1 00:00:00,000 --> 00:00:01,490 INTRODUCTION: The following content 2 00:00:01,490 --> 00:00:04,410 is provided by MIT OpenCourseWare under a Creative 3 00:00:04,410 --> 00:00:06,410 Commons license. 4 00:00:06,410 --> 00:00:08,200 Additional information about our license 5 00:00:08,200 --> 00:00:10,570 and MIT OpenCourseWare in general 6 00:00:10,570 --> 00:00:12,240 is available at OCW.MIT.edu. 7 00:00:15,397 --> 00:00:15,980 PROFESSOR: OK. 8 00:00:15,980 --> 00:00:20,500 Ready for today's discussion, which 9 00:00:20,500 --> 00:00:21,940 I'm quite sort of happy about. 10 00:00:21,940 --> 00:00:24,180 I hadn't really seen this coming. 11 00:00:27,720 --> 00:00:29,720 First, we haven't said anything about how 12 00:00:29,720 --> 00:00:33,730 do you estimate the error in a problem. 13 00:00:33,730 --> 00:00:40,070 Because we're often taking a continuous problem coming from 14 00:00:40,070 --> 00:00:47,720 a differential equation, typically, and discretizing it, 15 00:00:47,720 --> 00:00:50,040 and solving it -- solving the discrete problem. 16 00:00:50,040 --> 00:00:52,070 And we want some idea. 17 00:00:52,070 --> 00:00:54,600 It's not just some math question of course, 18 00:00:54,600 --> 00:01:03,470 because, you know, in engineering design or analysis, 19 00:01:03,470 --> 00:01:06,670 an idea of what the error is critical. 20 00:01:06,670 --> 00:01:08,690 And secondly, it gives us an idea 21 00:01:08,690 --> 00:01:12,730 of where we can improve it. 22 00:01:12,730 --> 00:01:17,340 You know, if we see what's controlling the error then 23 00:01:17,340 --> 00:01:18,830 we know what's going on. 24 00:01:21,700 --> 00:01:29,170 So I'm speaking now about steady-state problems. 25 00:01:29,170 --> 00:01:34,130 We discussed error for initial value problems, 26 00:01:34,130 --> 00:01:38,590 and we realized there that the error depended on -- 27 00:01:38,590 --> 00:01:40,720 usually finite differences. 28 00:01:40,720 --> 00:01:43,280 So the error we knew we could figure out 29 00:01:43,280 --> 00:01:47,490 locally from the finite difference that we chose. 30 00:01:47,490 --> 00:01:51,400 You know, we chose second order accurate differences 31 00:01:51,400 --> 00:01:54,550 frequently, but we could've moved up to fourth. 32 00:01:54,550 --> 00:01:56,140 And then the other ingredient you 33 00:01:56,140 --> 00:02:00,230 need is stability, to know that those local errors don't 34 00:02:00,230 --> 00:02:04,400 explode as time goes forward. 35 00:02:04,400 --> 00:02:04,900 OK. 36 00:02:04,900 --> 00:02:09,380 So we had something to say about error and accuracy 37 00:02:09,380 --> 00:02:12,650 at that time about that topic. 38 00:02:12,650 --> 00:02:17,730 And now I'm realizing that many of our other topic 39 00:02:17,730 --> 00:02:19,250 fall in this category. 40 00:02:19,250 --> 00:02:21,820 And I put three of the topics here. 41 00:02:21,820 --> 00:02:30,110 In the idea sort of where the underlying discretization is 42 00:02:30,110 --> 00:02:31,530 a projection. 43 00:02:31,530 --> 00:02:35,720 It's somehow a projection between the true solution 44 00:02:35,720 --> 00:02:41,470 and the discrete, computed solution. 45 00:02:41,470 --> 00:02:43,700 Somehow the discrete, computed solution 46 00:02:43,700 --> 00:02:49,430 is in some finite-dimensional family where we can compute, 47 00:02:49,430 --> 00:02:52,080 and we're projecting into that family. 48 00:02:52,080 --> 00:02:53,464 And here are three examples. 49 00:02:53,464 --> 00:02:55,630 So you'll see that, actually, what I'm talking about 50 00:02:55,630 --> 00:03:01,650 is central to the whole semester. 51 00:03:01,650 --> 00:03:08,970 Well, finite elements actually appeared more in 18.085. 52 00:03:08,970 --> 00:03:12,800 So I'll have to recap a little bit there. 53 00:03:12,800 --> 00:03:14,720 But they would be the natural tool 54 00:03:14,720 --> 00:03:19,100 for solving that list of continuous problems 55 00:03:19,100 --> 00:03:21,980 from calculus of variations that I talked about last time, 56 00:03:21,980 --> 00:03:27,160 minimizing some energy, where the energy is, typically, 57 00:03:27,160 --> 00:03:31,870 an integral and we're in the continuous case. 58 00:03:31,870 --> 00:03:36,180 And what's the finite element central idea? 59 00:03:36,180 --> 00:03:42,310 It is choose these guys, choose some basis functions, 60 00:03:42,310 --> 00:03:47,280 and look at their combinations. 61 00:03:47,280 --> 00:03:49,160 Maybe choose N basis functions. 62 00:03:49,160 --> 00:03:52,410 I guess I better get a chalk or it's 63 00:03:52,410 --> 00:03:54,860 going to be a short lecture. 64 00:03:54,860 --> 00:03:55,360 OK. 65 00:03:55,360 --> 00:04:00,420 So we have a continuous problem, continuous ODE, 66 00:04:00,420 --> 00:04:03,420 ordinary differential equation or a partial differential 67 00:04:03,420 --> 00:04:07,170 equation, with boundary conditions, as always, 68 00:04:07,170 --> 00:04:09,870 that we want to make discrete. 69 00:04:09,870 --> 00:04:12,800 And the point is I am not making a discrete 70 00:04:12,800 --> 00:04:17,440 by finite differences. 71 00:04:17,440 --> 00:04:22,170 This is a different route from finite differences. 72 00:04:22,170 --> 00:04:28,330 And in Laplace's equation and these, in many, many problems, 73 00:04:28,330 --> 00:04:31,050 this is the preferred route. 74 00:04:31,050 --> 00:04:34,870 And finite elements are a particular choice 75 00:04:34,870 --> 00:04:36,210 of these guys. 76 00:04:36,210 --> 00:04:42,675 The Galerkin idea -- so I'll use his name for the overall idea 77 00:04:42,675 --> 00:04:48,160 -- is choose these, and look for the best combination, 78 00:04:48,160 --> 00:04:51,890 and we have to say what does best mean. 79 00:04:51,890 --> 00:04:56,190 Well, best is going to mean -- in our minimization problems, 80 00:04:56,190 --> 00:05:00,730 it will be the combination that gives the minimum. 81 00:05:00,730 --> 00:05:05,760 The combination of these n trial functions. 82 00:05:05,760 --> 00:05:11,290 Now the exact solution is not going 83 00:05:11,290 --> 00:05:15,170 to be in that little n-dimensional space. 