682 1.33147 95 10 6 0.56 0.31 1.38289 1.37799 96 11 6 0.63 0.31 1.39744 1.39318 97 12 6 0.69 0.31 1.38229 1.37884 98 13 6 0.75 0.31 1.34047 1.33794 99 14 6 0.81 0.31 1.27587 1.27444 100 15 6 0.88 0.31 1.19375 1.19319 101 16 6 0.94 0.31 1.09845 1.09971 102 17 6 1.00 0.31 1.00000 1.00000 103 1 7 0.00 0.38 0.00000 0.00000 104 2 7 0.06 0.38 0.245360 0.242740 105 3 7 0.13 0.38 0.481923 0.478553 106 4 7 0.19 0.38 0.705116 0.700780 107 5 7 0.25 0.38 0.908359 0.903281 108 6 7 0.31 0.38 1.08631 1.08068 109 7 7 0.38 0.38 1.23452 1.22855 110 8 7 0.44 0.38 1.34970 1.34363 111 9 7 0.50 0.38 1.42983 1.42388 112 10 7 0.56 0.38 1.47422 1.46863 113 11 7 0.63 0.38 1.48358 1.47855 114 12 7 0.69 0.38 1.45994 1.45568 115 13 7 0.75 0.38 1.40663 1.40328 116 14 7 0.81 0.38 1.32802 1.32578 117 15 7 0.88 0.38 1.22982 1.22855 118 16 7 0.94 0.38 1.11728 1.11774 119 17 7 1.00 0.38 1.00000 1.00000 120 1 8 0.00 0.44 0.00000 0.00000 121 2 8 0.06 0.44 0.255775 0.253842 122 3 8 0.13 0.44 0.503328 0.500330 123 4 8 0.19 0.44 0.736420 0.732395 124 5 8 0.25 0.44 0.948433 0.943520 125 6 8 0.31 0.44 1.13359 1.12799 126 7 8 0.38 0.44 1.28720 1.28113 127 8 8 0.44 0.44 1.40576 1.39944 128 9 8 0.50 0.44 1.48710 1.48079 129 10 8 0.56 0.44 1.53051 1.52444 130 11 8 0.63 0.44 1.53672 1.53113 131 12 8 0.69 0.44 1.50789 1.50299 132 13 8 0.75 0.44 1.44755 1.44352 133 14 8 0.81 0.44 1.36035 1.35740 134 15 8 0.88 0.44 1.25225 1.25033 135 16 8 0.94 0.44 1.12921 1.12884 136 17 8 1.00 0.44 1.00000 1.00000 137 1 9 0.00 0.50 0.00000 0.00000 138 2 9 0.06 0.50 0.258762 0.257590 139 3 9 0.13 0.50 0.510192 0.507683 140 4 9 0.19 0.50 0.746629 0.743070 141 5 9 0.25 0.50 0.961666 0.957107 142 6 9 0.31 0.50 1.14932 1.14397 143 7 9 0.38 0.50 1.30483 1.29888 144 8 9 0.44 0.50 1.42460 1.41829 145 9 9 0.50 0.50 1.50644 1.50000 146 10 9 0.56 0.50 1.54960 1.54329 147 11 9 0.63 0.50 1.55483 1.54888 148 12 9 0.69 0.50 1.52432 1.51897 149 13 9 0.75 0.50 1.46167 1.45711 150 14 9 0.81 0.50 1.37163 1.36807 151 15 9 0.88 0.50 1.26019 1.25768 152 16 9 0.94 0.50 1.13376 1.13259 153 17 9 1.00 0.50 1.00000 1.00000 154 1 10 0.00 0.56 0.00000 0.00000 155 2 10 0.06 0.56 0.254206 0.253842 156 3 10 0.13 0.56 0.502254 0.500330 157 4 10 0.19 0.56 0.735351 0.732395 158 5 10 0.25 0.56 0.947549 0.943520 159 6 10 0.31 0.56 1.13289 1.12799 160 7 10 0.38 0.56 1.28672 1.28113 161 8 10 0.44 0.56 1.40551 1.39944 162 9 10 0.50 0.56 