>> fem2d_leastsquares ( 17,17 ) FEM2D_LEASTSQUARES MATLAB version Seek U(x,y), the solution of the least squares equation: ( 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: 0.000000 = XL<= X <= XR = 1.000000 0.000000 = YB<= Y <= YT = 1.000000 A finite element formulation 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 0.000000 0.000000 0.000000e+00 0.000000e+00 2 2 1 0.062500 0.000000 6.250000e-02 6.250000e-02 3 3 1 0.125000 0.000000 1.250000e-01 1.250000e-01 4 4 1 0.187500 0.000000 1.875000e-01 1.875000e-01 5 5 1 0.250000 0.000000 2.500000e-01 2.500000e-01 6 6 1 0.312500 0.000000 3.125000e-01 3.125000e-01 7 7 1 0.375000 0.000000 3.750000e-01 3.750000e-01 8 8 1 0.437500 0.000000 4.375000e-01 4.375000e-01 9 9 1 0.500000 0.000000 5.000000e-01 5.000000e-01 10 10 1 0.562500 0.000000 5.625000e-01 5.625000e-01 11 11 1 0.625000 0.000000 6.250000e-01 6.250000e-01 12 12 1 0.687500 0.000000 6.875000e-01 6.875000e-01 13 13 1 0.750000 0.000000 7.500000e-01 7.500000e-01 14 14 1 0.812500 0.000000 8.125000e-01 8.125000e-01 15 15 1 0.875000 0.000000 8.750000e-01 8.750000e-01 16 16 1 0.937500 0.000000 9.375000e-01 9.375000e-01 17 17 1 1.000000 0.000000 1.000000e+00 1.000000e+00 18 1 2 0.000000 0.062500 0.000000e+00 0.000000e+00 19 2 2 0.062500 0.062500 1.057896e-01 1.005602e-01 20 3 2 0.125000 0.062500 2.035449e-01 1.996578e-01 21 4 2 0.187500 0.062500 2.999509e-01 2.958864e-01 22 5 2 0.250000 0.062500 3.916027e-01 3.879497e-01 23 6 2 0.312500 0.062500 4.779237e-01 4.747117e-01 24 7 2 0.375000 0.062500 5.578595e-01 5.552400e-01 25 8 2 0.437500 0.062500 6.307752e-01 6.288417e-01 26 9 2 0.500000 0.062500 6.962616e-01 6.950903e-01 27 10 2 0.562500 0.062500 7.542063e-01 7.538417e-01 28 11 2 0.625000 0.062500 8.047836e-01 8.052400e-01 29 12 2 0.687500 0.062500 8.484522e-01 8.497117e-01 30 13 2 0.750000 0.062500 8.859351e-01 8.879497e-01 31 14 2 0.812500 0.062500 9.181898e-01 9.208864e-01 32 15 2 0.875000 0.062500 9.464535e-01 9.496578e-01 33 16 2 0.937500 0.062500 9.712937e-01 9.755602e-01 34 17 2 1.000000 0.062500 1.000000e+00 1.000000e+00 35 1 3 0.000000 0.125000 0.000000e+00 0.000000e+00 36 2 3 0.062500 0.125000 1.410449e-01 1.371578e-01 37 3 3 0.125000 0.125000 2.747722e-01 2.714466e-01 38 4 3 0.187500 0.125000 4.038319e-01 4.001075e-01 39 5 3 0.250000 0.125000 5.243065e-01 5.205981e-01 40 6 3 0.312500 0.125000 6.343064e-01 6.306896e-01 41 7 3 0.375000 0.125000 7.319233e-01 7.285534e-01 42 8 3 0.437500 0.125000 8.158277e-01 8.128303e-01 43 9 3 0.500000 0.125000 8.851923e-01 8.826834e-01 44 10 3 0.562500 0.125000 9.397544e-01 9.378303e-01 45 11 3 0.625000 0.125000 