84 00:05:15,170 --> 00:05:17,660 Those functions for finite elements 85 00:05:17,660 --> 00:05:22,080 might be piecewise linear, or if you want to upgrade it, 86 00:05:22,080 --> 00:05:24,270 they might be piecewise parabolas, 87 00:05:24,270 --> 00:05:26,790 piecewise cubics, whatever. 88 00:05:26,790 --> 00:05:29,240 That was the brilliant finite element idea, 89 00:05:29,240 --> 00:05:32,030 to choose simple functions. 90 00:05:32,030 --> 00:05:36,080 And then you could take N quite large, 91 00:05:36,080 --> 00:05:39,490 but you're still not getting the exact solution of course, 92 00:05:39,490 --> 00:05:42,850 and it's to estimate how far off you are. 93 00:05:42,850 --> 00:05:46,900 But then also in multigrid, what was the idea in multigrid? 94 00:05:46,900 --> 00:05:51,600 We started with a system at level h. 95 00:05:51,600 --> 00:05:57,830 We started with some problem A_h*u_h equal f_h, 96 00:05:57,830 --> 00:06:01,730 which was a big system. 97 00:06:01,730 --> 00:06:05,500 If we did ordinary Jacobi or Gauss-Seidel, 98 00:06:05,500 --> 00:06:07,220 that was too slow. 99 00:06:07,220 --> 00:06:09,970 The multigrid idea was project, there's 100 00:06:09,970 --> 00:06:15,940 that magic word project, onto to a coarse grid, 101 00:06:15,940 --> 00:06:19,420 where the problem is smaller. 102 00:06:19,420 --> 00:06:21,540 Here, we started with a continuous problem, 103 00:06:21,540 --> 00:06:24,890 and we got it down to size N. Here we 104 00:06:24,890 --> 00:06:27,780 start with a discrete problem and we 105 00:06:27,780 --> 00:06:32,610 cut its size in half or in quarter or in eighth, again. 106 00:06:32,610 --> 00:06:36,750 So projection is producing a smaller problem, 107 00:06:36,750 --> 00:06:40,580 and the question is what are you losing? 108 00:06:40,580 --> 00:06:42,330 And actually conjugate gradients. 109 00:06:47,160 --> 00:06:48,520 You remember these spaces? 110 00:06:48,520 --> 00:06:54,400 The spaces span the combinations of b, A*b, 111 00:06:54,400 --> 00:06:56,840 A squared b and so on. 112 00:06:56,840 --> 00:07:03,750 That was the computationally convenient subspace 113 00:07:03,750 --> 00:07:05,600 to project onto. 114 00:07:05,600 --> 00:07:14,730 And so we've got important examples. 115 00:07:14,730 --> 00:07:19,300 Now what I realized, I'm pleased about this, 116 00:07:19,300 --> 00:07:22,810 is that they all fit. 117 00:07:22,810 --> 00:07:27,440 So here's my problem, what's the error? 118 00:07:27,440 --> 00:07:31,010 So I'm going to u star for the correct answer. 119 00:07:31,010 --> 00:07:32,620 This is the true. 120 00:07:35,230 --> 00:07:39,320 And I'm going to use U star, capital U star, 121 00:07:39,320 --> 00:07:44,360 for the approximate, the one that we 122 00:07:44,360 --> 00:07:51,680 get by any of those key ideas in numerical analysis. 123 00:07:51,680 --> 00:07:54,170 And I'm trying to estimate the difference. 124 00:07:54,170 --> 00:07:55,100 OK. 125 00:07:55,100 --> 00:07:57,220 And I just put up here that I'm dealing 126 00:07:57,220 --> 00:08:00,260 with positive definite problems. 127 00:08:00,260 --> 00:08:04,640 In fact, the matrix K, the symmetric positive definite 128 00:08:04,640 --> 00:08:09,740 problem you have, often has this A transposed C*A form, 129 00:08:09,740 --> 00:08:15,830 or in the continuous case, K might stand for -- 130 00:08:15,830 --> 00:08:20,480 the A transpose might be minus the derivative; 131 00:08:20,480 --> 00:08:26,150 the C might be a variable or constant, 132 00:08:26,150 --> 00:08:31,510 the physical coefficient; A would be d by dx, 133 00:08:31,510 --> 00:08:34,460 that would be the K thing. 134 00:08:38,470 --> 00:08:42,470 The equation I want to get is K*u equal f. 135 00:08:42,470 --> 00:08:46,080 That's the strong form. 136 00:08:46,080 --> 00:08:48,930 Strong form will be K*u equal f. 137 00:08:48,930 --> 00:08:51,920 But the whole point of last lecture and this one 138 00:08:51,920 --> 00:08:55,410 is that we don't get to the strong form. 139 00:08:55,410 --> 00:08:59,040 This projection starts with a minimum problem, 140 00:08:59,040 --> 00:09:02,090 and gets to a weak form. 141 00:09:02,090 --> 00:09:05,830 And it's the minimum problem or the weak form 142 00:09:05,830 --> 00:09:09,130 that we want to think about, not the strong form. 143 00:09:09,130 --> 00:09:10,150 OK. 144 00:09:10,150 --> 00:09:11,490 Now. 145 00:09:11,490 --> 00:09:16,120 Up there I wrote an identity. 146 00:09:16,120 --> 00:09:19,240 If you just multiply that right hand side out, 147 00:09:19,240 --> 00:09:20,790 I believe it comes out right. 148 00:09:23,430 --> 00:09:29,370 And it's very valuable for our purposes here. 149 00:09:29,370 --> 00:09:30,290 OK. 150 00:09:30,290 --> 00:09:32,630 So what's on the left side? 151 00:09:32,630 --> 00:09:35,310 This is the quantity that we're minimizing, 152 00:09:35,310 --> 00:09:42,520 we're minimizing this. 153 00:09:42,520 --> 00:09:48,930 And by just manipulation, we wrote it this way. 154 00:09:51,580 --> 00:09:56,320 So now I can identify what u star is. 155 00:09:56,320 --> 00:09:59,030 So I'm going to minimize this over all u. 156 00:10:02,770 --> 00:10:06,040 That certainly looks like a discrete problem, right? 157 00:10:06,040 --> 00:10:12,040 So think of u as a vector, f as a vector, 158 00:10:12,040 --> 00:10:15,270 K as a positive definite symmetric matrix. 159 00:10:15,270 --> 00:10:16,820 Just think about that. 160 00:10:16,820 --> 00:10:23,150 But I want it to apply very much to the continuous case 161 00:10:23,150 --> 00:10:24,980 too, to differential equations. 162 00:10:24,980 --> 00:10:29,640 But here's a point then, what's the winner? 163 00:10:29,640 --> 00:10:33,780 You can't immediately see -- well maybe you can, somehow, 164 00:10:33,780 --> 00:10:37,870 see that when I set the derivative of this thing to 0, 165 00:10:37,870 --> 00:10:38,970 I get that equation. 166 00:10:42,210 --> 00:10:46,280 You can kind of believe it, even if it's vectors and matrices. 167 00:10:46,280 --> 00:10:50,240 But here you can see right away that -- 168 00:10:50,240 --> 00:10:52,460 how do I make this small? 