1.48710 1.48079 163 10 10 0.56 0.56 1.53076 1.52444 164 11 10 0.63 0.56 1.53720 1.53113 165 12 10 0.69 0.56 1.50859 1.50299 166 13 10 0.75 0.56 1.44843 1.44352 167 14 10 0.81 0.56 1.36142 1.35740 168 15 10 0.88 0.56 1.25333 1.25033 169 16 10 0.94 0.56 1.13078 1.12884 170 17 10 1.00 0.56 1.00000 1.00000 171 1 11 0.00 0.63 0.00000 0.00000 172 2 11 0.06 0.63 0.242284 0.242740 173 3 11 0.13 0.63 0.479819 0.478553 174 4 11 0.19 0.63 0.703020 0.700780 175 5 11 0.25 0.63 0.906628 0.903281 176 6 11 0.31 0.63 1.08494 1.08068 177 7 11 0.38 0.63 1.23358 1.22855 178 8 11 0.44 0.63 1.34922 1.34363 179 9 11 0.50 0.63 1.42983 1.42388 180 10 11 0.56 0.63 1.47470 1.46863 181 11 11 0.63 0.63 1.48452 1.47855 182 12 11 0.69 0.63 1.46131 1.45568 183 13 11 0.75 0.63 1.40836 1.40328 184 14 11 0.81 0.63 1.33012 1.32578 185 15 11 0.88 0.63 1.23192 1.22855 186 16 11 0.94 0.63 1.12036 1.11774 187 17 11 1.00 0.63 1.00000 1.00000 188 1 12 0.00 0.69 0.00000 0.00000 189 2 12 0.06 0.69 0.223452 0.224712 190 3 12 0.13 0.69 0.443747 0.443190 191 4 12 0.19 0.69 0.650874 0.649440 192 5 12 0.25 0.69 0.840466 0.837938 193 6 12 0.31 0.69 1.00729 1.00384 194 7 12 0.38 0.69 1.14744 1.14318 195 8 12 0.44 0.69 1.25789 1.25299 196 9 12 0.50 0.69 1.33682 1.33147 197 10 12 0.56 0.69 1.38359 1.37799 198 11 12 0.63 0.69 1.39881 1.39318 199 12 12 0.69 0.69 1.38429 1.37884 200 13 12 0.75 0.69 1.34299 1.33794 201 14 12 0.81 0.69 1.27892 1.27444 202 15 12 0.88 0.69 1.19681 1.19319 203 16 12 0.94 0.69 1.10292 1.09971 204 17 12 1.00 0.69 1.00000 1.00000 205 1 13 0.00 0.75 0.00000 0.00000 206 2 13 0.06 0.75 0.198435 0.200450 207 3 13 0.13 0.75 0.395433 0.395598 208 4 13 0.19 0.75 0.580942 0.580347 209 5 13 0.25 0.75 0.751640 0.750000 210 6 13 0.31 0.75 0.902966 0.900438 211 7 13 0.38 0.75 1.03163 1.02828 212 8 13 0.44 0.75 1.13505 1.13102 213 9 13 0.50 0.75 1.21167 1.20711 214 10 13 0.56 0.75 1.26093 1.25602 215 11 13 0.63 0.75 1.28336 1.27828 216 12 13 0.69 0.75 1.28049 1.27544 217 13 13 0.75 0.75 1.25482 1.25000 218 14 13 0.81 0.75 1.20979 1.20535 219 15 13 0.88 0.75 1.14931 1.14560 220 16 13 0.94 0.75 1.07910 1.07545 221 17 13 1.00 0.75 1.00000 1.00000 222 1 14 0.00 0.81 0.00000 0.00000 223 2 14 0.06 0.81 0.168190 0.170886 224 3 14 0.13 0.81 0.336675 0.337608 225 4 14 0.19 0.81 0.495792 0.496158 226 5 14 0.25 0.81 0.643442 0.642847 227 6 14 0.31 0.81 0.775874 0.774440 228 7 14 0.38 0.81 