9.798187e-01 9.785534e-01 46 12 3 0.687500 0.125000 1.006247e+00 1.005690e+00 47 13 3 0.750000 0.125000 1.020433e+00 1.020598e+00 48 14 3 0.812500 0.125000 1.024175e+00 1.025108e+00 49 15 3 0.875000 0.125000 1.020092e+00 1.021447e+00 50 16 3 0.937500 0.125000 1.008953e+00 1.012158e+00 51 17 3 1.000000 0.125000 1.000000e+00 1.000000e+00 52 1 4 0.000000 0.187500 0.000000e+00 0.000000e+00 53 2 4 0.062500 0.187500 1.749509e-01 1.708864e-01 54 3 4 0.125000 0.187500 3.413319e-01 3.376075e-01 55 4 4 0.187500 0.187500 5.004534e-01 4.961583e-01 56 5 4 0.250000 0.187500 6.472912e-01 6.428475e-01 57 6 4 0.312500 0.187500 7.789227e-01 7.744398e-01 58 7 4 0.375000 0.187500 8.926156e-01 8.882800e-01 59 8 4 0.437500 0.187500 9.864200e-01 9.823951e-01 60 9 4 0.500000 0.187500 1.059129e+00 1.055570e+00 61 10 4 0.562500 0.187500 1.110351e+00 1.107395e+00 62 11 4 0.625000 0.187500 1.140520e+00 1.138280e+00 63 12 4 0.687500 0.187500 1.150874e+00 1.149440e+00 64 13 4 0.750000 0.187500 1.143442e+00 1.142847e+00 65 14 4 0.812500 0.187500 1.120792e+00 1.121158e+00 66 15 4 0.875000 0.187500 1.086675e+00 1.087608e+00 67 16 4 0.937500 0.187500 1.043190e+00 1.045886e+00 68 17 4 1.000000 0.187500 1.000000e+00 1.000000e+00 69 1 5 0.000000 0.250000 0.000000e+00 0.000000e+00 70 2 5 0.062500 0.250000 2.041027e-01 2.004497e-01 71 3 5 0.125000 0.250000 3.993065e-01 3.955981e-01 72 4 5 0.187500 0.250000 5.847912e-01 5.803475e-01 73 5 5 0.250000 0.250000 7.548179e-01 7.500000e-01 74 6 5 0.312500 0.250000 9.054872e-01 9.004378e-01 75 7 5 0.375000 0.250000 1.033359e+00 1.028281e+00 76 8 5 0.437500 0.250000 1.135933e+00 1.131020e+00 77 9 5 0.500000 0.250000 1.211666e+00 1.207107e+00 78 10 5 0.562500 0.250000 1.260049e+00 1.256020e+00 79 11 5 0.625000 0.250000 1.281628e+00 1.278281e+00 80 12 5 0.687500 0.250000 1.277966e+00 1.275438e+00 81 13 5 0.750000 0.250000 1.251640e+00 1.250000e+00 82 14 5 0.812500 0.250000 1.205942e+00 1.205347e+00 83 15 5 0.875000 0.250000 1.145433e+00 1.145598e+00 84 16 5 0.937500 0.250000 1.073435e+00 1.075450e+00 85 17 5 1.000000 0.250000 1.000000e+00 1.000000e+00 86 1 6 0.000000 0.312500 0.000000e+00 0.000000e+00 87 2 6 0.062500 0.312500 2.279237e-01 2.247117e-01 88 3 6 0.125000 0.312500 4.468064e-01 4.431896e-01 89 4 6 0.187500 0.312500 6.539227e-01 6.494398e-01 90 5 6 0.250000 0.312500 8.429872e-01 8.379378e-01 91 6 6 0.312500 0.312500 1.009288e+00 1.003842e+00 92 7 6 0.375000 0.312500 1.148808e+00 1.143178e+00 93 8 6 0.437500 0.312500 1.258591e+00 1.252993e+00 94 9 6 0.500000 0.312500 1.336821e+00 1.331470e+00 95 10 6 0.562500 0.312500 1.382891e+00 1.377993e+00 96 