169 00:10:52,460 --> 00:10:55,050 This is a constant here. 170 00:10:55,050 --> 00:10:59,830 That's just a constant, so it doesn't depend on u. 171 00:10:59,830 --> 00:11:04,880 So what choice of u makes that term small? 172 00:11:04,880 --> 00:11:08,290 Well of course it's the choice of u 173 00:11:08,290 --> 00:11:12,190 is the one that makes this thing 0, because that's 174 00:11:12,190 --> 00:11:14,850 a positive definite matrix. 175 00:11:14,850 --> 00:11:18,800 No way anything is going to get negative here. 176 00:11:18,800 --> 00:11:21,320 This is a something transposed K something. 177 00:11:21,320 --> 00:11:24,380 You remember what positive definite means. 178 00:11:24,380 --> 00:11:32,660 It means that x transpose K*x, for every vector x, 179 00:11:32,660 --> 00:11:34,740 is never negative. 180 00:11:34,740 --> 00:11:36,680 So the best we could do would be to bring 181 00:11:36,680 --> 00:11:38,920 this to 0, and of course to bring it to 0 182 00:11:38,920 --> 00:11:42,410 is to bring the x to 0, to bring this thing to 0, 183 00:11:42,410 --> 00:11:44,580 so this thing should be 0. 184 00:11:44,580 --> 00:11:47,700 So u minus K inverse f should be 0. 185 00:11:47,700 --> 00:11:52,810 And of course, that leads us back to the same conclusion 186 00:11:52,810 --> 00:11:55,930 that we reached from the -- the strong form. 187 00:11:55,930 --> 00:11:56,950 OK. 188 00:11:56,950 --> 00:11:58,310 Good. 189 00:11:58,310 --> 00:12:00,870 So that an identity, but now I want 190 00:12:00,870 --> 00:12:08,010 to use this identity to answer the question about what if I 191 00:12:08,010 --> 00:12:11,200 minimize only on a subspace? 192 00:12:11,200 --> 00:12:13,700 That's the problem for today. 193 00:12:13,700 --> 00:12:15,270 So that's a question. 194 00:12:15,270 --> 00:12:17,670 Maybe I'll write it on this fresh board. 195 00:12:17,670 --> 00:12:20,870 Now I minimize only over some subspace. 196 00:12:20,870 --> 00:12:27,290 Now can I can I use capital U for the -- 197 00:12:27,290 --> 00:12:34,530 I used little u for, in here, allowing it to be any vector, 198 00:12:34,530 --> 00:12:36,450 and I found a winner. 199 00:12:36,450 --> 00:12:42,610 And let me give the winner a name, u star. 200 00:12:42,610 --> 00:12:50,800 It's a real headache in this subject of what -- 201 00:12:50,800 --> 00:12:52,500 just the notation. 202 00:12:52,500 --> 00:12:54,540 What should we call the winner? 203 00:12:54,540 --> 00:12:58,200 So sometimes I call it u hat, that 204 00:12:58,200 --> 00:13:01,780 was a familiar notation in estimation theory, 205 00:13:01,780 --> 00:13:03,210 least squares. 206 00:13:03,210 --> 00:13:06,990 Today I'm going to call it u star. 207 00:13:06,990 --> 00:13:10,100 So it's u star is the winner, small u star. 208 00:13:13,880 --> 00:13:21,910 Let me take time-out, one minute time-out, on a board -- I mean, 209 00:13:21,910 --> 00:13:25,490 optimization is about the problem of minimizing some 210 00:13:25,490 --> 00:13:31,356 function F of x. 211 00:13:31,356 --> 00:13:32,730 I'm just going to take one minute 212 00:13:32,730 --> 00:13:35,940 on the problems of an author. 213 00:13:35,940 --> 00:13:39,830 What do you call the winning function, winning vector, 214 00:13:39,830 --> 00:13:41,416 or the winning x? 215 00:13:41,416 --> 00:13:42,790 I mean it could be just a scalar, 216 00:13:42,790 --> 00:13:45,160 we could be doing just calculus. 217 00:13:45,160 --> 00:13:49,620 What do I call the x that gives the minimum? 218 00:13:49,620 --> 00:13:52,960 You may have a favorite, you can't call it x right? 219 00:13:52,960 --> 00:13:55,520 I mean, because that's just confusing it 220 00:13:55,520 --> 00:13:57,990 with the variable x. 221 00:13:57,990 --> 00:14:00,380 So I'm saying, well, you could call it x hat, 222 00:14:00,380 --> 00:14:10,440 you could call it x star, you call it capital X. 223 00:14:10,440 --> 00:14:12,600 So I'll just write a few of those down -- 224 00:14:12,600 --> 00:14:15,930 x star would be a possibility, x hat would be a possibility, 225 00:14:15,930 --> 00:14:20,420 x minimizing would be a possibility. 226 00:14:20,420 --> 00:14:21,890 I'm doing this just because I want 227 00:14:21,890 --> 00:14:25,220 to write down the thing that you often see, 228 00:14:25,220 --> 00:14:34,750 which is argmin of F of x. 229 00:14:34,750 --> 00:14:37,990 And I write that down just so that if you ever see it, 230 00:14:37,990 --> 00:14:40,460 you know what the heck it means. 231 00:14:40,460 --> 00:14:44,270 It has the same meaning as any of these. 232 00:14:44,270 --> 00:14:48,240 I am not a big fan of that. 233 00:14:48,240 --> 00:14:51,150 But what does this mean? 234 00:14:51,150 --> 00:14:57,960 It means the argument that gives the minimum of F of x, right. 235 00:14:57,960 --> 00:15:04,630 Argument is a fancy word for the variable in the function. 236 00:15:04,630 --> 00:15:06,780 So this says the argument that gives 237 00:15:06,780 --> 00:15:08,680 the minimum of F of x, and that's 238 00:15:08,680 --> 00:15:11,530 what we're looking for to name. 239 00:15:11,530 --> 00:15:17,290 But I'm sure not happy about writing that name down. 240 00:15:17,290 --> 00:15:23,590 So here it goes, it disappears. 241 00:15:23,590 --> 00:15:28,130 But you'll see it, and now you now want it means. 242 00:15:28,130 --> 00:15:30,100 OK. 243 00:15:30,100 --> 00:15:36,230 So this is now the central issue here. 244 00:15:36,230 --> 00:15:45,880 I want to minimize my same guy, 1/2 u transpose K*u minus u 245 00:15:45,880 --> 00:15:47,120 transpose f. 246 00:15:47,120 --> 00:15:52,640 Or, which is exactly the same, that same right-hand side, 247 00:15:52,640 --> 00:15:57,030 because the two are equal, I want to minimize over some 248 00:15:57,030 --> 00:16:00,150 capital U's, not all U's. 249 00:16:00,150 --> 00:16:04,510 If I minimize over all U's, then I get the exact answer. 250 00:16:04,510 --> 00:16:07,380 But the idea of all these numerical methods 251 00:16:07,380 --> 00:16:11,370 is minimize over some finite-dimensional, 252 00:16:11,370 --> 00:16:15,100 smaller-dimensional, subspace of trial functions. 253 00:16:15,100 --> 00:16:16,100 I don't know. 254 00:16:16,100 --> 00:16:21,770 I'll say minimize over U in the trial space, 255 00:16:21,770 --> 00:16:24,260 just to write it out in words. 