0.890520 0.888280 229 8 14 0.44 0.81 0.985351 0.982395 230 9 14 0.50 0.81 1.05913 1.05557 231 10 14 0.56 0.81 1.11142 1.10740 232 11 14 0.63 0.81 1.14262 1.13828 233 12 14 0.69 0.81 1.15392 1.14944 234 13 14 0.75 0.81 1.14729 1.14285 235 14 14 0.81 0.81 1.12545 1.12116 236 15 14 0.88 0.81 1.09133 1.08761 237 16 14 0.94 0.81 1.04995 1.04589 238 17 14 1.00 0.81 1.00000 1.00000 239 1 15 0.00 0.88 0.00000 0.00000 240 2 15 0.06 0.88 0.133953 0.137158 241 3 15 0.13 0.88 0.270092 0.271447 242 4 15 0.19 0.88 0.399175 0.400108 243 5 15 0.25 0.88 0.520433 0.520598 244 6 15 0.31 0.88 0.631247 0.630690 245 7 15 0.38 0.88 0.729819 0.728553 246 8 15 0.44 0.88 0.814754 0.812830 247 9 15 0.50 0.88 0.885192 0.882683 248 10 15 0.56 0.88 0.940828 0.937830 249 11 15 0.63 0.88 0.981923 0.978553 250 12 15 0.69 0.88 1.00931 1.00569 251 13 15 0.75 0.88 1.02431 1.02060 252 14 15 0.81 0.88 1.02883 1.02511 253 15 15 0.88 0.88 1.02477 1.02145 254 16 15 0.94 0.88 1.01604 1.01216 255 17 15 1.00 0.88 1.00000 1.00000 256 1 16 0.00 0.94 0.00000 0.00000 257 2 16 0.06 0.94 0.962937E-01 0.100560 258 3 16 0.13 0.94 0.196453 0.199658 259 4 16 0.19 0.94 0.293190 0.295886 260 5 16 0.25 0.94 0.385935 0.387950 261 6 16 0.31 0.94 0.473452 0.474712 262 7 16 0.38 0.94 0.554784 0.555240 263 8 16 0.44 0.94 0.629206 0.628842 264 9 16 0.50 0.94 0.696262 0.695090 265 10 16 0.56 0.94 0.755775 0.753842 266 11 16 0.63 0.94 0.807860 0.805240 267 12 16 0.69 0.94 0.852924 0.849712 268 13 16 0.75 0.94 0.891603 0.887950 269 14 16 0.81 0.94 0.924951 0.920886 270 15 16 0.88 0.94 0.953545 0.949658 271 16 16 0.94 0.94 0.980790 0.975560 272 17 16 1.00 0.94 1.00000 1.00000 273 1 17 0.00 1.00 0.00000 0.00000 274 2 17 0.06 1.00 0.625000E-01 0.625000E-01 275 3 17 0.13 1.00 0.125000 0.125000 276 4 17 0.19 1.00 0.187500 0.187500 277 5 17 0.25 1.00 0.250000 0.250000 278 6 17 0.31 1.00 0.312500 0.312500 279 7 17 0.38 1.00 0.375000 0.375000 280 8 17 0.44 1.00 0.437500 0.437500 281 9 17 0.50 1.00 0.500000 0.500000 282 10 17 0.56 1.00 0.562500 0.562500 283 11 17 0.63 1.00 0.625000 0.625000 284 12 17 0.69 1.00 0.687500 0.687500 285 13 17 0.75 1.00 0.750000 0.750000 286 14 17 0.81 1.00 0.812500 0.812500 287 15 17 0.88 1.00 0.875000 0.875000 288 16 17 0.94 1.00 0.937500 0.937500 289 17 17 1.00 1.00 1.00000 1.00000 ||U|| = 0.996235 ||W|| = 0.994293 ||U-W|| = 0.623472E-01 FEM2D_LEASTSQUARES: Normal end of execution.