11 6 0.625000 0.312500 1.397436e+00 1.393178e+00 97 12 6 0.687500 0.312500 1.382290e+00 1.378842e+00 98 13 6 0.750000 0.312500 1.340466e+00 1.337938e+00 99 14 6 0.812500 0.312500 1.275874e+00 1.274440e+00 100 15 6 0.875000 0.312500 1.193747e+00 1.193190e+00 101 16 6 0.937500 0.312500 1.098452e+00 1.099712e+00 102 17 6 1.000000 0.312500 1.000000e+00 1.000000e+00 103 1 7 0.000000 0.375000 0.000000e+00 0.000000e+00 104 2 7 0.062500 0.375000 2.453595e-01 2.427400e-01 105 3 7 0.125000 0.375000 4.819233e-01 4.785534e-01 106 4 7 0.187500 0.375000 7.051156e-01 7.007800e-01 107 5 7 0.250000 0.375000 9.083593e-01 9.032815e-01 108 6 7 0.312500 0.375000 1.086308e+00 1.080678e+00 109 7 7 0.375000 0.375000 1.234521e+00 1.228553e+00 110 8 7 0.437500 0.375000 1.349702e+00 1.343627e+00 111 9 7 0.500000 0.375000 1.429828e+00 1.423880e+00 112 10 7 0.562500 0.375000 1.474221e+00 1.468627e+00 113 11 7 0.625000 0.375000 1.483579e+00 1.478553e+00 114 12 7 0.687500 0.375000 1.459936e+00 1.455678e+00 115 13 7 0.750000 0.375000 1.406628e+00 1.403281e+00 116 14 7 0.812500 0.375000 1.328020e+00 1.325780e+00 117 15 7 0.875000 0.375000 1.229819e+00 1.228553e+00 118 16 7 0.937500 0.375000 1.117284e+00 1.117740e+00 119 17 7 1.000000 0.375000 1.000000e+00 1.000000e+00 120 1 8 0.000000 0.437500 0.000000e+00 0.000000e+00 121 2 8 0.062500 0.437500 2.557752e-01 2.538417e-01 122 3 8 0.125000 0.437500 5.033277e-01 5.003303e-01 123 4 8 0.187500 0.437500 7.364200e-01 7.323951e-01 124 5 8 0.250000 0.437500 9.484328e-01 9.435199e-01 125 6 8 0.312500 0.437500 1.133591e+00 1.127993e+00 126 7 8 0.375000 0.437500 1.287202e+00 1.281127e+00 127 8 8 0.437500 0.437500 1.405755e+00 1.399440e+00 128 9 8 0.500000 0.437500 1.487100e+00 1.480785e+00 129 10 8 0.562500 0.437500 1.530510e+00 1.524440e+00 130 11 8 0.625000 0.437500 1.536721e+00 1.531127e+00 131 12 8 0.687500 0.437500 1.507891e+00 1.502993e+00 132 13 8 0.750000 0.437500 1.447549e+00 1.443520e+00 133 14 8 0.812500 0.437500 1.360351e+00 1.357395e+00 134 15 8 0.875000 0.437500 1.252254e+00 1.250330e+00 135 16 8 0.937500 0.437500 1.129206e+00 1.128842e+00 136 17 8 1.000000 0.437500 1.000000e+00 1.000000e+00 137 1 9 0.000000 0.500000 0.000000e+00 0.000000e+00 138 2 9 0.062500 0.500000 2.587616e-01 2.575903e-01 139 3 9 0.125000 0.500000 5.101923e-01 5.076834e-01 140 4 9 0.187500 0.500000 7.466291e-01 7.430702e-01 141 5 9 0.250000 0.500000 9.616656e-01 9.571068e-01 142 6 9 0.312500 0.500000 1.149321e+00 1.143970e+00 143 7 9 0.375000 0.500000 1.304828e+00 1.298880e+00 144 8 9 0.437500 0.500000 1.424600e+00 1.418285e+00 145 9 9 0.500000 0.500000 1.506438e+00 