256 00:16:24,260 --> 00:16:24,760 OK. 257 00:16:30,410 --> 00:16:34,900 Well, now I guess my name for the winner in the trial space 258 00:16:34,900 --> 00:16:36,810 is going to be U star. 259 00:16:36,810 --> 00:16:48,810 So the winner in the trial space will be U star. 260 00:16:48,810 --> 00:16:52,920 So that's my finite element solution, my conjugate gradient 261 00:16:52,920 --> 00:16:57,190 solution, my multigrid solution. 262 00:16:57,190 --> 00:17:04,230 All these problems are reducing to a smaller trial space, 263 00:17:04,230 --> 00:17:09,010 and picking the winner there, computing the winner there. 264 00:17:09,010 --> 00:17:14,760 And then the question of today is how far apart are the two? 265 00:17:14,760 --> 00:17:16,700 OK. 266 00:17:16,700 --> 00:17:18,930 And now there's this little formula 267 00:17:18,930 --> 00:17:21,790 that gives us a good idea. 268 00:17:21,790 --> 00:17:25,000 This little formula gives us a handle on that. 269 00:17:25,000 --> 00:17:34,940 So now, the point is -- So now, can I look at this formula? 270 00:17:34,940 --> 00:17:37,290 Maybe maybe I'll copy the formula here. 271 00:17:37,290 --> 00:17:44,790 This is the minimum over all these trial guys, of 1/2 -- oh, 272 00:17:44,790 --> 00:17:50,040 but I'm going to write it that way -- 1/2 U, 273 00:17:50,040 --> 00:17:53,250 that's my trial guy, minus K inverse f, 274 00:17:53,250 --> 00:17:55,380 that's my u star now. 275 00:17:55,380 --> 00:17:58,530 Am I OK to give that name to K inverse f? 276 00:17:58,530 --> 00:18:02,410 Yeah, I already gave it, I guess. u star is K inverse f. 277 00:18:02,410 --> 00:18:04,331 So that's cool. 278 00:18:04,331 --> 00:18:04,830 Shorter. 279 00:18:04,830 --> 00:18:06,530 It's shorter and better. 280 00:18:06,530 --> 00:18:15,400 K -- or its transpose -- U minus u star plus a constant. 281 00:18:15,400 --> 00:18:17,470 So I can forget that. 282 00:18:17,470 --> 00:18:20,130 I can forget this constant part. 283 00:18:20,130 --> 00:18:27,400 That's not important to us. 284 00:18:27,400 --> 00:18:33,500 So this simple identity then has expressed 285 00:18:33,500 --> 00:18:45,820 our key discretization approach here as finding the U, 286 00:18:45,820 --> 00:18:51,720 and we're going to call it U star, that's nearest to u star. 287 00:18:51,720 --> 00:18:58,361 That's the great fact that makes the whole subject pleasant. 288 00:18:58,361 --> 00:18:58,860 OK. 289 00:18:58,860 --> 00:19:04,160 So the winner U star is then, by this, 290 00:19:04,160 --> 00:19:06,570 since I'm minimizing that expression, 291 00:19:06,570 --> 00:19:16,510 it's certainly the nearest in the K norm -- 292 00:19:16,510 --> 00:19:23,090 it's the nearest weighted by K, somehow that K is important. 293 00:19:23,090 --> 00:19:30,090 That K is reflecting the problem we're solving. 294 00:19:30,090 --> 00:19:38,490 That K is the energy or whatever -- to U star. 295 00:19:38,490 --> 00:19:39,210 OK. 296 00:19:39,210 --> 00:19:41,940 That's the great conclusion. 297 00:19:41,940 --> 00:19:43,880 You could say that's the fundamental theorem 298 00:19:43,880 --> 00:19:51,700 of this projection, error estimates and stuff. 299 00:19:51,700 --> 00:19:54,110 So do you see that we got to that? 300 00:19:54,110 --> 00:19:58,130 The thing that minimizes makes this as small as we can. 301 00:19:58,130 --> 00:20:04,340 So nearest is simply a translation of what that says. 302 00:20:04,340 --> 00:20:07,840 Pick the capital U that's closest to u star. 303 00:20:07,840 --> 00:20:08,770 OK. 304 00:20:08,770 --> 00:20:11,710 Now how does that help? 305 00:20:11,710 --> 00:20:15,440 That helps because I want to estimate the difference. 306 00:20:15,440 --> 00:20:26,950 So now I estimate the difference of U star minus u star. 307 00:20:32,480 --> 00:20:36,010 So I'm trying to get a handle on how different those are. 308 00:20:36,010 --> 00:20:44,420 How different is the -- so let me take, in this case of, say, 309 00:20:44,420 --> 00:20:49,910 Laplace's equation or the 1D case -- again, 310 00:20:49,910 --> 00:20:53,560 I'm speaking steady-state boundary value problem. 311 00:20:53,560 --> 00:20:58,150 So U star is the best combination of these trial 312 00:20:58,150 --> 00:21:05,440 functions, capital U, and little u star is the exact solution. 313 00:21:05,440 --> 00:21:06,480 Let me draw a picture. 314 00:21:09,070 --> 00:21:13,600 So suppose I'm in 1D, 0 to 1. 315 00:21:13,600 --> 00:21:14,100 OK. 316 00:21:14,100 --> 00:21:18,280 Yeah, let me pick that model problem again. 317 00:21:18,280 --> 00:21:29,560 So minus the derivative of c*du/dx is f of x. 318 00:21:29,560 --> 00:21:35,040 With boundary conditions. 319 00:21:35,040 --> 00:21:38,870 And I guess if I'm consistent now, I should write u star, 320 00:21:38,870 --> 00:21:41,180 because it's the winning solution. 321 00:21:41,180 --> 00:21:44,630 And suppose the boundary conditions are 0 at both ends, 322 00:21:44,630 --> 00:21:46,150 and it does something. 323 00:21:46,150 --> 00:21:47,990 OK. 324 00:21:47,990 --> 00:21:48,970 So that's u star. 325 00:21:52,040 --> 00:21:54,270 OK. 326 00:21:54,270 --> 00:21:57,020 So that's a continuous case, which 327 00:21:57,020 --> 00:22:01,500 I'm drawing a picture to represent what 328 00:22:01,500 --> 00:22:03,170 the solution might look like. 329 00:22:03,170 --> 00:22:03,890 OK. 330 00:22:03,890 --> 00:22:08,160 Now what about this finite element stuff? 331 00:22:08,160 --> 00:22:14,450 Suppose the phi_i's are linear pieces. 332 00:22:14,450 --> 00:22:20,450 I'm going to do linear finite element method. 333 00:22:20,450 --> 00:22:24,610 Then any combination of these linear guys 334 00:22:24,610 --> 00:22:27,810 is going to be piecewise linear, there's going to be a mesh. 335 00:22:27,810 --> 00:22:30,200 You know the set-up. 336 00:22:30,200 --> 00:22:33,940 It might look like that, and have a value there. 337 00:22:33,940 --> 00:22:36,050 It might have a value there, might have a value 338 00:22:36,050 --> 00:22:39,710 there, there, there, and there. 