1.500000e+00 146 10 9 0.562500 0.500000 1.549600e+00 1.543285e+00 147 11 9 0.625000 0.500000 1.554828e+00 1.548880e+00 148 12 9 0.687500 0.500000 1.524321e+00 1.518970e+00 149 13 9 0.750000 0.500000 1.461666e+00 1.457107e+00 150 14 9 0.812500 0.500000 1.371629e+00 1.368070e+00 151 15 9 0.875000 0.500000 1.260192e+00 1.257683e+00 152 16 9 0.937500 0.500000 1.133762e+00 1.132590e+00 153 17 9 1.000000 0.500000 1.000000e+00 1.000000e+00 154 1 10 0.000000 0.562500 0.000000e+00 0.000000e+00 155 2 10 0.062500 0.562500 2.542063e-01 2.538417e-01 156 3 10 0.125000 0.562500 5.022544e-01 5.003303e-01 157 4 10 0.187500 0.562500 7.353512e-01 7.323951e-01 158 5 10 0.250000 0.562500 9.475494e-01 9.435199e-01 159 6 10 0.312500 0.562500 1.132891e+00 1.127993e+00 160 7 10 0.375000 0.562500 1.286721e+00 1.281127e+00 161 8 10 0.437500 0.562500 1.405510e+00 1.399440e+00 162 9 10 0.500000 0.562500 1.487100e+00 1.480785e+00 163 10 10 0.562500 0.562500 1.530755e+00 1.524440e+00 164 11 10 0.625000 0.562500 1.537202e+00 1.531127e+00 165 12 10 0.687500 0.562500 1.508591e+00 1.502993e+00 166 13 10 0.750000 0.562500 1.448433e+00 1.443520e+00 167 14 10 0.812500 0.562500 1.361420e+00 1.357395e+00 168 15 10 0.875000 0.562500 1.253328e+00 1.250330e+00 169 16 10 0.937500 0.562500 1.130775e+00 1.128842e+00 170 17 10 1.000000 0.562500 1.000000e+00 1.000000e+00 171 1 11 0.000000 0.625000 0.000000e+00 0.000000e+00 172 2 11 0.062500 0.625000 2.422836e-01 2.427400e-01 173 3 11 0.125000 0.625000 4.798187e-01 4.785534e-01 174 4 11 0.187500 0.625000 7.030202e-01 7.007800e-01 175 5 11 0.250000 0.625000 9.066281e-01 9.032815e-01 176 6 11 0.312500 0.625000 1.084936e+00 1.080678e+00 177 7 11 0.375000 0.625000 1.233579e+00 1.228553e+00 178 8 11 0.437500 0.625000 1.349221e+00 1.343627e+00 179 9 11 0.500000 0.625000 1.429828e+00 1.423880e+00 180 10 11 0.562500 0.625000 1.474702e+00 1.468627e+00 181 11 11 0.625000 0.625000 1.484521e+00 1.478553e+00 182 12 11 0.687500 0.625000 1.461308e+00 1.455678e+00 183 13 11 0.750000 0.625000 1.408359e+00 1.403281e+00 184 14 11 0.812500 0.625000 1.330116e+00 1.325780e+00 185 15 11 0.875000 0.625000 1.231923e+00 1.228553e+00 186 16 11 0.937500 0.625000 1.120360e+00 1.117740e+00 187 17 11 1.000000 0.625000 1.000000e+00 1.000000e+00 188 1 12 0.000000 0.687500 0.000000e+00 0.000000e+00 189 2 12 0.062500 0.687500 2.234522e-01 2.247117e-01 190 3 12 0.125000 0.687500 4.437471e-01 4.431896e-01 191 4 12 0.187500 0.687500 6.508743e-01 6.494398e-01 192 5 12 0.250000 0.687500 8.404662e-01 8.379378e-01 193 6 12 0.312500 0.687500 1.007290e+00 1.003842e+00 194 7 12 0.375000 0.687500 1.147436e+00 1.143178e+00 195 8 12 0.437500 0.687500 1.257891e+00 1.252993e+00 196 9 12 0.500000 0.687500 1.336821e+00 1.331470e+00 197 10 12 0.562500 0.687500 1.383591e+00 1.377993e+00 198 11 12 0.625000 0.687500 1.398808e+00 1.393178e+00 199 12 12 0.687500 0.687500 1.384288e+00 1.378842e+00 200 13 12 0.750000 0.687500 1.342987e+00 1.337938e+00 201 14 12 0.812500 0.687500 1.278923e+00 1.274440e+00 202 15 12 0.875000 0.687500 1.196806e+00 1.193190e+00 203 16 12 0.937500 0.687500 1.102924e+00 1.099712e+00 204 17 12 1.000000 0.687500 1.000000e+00 1.000000e+00 205 1 13 0.000000 0.750000 0.000000e+00 0.000000e+00 206 2 13 0.062500 0.750000 1.984351e-01 2.004497e-01 207 3 13 0.125000 0.750000 3.954326e-01 3.955981e-01 208 4 13 0.187500 0.750000 5.809423e-01 5.803475e-01 209 5 13 0.250000 0.750000 7.516401e-01 7.500000e-01 210 6 13 0.312500 0.750000 9.029662e-01 9.004378e-01 211 7 13 0.375000 0.750000 1.031628e+00 1.028281e+00 212 8 13 0.437500 0.750000 1.135049e+00 1.131020e+00 213 9 13 0.500000 0.750000 1.211666e+00 1.207107e+00 214 10 13 0.562500 0.750000 1.260933e+00 1.256020e+00 215 11 13 0.625000 0.750000 1.283359e+00 1.278281e+00 216 12 13 0.687500 0.750000 1.280487e+00 1.275438e+00 217 13 13 0.750000 0.750000 1.254818e+00 1.250000e+00 218 14 13 0.812500 0.750000 1.209791e+00 1.205347e+00 219 15 13 0.875000 0.750000 1.149307e+00 1.145598e+00 220 16 13 0.937500 0.750000 1.079103e+00 1.075450e+00 221 17 13 1.000000 0.750000 1.000000e+00 1.000000e+00 222 1 14 0.000000 0.812500 0.000000e+00 0.000000e+00 223 2 14 0.062500 0.812500 1.681898e-01 1.708864e-01 224 3 14 0.125000 0.812500 3.366748e-01 3.376075e-01 225 4 14 0.187500 0.812500 4.957916e-01 4.961583e-01 226 5 14 0.250000 0.812500 6.434423e-01 6.428475e-01 227 6 14 0.312500 0.812500 7.758743e-01 7.744398e-01 228 7 14 0.375000 0.812500 8.905202e-01 8.882800e-01 229 8 14 0.437500 0.812500 9.853512e-01 9.823951e-01 230 9 14 0.500000 0.812500 1.059129e+00 1.055570e+00 231 10 14 0.562500 0.812500 1.111420e+00 1.107395e+00 232 11 14 0.625000 0.812500 1.142616e+00 1.138280e+00 233 12 14 0.687500 0.812500 1.153923e+00 1.149440e+00 234 13 14 0.750000 0.812500 1.147291e+00 1.142847e+00 235 14 14 0.812500 0.812500 1.125453e+00 1.121158e+00 236 15 14 0.875000 0.812500 1.091332e+00 1.087608e+00 237 16 14 0.937500 0.812500 1.049951e+00 1.045886e+00 238 17 14 1.000000 0.812500 1.000000e+00 1.000000e+00 239 1 15 0.000000 0.875000 0.000000e+00 0.000000e+00 240 2 15 0.062500 0.875000 1.339535e-01 1.371578e-01 241 3 15 0.125000 0.875000 2.700916e-01 2.714466e-01 242 4 15 0.187500 0.875000 3.991748e-01 4.001075e-01 243 5 15 