339 00:22:39,710 --> 00:22:41,770 OK. 340 00:22:41,770 --> 00:22:47,650 And let's suppose this is the winning -- 341 00:22:47,650 --> 00:22:52,880 doesn't look like a winner to me, 342 00:22:52,880 --> 00:22:55,690 because I think it could probably do better, but -- 343 00:22:55,690 --> 00:22:57,390 capital U star. 344 00:22:57,390 --> 00:23:01,070 But remember that we're measuring -- 345 00:23:01,070 --> 00:23:04,210 what's the K norm stuff? 346 00:23:04,210 --> 00:23:07,150 I'm measuring the difference between these two not 347 00:23:07,150 --> 00:23:11,860 pointwise, which would be of course pleasant 348 00:23:11,860 --> 00:23:14,530 to say, OK, the distance is just, you know, 349 00:23:14,530 --> 00:23:21,200 maybe the maximum distance or the mean square error. 350 00:23:21,200 --> 00:23:24,200 That would be quite pleasant. 351 00:23:24,200 --> 00:23:30,280 But here the measure of distance involves this K, 352 00:23:30,280 --> 00:23:35,740 which comes with the problem. 353 00:23:35,740 --> 00:23:41,510 So by distance here I mean the distance 354 00:23:41,510 --> 00:23:45,320 between U star and u star. 355 00:23:45,320 --> 00:23:51,360 Can I write it with a capital K there to indicate 356 00:23:51,360 --> 00:23:53,080 that that's the K norm? 357 00:23:53,080 --> 00:23:59,500 That's the norm in which this is small, as small as can be made. 358 00:23:59,500 --> 00:24:02,840 And what is the K norm for this particular problem? 359 00:24:02,840 --> 00:24:08,450 Well it's the integral, and involves the c, 360 00:24:08,450 --> 00:24:15,680 and it involves the U star prime minus the u star prime square 361 00:24:15,680 --> 00:24:22,810 dx, integrated from 0 to 1. 362 00:24:22,810 --> 00:24:25,440 I'm just picking this example problem 363 00:24:25,440 --> 00:24:31,270 so that you get some idea of how we're measuring the error. 364 00:24:31,270 --> 00:24:33,090 It's the natural measure for the error. 365 00:24:33,090 --> 00:24:34,240 It's the energy measure. 366 00:24:34,240 --> 00:24:36,680 We're measuring error in energy. 367 00:24:36,680 --> 00:24:40,410 This is an energy expression. 368 00:24:40,410 --> 00:24:44,320 This thing, you know, represents some kind 369 00:24:44,320 --> 00:24:50,140 of elastic bar or something, so we're measuring 370 00:24:50,140 --> 00:24:51,910 the internal energy here. 371 00:24:51,910 --> 00:24:55,390 And notice in particular, maybe the most important point 372 00:24:55,390 --> 00:25:00,180 is not the c of x, which just comes along for the ride, 373 00:25:00,180 --> 00:25:08,990 but the fact that our measure of the error is in the derivative. 374 00:25:08,990 --> 00:25:12,610 We're measuring error in slopes, because those 375 00:25:12,610 --> 00:25:17,370 are the stresses in the bar, and that's 376 00:25:17,370 --> 00:25:23,410 where the energy comes from, internal strain energy. 377 00:25:23,410 --> 00:25:24,290 OK. 378 00:25:24,290 --> 00:25:32,470 So in other words, I have this function u star, 379 00:25:32,470 --> 00:25:39,020 this curved guy, and I have this function, which 380 00:25:39,020 --> 00:25:41,220 I'm thinking to be the winner. 381 00:25:41,220 --> 00:25:43,440 And again, it's the winner in the sense 382 00:25:43,440 --> 00:25:45,670 that it minimizes this. 383 00:25:45,670 --> 00:25:47,470 This is the minimum. 384 00:25:47,470 --> 00:25:51,800 This is the expression that we have some handle on, 385 00:25:51,800 --> 00:25:56,100 because we know that U star, capital U star, 386 00:25:56,100 --> 00:25:59,080 will make that as small as it can. 387 00:25:59,080 --> 00:26:01,610 It does not make small the pointwise error. 388 00:26:04,850 --> 00:26:08,700 It might try, it might accidentally -- 389 00:26:08,700 --> 00:26:11,620 we hope it does of course -- get pointwise error right. 390 00:26:11,620 --> 00:26:17,060 But what it is constructed to get right is energy error. 391 00:26:17,060 --> 00:26:17,960 Make that small. 392 00:26:17,960 --> 00:26:18,630 OK. 393 00:26:18,630 --> 00:26:22,860 So now the question is how do we estimate the difference? 394 00:26:22,860 --> 00:26:23,550 OK. 395 00:26:23,550 --> 00:26:25,750 So now, here's this key point, how 396 00:26:25,750 --> 00:26:29,060 do we estimate the difference. 397 00:26:29,060 --> 00:26:32,310 Again we're looking at the error. 398 00:26:32,310 --> 00:26:39,330 By the way, I should have, maybe I did, put a square there. 399 00:26:39,330 --> 00:26:42,380 That was the norm squared, but of course, 400 00:26:42,380 --> 00:26:46,610 minimizing the norm -- you realize why I don't want a big 401 00:26:46,610 --> 00:26:48,990 square root sign, it's just clumsy. 402 00:26:48,990 --> 00:26:51,400 So I'm looking at the square there. 403 00:26:51,400 --> 00:26:52,115 OK. 404 00:26:52,115 --> 00:26:53,740 So now, how do you estimate that thing? 405 00:26:53,740 --> 00:26:58,280 How do you estimate the thing, knowing 406 00:26:58,280 --> 00:27:05,060 that this piecewise linear, that came out of some finite element 407 00:27:05,060 --> 00:27:10,940 calculation or some giant code, is best possible? 408 00:27:10,940 --> 00:27:13,670 Well here's the idea. 409 00:27:13,670 --> 00:27:28,980 Look at a convenient candidate that might not be the winner. 410 00:27:31,570 --> 00:27:36,340 Let me put, because this was as small as possible, 411 00:27:36,340 --> 00:27:43,330 this is less than or equal to U minus u star -- 412 00:27:43,330 --> 00:27:54,130 always in the correct measure -- for every trial function U. 413 00:27:54,130 --> 00:27:59,380 This just says what I've said now three ways, 414 00:27:59,380 --> 00:28:02,900 that U star's the best in the K norm. 415 00:28:02,900 --> 00:28:10,940 Now I -- to get some bound on this, I can take any U, 416 00:28:10,940 --> 00:28:15,530 I can take any U and estimate its difference from U star, 417 00:28:15,530 --> 00:28:17,410 and that will give me a bound. 418 00:28:17,410 --> 00:28:21,800 In other words, I don't know what this particular guy 419 00:28:21,800 --> 00:28:22,800 happened to be. 420 00:28:28,400 --> 00:28:31,770 Let me just jump to the key idea. 421 00:28:31,770 --> 00:28:35,730 I know that that one is better than the one -- 422 00:28:35,730 --> 00:28:39,040 and if I had a different color -- 423 00:28:39,040 --> 00:28:43,340 maybe this thing has got another color. 424 00:28:48,500 --> 00:28:52,500 Okay, here's another color. 425 00:28:52,500 --> 00:28:54,490 I know it's better than, for example -- 426 00:28:54,490 --> 00:29:00,050 I'm just going to pick one piecewise linear trial function 427 00:29:00,050 --> 00:29:01,520 that's quite convenient. 428 00:29:01,520 --> 00:29:08,780 Pick the one that interpolates the exact one. 429 00:29:08,780 --> 00:29:10,150 OK. 430 00:29:10,150 --> 00:29:15,040 Now for some reason, known only to finite elements, 431 00:29:15,040 --> 00:29:16,970 that wasn't the finite element winner. 432 00:29:16,970 --> 00:29:22,500 That wasn't U star, that was another U. For example -- 433 00:29:22,500 --> 00:29:29,620 so like, I'll put it in blue here -- for example, 434 00:29:29,620 --> 00:29:38,760 take U to be the function that interpolates u star, 435 00:29:38,760 --> 00:29:43,500 little u star, the function that I've drawn here. 436 00:29:43,500 --> 00:29:50,020 Since our question is how close can we get to the curved guy 437 00:29:50,020 --> 00:29:54,825 by piecewise linear, well, one choice 438 00:29:54,825 --> 00:29:58,455 is how close does the interpolate come. 439 00:29:58,455 --> 00:30:00,160 It doesn't necessarily come the closest. 440 00:30:00,160 --> 00:30:00,980 Probably not. 441 00:30:00,980 --> 00:30:04,350 But it's in the right ballpark. 442 00:30:04,350 --> 00:30:06,590 So now I just ask the question, and let 443 00:30:06,590 --> 00:30:11,660 me draw the same picture again. 444 00:30:11,660 --> 00:30:16,300 I have a function and I interpolate it 445 00:30:16,300 --> 00:30:19,990 by piecewise linear guys. 446 00:30:22,530 --> 00:30:27,330 So piecewise linear function there. 447 00:30:27,330 --> 00:30:32,520 And so this is a comparison between the u star 448 00:30:32,520 --> 00:30:41,840 and its interpolate, which is my candidate U that I'm 449 00:30:41,840 --> 00:30:45,670 recommending as a trial. 450 00:30:45,670 --> 00:30:49,260 And now it's a pure approximation question. 451 00:30:49,260 --> 00:30:53,880 You see, we no longer have to know all about finite elements, 452 00:30:53,880 --> 00:30:56,470 we don't have to know anything about finite elements. 453 00:30:56,470 --> 00:31:02,540 We're just asking the question: if you give me a function, 454 00:31:02,540 --> 00:31:07,750 and you compare it with the piecewise linear interpolate, 455 00:31:07,750 --> 00:31:09,940 how far apart are they? 456 00:31:09,940 --> 00:31:12,980 How far apart? 457 00:31:12,980 --> 00:31:21,150 And let me call the step size h, and of course, 458 00:31:21,150 --> 00:31:22,760 could be unequal steps. 459 00:31:22,760 --> 00:31:25,270 It could be unstructured mesh. 460 00:31:25,270 --> 00:31:28,860 Everything works here. 461 00:31:28,860 --> 00:31:34,410 Now we come to just a basic sort of understanding of calculus. 462 00:31:34,410 --> 00:31:41,540 How close does a curve come from the chord? 463 00:31:41,540 --> 00:31:43,720 Really, that's what it's come down to. 464 00:31:43,720 --> 00:31:49,210 How close is a curved function from a chord? 465 00:31:49,210 --> 00:31:51,490 I can even blow that up. 466 00:31:51,490 --> 00:31:55,050 So here I have some curve going up, 467 00:31:55,050 --> 00:31:57,450 and compare that with the chord. 468 00:32:01,430 --> 00:32:04,010 And this distance here is h. 469 00:32:04,010 --> 00:32:12,950 Anybody know? 470 00:32:12,950 --> 00:32:17,230 So I'm just looking for something like, 471 00:32:17,230 --> 00:32:21,930 is the difference of order h, is it of order h square, 472 00:32:21,930 --> 00:32:23,830 is it of order e to the h. 473 00:32:23,830 --> 00:32:27,360 What the distance between the two? 474 00:32:31,600 --> 00:32:34,560 And you could imagine it's a parabola, because I'm focusing 475 00:32:34,560 --> 00:32:36,270 down on just a little piece. 476 00:32:36,270 --> 00:32:39,690 I take a little parabola, a little h piece of it, 477 00:32:39,690 --> 00:32:43,050 and I compare it with a chord, what's 478 00:32:43,050 --> 00:32:45,000 the distance between the two? 479 00:32:45,000 --> 00:32:47,650 That distance there. 480 00:32:47,650 --> 00:32:53,220 Well your eye probably tells you that it's smaller than h, 481 00:32:53,220 --> 00:32:56,750 because h was this big, and I'm only 482 00:32:56,750 --> 00:32:59,190 looking this big vertically. 483 00:32:59,190 --> 00:33:06,250 So this distance, that distance there, maximum distance, 484 00:33:06,250 --> 00:33:08,240 is of order h square. 485 00:33:12,730 --> 00:33:14,100 But that's not the question. 486 00:33:16,870 --> 00:33:20,740 The question is how far apart are these, 487 00:33:20,740 --> 00:33:24,570 how far apart of the these in this measure? 488 00:33:27,230 --> 00:33:30,170 How far apart are the slopes? 489 00:33:30,170 --> 00:33:36,620 Because the K norm is dealing with the slopes, and not 490 00:33:36,620 --> 00:33:38,420 the function itself. 491 00:33:38,420 --> 00:33:43,500 So more important than the distance is the error in slope, 492 00:33:43,500 --> 00:33:47,770 and would you want to guess what that's like? 493 00:33:52,380 --> 00:33:55,860 What's your guess on that, the error in the slope? 494 00:33:55,860 --> 00:33:58,470 Now the slopes are not going to be as good as the function, 495 00:33:58,470 --> 00:33:59,700 as always. 496 00:33:59,700 --> 00:34:03,420 Slopes are one derivative higher, you lose something, 497 00:34:03,420 --> 00:34:07,930 you get order h. 498 00:34:07,930 --> 00:34:11,590 So that's the error. 499 00:34:14,310 --> 00:34:16,830 Let's see, I hope I'm right here. 500 00:34:16,830 --> 00:34:17,500 Yeah. 501 00:34:17,500 --> 00:34:19,370 I think that's right. 502 00:34:19,370 --> 00:34:25,440 There I was speaking pointwise, and then I've 503 00:34:25,440 --> 00:34:28,610 got to integrate it over the whole interval 504 00:34:28,610 --> 00:34:30,670 here, a unit interval. 505 00:34:30,670 --> 00:34:38,450 So if I square it, of course, I'm going to get h square, 506 00:34:38,450 --> 00:34:41,192 and then if I integrated it, I still have h square, 507 00:34:41,192 --> 00:34:43,150 and then when I take the square root, I have h. 508 00:34:43,150 --> 00:34:45,530 So h is the right quantity. 