0.250000 0.875000 5.204326e-01 5.205981e-01 244 6 15 0.312500 0.875000 6.312471e-01 6.306896e-01 245 7 15 0.375000 0.875000 7.298187e-01 7.285534e-01 246 8 15 0.437500 0.875000 8.147544e-01 8.128303e-01 247 9 15 0.500000 0.875000 8.851923e-01 8.826834e-01 248 10 15 0.562500 0.875000 9.408277e-01 9.378303e-01 249 11 15 0.625000 0.875000 9.819233e-01 9.785534e-01 250 12 15 0.687500 0.875000 1.009306e+00 1.005690e+00 251 13 15 0.750000 0.875000 1.024307e+00 1.020598e+00 252 14 15 0.812500 0.875000 1.028832e+00 1.025108e+00 253 15 15 0.875000 0.875000 1.024772e+00 1.021447e+00 254 16 15 0.937500 0.875000 1.016045e+00 1.012158e+00 255 17 15 1.000000 0.875000 1.000000e+00 1.000000e+00 256 1 16 0.000000 0.937500 0.000000e+00 0.000000e+00 257 2 16 0.062500 0.937500 9.629370e-02 1.005602e-01 258 3 16 0.125000 0.937500 1.964535e-01 1.996578e-01 259 4 16 0.187500 0.937500 2.931898e-01 2.958864e-01 260 5 16 0.250000 0.937500 3.859351e-01 3.879497e-01 261 6 16 0.312500 0.937500 4.734522e-01 4.747117e-01 262 7 16 0.375000 0.937500 5.547836e-01 5.552400e-01 263 8 16 0.437500 0.937500 6.292063e-01 6.288417e-01 264 9 16 0.500000 0.937500 6.962616e-01 6.950903e-01 265 10 16 0.562500 0.937500 7.557752e-01 7.538417e-01 266 11 16 0.625000 0.937500 8.078595e-01 8.052400e-01 267 12 16 0.687500 0.937500 8.529237e-01 8.497117e-01 268 13 16 0.750000 0.937500 8.916027e-01 8.879497e-01 269 14 16 0.812500 0.937500 9.249509e-01 9.208864e-01 270 15 16 0.875000 0.937500 9.535449e-01 9.496578e-01 271 16 16 0.937500 0.937500 9.807896e-01 9.755602e-01 272 17 16 1.000000 0.937500 1.000000e+00 1.000000e+00 273 1 17 0.000000 1.000000 0.000000e+00 0.000000e+00 274 2 17 0.062500 1.000000 6.250000e-02 6.250000e-02 275 3 17 0.125000 1.000000 1.250000e-01 1.250000e-01 276 4 17 0.187500 1.000000 1.875000e-01 1.875000e-01 277 5 17 0.250000 1.000000 2.500000e-01 2.500000e-01 278 6 17 0.312500 1.000000 3.125000e-01 3.125000e-01 279 7 17 0.375000 1.000000 3.750000e-01 3.750000e-01 280 8 17 0.437500 1.000000 4.375000e-01 4.375000e-01 281 9 17 0.500000 1.000000 5.000000e-01 5.000000e-01 282 10 17 0.562500 1.000000 5.625000e-01 5.625000e-01 283 11 17 0.625000 1.000000 6.250000e-01 6.250000e-01 284 12 17 0.687500 1.000000 6.875000e-01 6.875000e-01 285 13 17 0.750000 1.000000 7.500000e-01 7.500000e-01 286 14 17 0.812500 1.000000 8.125000e-01 8.125000e-01 287 15 17 0.875000 1.000000 8.750000e-01 8.750000e-01 288 16 17 0.937500 1.000000 9.375000e-01 9.375000e-01 289 17 17 1.000000 1.000000 1.000000e+00 1.000000e+00 ||U|| = 0.996235 ||W|| = 0.994293 ||U-W|| = 0.062347 FEM2D_LEASTSQUARES: Normal end of execution. >>