509 00:34:49,150 --> 00:34:54,940 If I can put down here what our conclusion was from this method 510 00:34:54,940 --> 00:35:03,640 by taking capital U to be this convenient -- 511 00:35:03,640 --> 00:35:06,910 not the only choice, but a convenient choice, 512 00:35:06,910 --> 00:35:09,290 just to get an idea what the error might be. 513 00:35:09,290 --> 00:35:11,760 And then the error from the actual U star 514 00:35:11,760 --> 00:35:14,100 is got to be better, our estimate 515 00:35:14,100 --> 00:35:22,310 is order of h for this particular application. 516 00:35:22,310 --> 00:35:23,810 This particular application. 517 00:35:23,810 --> 00:35:32,100 That's the error in energy norm in the slopes. 518 00:35:37,690 --> 00:35:41,380 So the theory of finite elements would go ahead 519 00:35:41,380 --> 00:35:47,540 to try to show that the error in the distance between them, 520 00:35:47,540 --> 00:35:52,320 in the displacement you could say, is h square. 521 00:35:52,320 --> 00:35:56,710 But that's not so easy, and I won't be able to do it here. 522 00:35:56,710 --> 00:36:02,250 Why is it not easy to say that the error in the displacement 523 00:36:02,250 --> 00:36:03,750 is of order h square? 524 00:36:03,750 --> 00:36:09,350 It's certainly true for this interpolate, right. 525 00:36:09,350 --> 00:36:12,530 There's no question that the interpolate you could easily 526 00:36:12,530 --> 00:36:17,190 see is h square, order h square away from the function. 527 00:36:17,190 --> 00:36:25,950 But the point is that capital U star was not 528 00:36:25,950 --> 00:36:30,330 the best in staying near the function, 529 00:36:30,330 --> 00:36:33,600 it was only the best in staying near the slope. 530 00:36:33,600 --> 00:36:40,440 So I don't have this crutch to lean on here. 531 00:36:40,440 --> 00:36:45,760 This is in the K norm, and not in the mean square norm. 532 00:36:45,760 --> 00:36:49,960 It's in a norm that deals with slope, but not 533 00:36:49,960 --> 00:36:53,901 with just plain displacement distance. 534 00:36:53,901 --> 00:36:54,400 OK. 535 00:36:54,400 --> 00:36:57,330 Maybe that's made the point. 536 00:36:57,330 --> 00:37:03,650 And this is the part of finite element theory just to do that. 537 00:37:03,650 --> 00:37:05,280 OK. 538 00:37:05,280 --> 00:37:11,280 If I could take a minute about notation, 539 00:37:11,280 --> 00:37:19,650 because the notation that I've used here of a K 540 00:37:19,650 --> 00:37:21,130 is really a matrix notation. 541 00:37:24,750 --> 00:37:27,660 You don't truly see it in finite element papers. 542 00:37:27,660 --> 00:37:30,040 For me, it's clear, right? 543 00:37:30,040 --> 00:37:34,160 I mean that identity was quite clear 544 00:37:34,160 --> 00:37:36,280 from vectors and matrices. 545 00:37:36,280 --> 00:37:41,720 But with finite elements I'm really dealing with functions. 546 00:37:41,720 --> 00:37:49,610 So it's not fair to use matrix notation, you know, 547 00:37:49,610 --> 00:37:54,150 when it's integrals, and derivatives, and functions 548 00:37:54,150 --> 00:37:56,090 that are involve here. 549 00:37:56,090 --> 00:37:59,910 You know, these are functions, and the actual solution 550 00:37:59,910 --> 00:38:01,140 is a function. 551 00:38:01,140 --> 00:38:13,500 So I all I want to do is mention the notation that you now see, 552 00:38:13,500 --> 00:38:20,820 say in 16.920, the engineering course that would be, you know, 553 00:38:20,820 --> 00:38:26,690 quite related to this one of finite elements and Galerkin 554 00:38:26,690 --> 00:38:30,840 and all sorts of stuff, that we'll touch in the remaining 555 00:38:30,840 --> 00:38:31,880 weeks. 556 00:38:31,880 --> 00:38:35,900 How would they write the minimization problem? 557 00:38:39,240 --> 00:38:45,470 The original problem now, going back to the original problem, 558 00:38:45,470 --> 00:38:47,560 and just saying, wait a minute, we didn't really 559 00:38:47,560 --> 00:38:49,600 have a good notation for it. 560 00:38:52,530 --> 00:38:56,400 The thing I'm looking for is a notation -- it's like this. 561 00:38:56,400 --> 00:39:05,770 This is the kind of thing that I want, c of x u prime square dx. 562 00:39:05,770 --> 00:39:10,360 Well, it might also have a first-order term 563 00:39:10,360 --> 00:39:15,160 a d of x, u square of x, it could have that. 564 00:39:15,160 --> 00:39:16,620 It could have second derivatives. 565 00:39:16,620 --> 00:39:21,380 It could be in two variables. 566 00:39:21,380 --> 00:39:30,390 For Laplace we had minimum of du/dx squared and du/dx squared 567 00:39:30,390 --> 00:39:39,470 dxdy, well and we had the linear term too. 568 00:39:39,470 --> 00:39:44,040 I've just written sort of the left-hand side, 569 00:39:44,040 --> 00:39:47,110 the quadratic term. 570 00:39:47,110 --> 00:39:51,860 And all I want to say is that everybody, well not everybody, 571 00:39:51,860 --> 00:39:56,570 but a lot of people now, would use the notation 572 00:39:56,570 --> 00:40:02,840 a of u, u for the quadratic term. 573 00:40:02,840 --> 00:40:05,540 So that a of u, u, in an engineering 574 00:40:05,540 --> 00:40:09,910 paper represents the internal strain energy, 575 00:40:09,910 --> 00:40:14,450 whether it's this, whether it's this, a combination. 576 00:40:14,450 --> 00:40:19,950 It somehow suggest to us it's quadratic. 577 00:40:19,950 --> 00:40:27,800 And there's a linear term, and that's often written l of u. 578 00:40:27,800 --> 00:40:30,560 So I better put down what l of u typically is. 579 00:40:30,560 --> 00:40:35,600 This l of u might be the integral of f of x u of x dx. 580 00:40:39,520 --> 00:40:41,220 That would be the linear term. 581 00:40:44,370 --> 00:40:50,940 And I think I'd be happier to have the 1/2 there. 582 00:40:50,940 --> 00:40:58,780 So that -- really, it's just a match with this, 583 00:40:58,780 --> 00:41:01,830 but somehow it's a little cooler. 584 00:41:05,290 --> 00:41:08,530 This looks so much like vectors and matrices, 585 00:41:08,530 --> 00:41:11,900 that that and that is kind of neutral. 586 00:41:11,900 --> 00:41:12,850 OK. 587 00:41:12,850 --> 00:41:15,295 So I'm just speaking about notation here, 588 00:41:15,295 --> 00:41:17,430 and I could've mentioned this last time 589 00:41:17,430 --> 00:41:21,720 when I was speaking about all these examples from calculus 590 00:41:21,720 --> 00:41:22,640 of variations. 591 00:41:22,640 --> 00:41:23,450 OK. 592 00:41:23,450 --> 00:41:25,740 So that's the minimum problem. 593 00:41:28,610 --> 00:41:32,890 If you give me that notation for the minimum problem, 594 00:41:32,890 --> 00:41:37,780 what's the weak form in this notation? 595 00:41:37,780 --> 00:41:40,490 So I'm introducing this just because you see it elsewhere. 596 00:41:40,490 --> 00:41:45,620 It's it's exactly what we're doing all the time, 597 00:41:45,620 --> 00:41:47,680 so I just want you to recognize it. 598 00:41:47,680 --> 00:41:48,450 OK. 599 00:41:48,450 --> 00:41:51,120 So how do we get the weak form? 600 00:41:51,120 --> 00:41:54,890 Can I recap how you get the weak form? 601 00:41:54,890 --> 00:41:57,650 If u is the winner. 602 00:41:57,650 --> 00:41:58,160 Right. 603 00:41:58,160 --> 00:42:01,720 I'll just think of u as the winner. 604 00:42:01,720 --> 00:42:06,050 Then if I move it by v, move it a little, 605 00:42:06,050 --> 00:42:08,200 then this expression should go up. 606 00:42:08,200 --> 00:42:10,320 So what happens if I move it by v? 607 00:42:10,320 --> 00:42:11,820 So I'm going to compare the two. 608 00:42:11,820 --> 00:42:19,500 I'm going to compare that with 1/2 a of u plus v, you know, 609 00:42:19,500 --> 00:42:23,010 upped a little, minus l of u plus v. 610 00:42:23,010 --> 00:42:27,660 And the point is that this guy should be -- 611 00:42:27,660 --> 00:42:32,660 maybe I'll erase min and put in less or equal to. 612 00:42:32,660 --> 00:42:38,040 I'm just recapping that this should 613 00:42:38,040 --> 00:42:43,150 be less or equal to this one for all v. You 614 00:42:43,150 --> 00:42:44,580 see it's the weak stuff? 615 00:42:48,670 --> 00:42:50,140 u is the winner. 616 00:42:50,140 --> 00:42:54,310 I'm now using u and not u star, and I'm 617 00:42:54,310 --> 00:42:59,120 using v for the delta u, for the movement 618 00:42:59,120 --> 00:43:04,920 away from the winner, which raises the energy. 619 00:43:04,920 --> 00:43:05,920 OK. 620 00:43:05,920 --> 00:43:09,480 Now what do I plan to do? 621 00:43:09,480 --> 00:43:16,110 I plan to cancel common terms here, and see what's going on, 622 00:43:16,110 --> 00:43:19,940 and find that first variation. 623 00:43:19,940 --> 00:43:21,340 Just what I did last time. 624 00:43:21,340 --> 00:43:24,270 I'm just doing it in this new notation. 625 00:43:24,270 --> 00:43:25,900 So what's the point? 626 00:43:25,900 --> 00:43:27,440 This l of u is linear. 627 00:43:27,440 --> 00:43:36,140 So l of u plus v is the same as l of u and l of v, right. 628 00:43:36,140 --> 00:43:39,720 If I put in u plus v there, it splits 629 00:43:39,720 --> 00:43:43,310 into two integrals here they are, so when I subtract, 630 00:43:43,310 --> 00:43:45,690 these guys go. 631 00:43:45,690 --> 00:43:47,700 Now, what about this one? 632 00:43:47,700 --> 00:43:51,120 Well, just as last time when I put in u plus v 633 00:43:51,120 --> 00:43:56,820 here and expanded everything, now I have something squared. 634 00:43:56,820 --> 00:44:02,910 So this business here is going to be 1/2. 635 00:44:02,910 --> 00:44:07,010 I'll get something from the u alone. 636 00:44:07,010 --> 00:44:13,450 And then I'll get two something, canceling the 1/2, from u 637 00:44:13,450 --> 00:44:17,180 and v, that's the cross term. 638 00:44:17,180 --> 00:44:20,350 And then I'll get something from the v alone. 639 00:44:27,080 --> 00:44:32,510 I'm dodging a couple of bullets here just going 640 00:44:32,510 --> 00:44:33,631 to the main point. 641 00:44:33,631 --> 00:44:34,130 OK. 642 00:44:34,130 --> 00:44:37,580 So the main point is I'm going to subtract, 643 00:44:37,580 --> 00:44:39,750 I'm going to at the differences. 644 00:44:39,750 --> 00:44:42,610 So the zero-order terms are all gone, 645 00:44:42,610 --> 00:44:51,210 and it's this quantity that has to be greater or equal 0, 646 00:44:51,210 --> 00:44:54,380 right -- because I was left with that greater or equal sign -- 647 00:44:54,380 --> 00:44:57,130 for all v. OK. 648 00:45:07,090 --> 00:45:10,740 Don't let me leave minus l of v there. 649 00:45:15,410 --> 00:45:16,790 OK. 650 00:45:16,790 --> 00:45:19,450 So that's the thing that has to be greater or equal to 0. 651 00:45:19,450 --> 00:45:25,810 Now we're just going to repeat the same the discussion that we 652 00:45:25,810 --> 00:45:29,170 had last time with different letters. 653 00:45:29,170 --> 00:45:34,680 If this is going to be greater or equal 0 for all v, 654 00:45:34,680 --> 00:45:39,520 then I'm going to think of small v's, in which case 655 00:45:39,520 --> 00:45:43,820 this term is going to be smaller than the others and won't help. 656 00:45:43,820 --> 00:45:45,140 So what has to happen? 657 00:45:45,140 --> 00:45:49,820 What has to happen for this to be greater equal to 0 658 00:45:49,820 --> 00:45:51,020 for all v? 659 00:45:51,020 --> 00:45:52,970 You know like we're taking the derivative 660 00:45:52,970 --> 00:45:56,980 in the direction of v. We're moving in the direction of v. 661 00:45:56,980 --> 00:46:00,720 And this is the first-order term, 662 00:46:00,720 --> 00:46:02,710 that's the first variation. 663 00:46:02,710 --> 00:46:06,220 That's what has to be 0 for all of v. 664 00:46:06,220 --> 00:46:13,270 I guess I can bring it over here if you eye will follow it. 665 00:46:13,270 --> 00:46:24,100 I have a of u, v minus l of v, that's 666 00:46:24,100 --> 00:46:28,870 the sort of first-order term, and then the second-order term. 667 00:46:28,870 --> 00:46:33,120 Has to be greater or equal 0, all v. 668 00:46:33,120 --> 00:46:36,450 And the question is so what? 669 00:46:36,450 --> 00:46:37,840 What do we get out of that? 670 00:46:37,840 --> 00:46:40,830 Well what I'm saying is this stuff 671 00:46:40,830 --> 00:46:47,000 has to be 0, because v could have either sign. 672 00:46:47,000 --> 00:46:51,390 So if this was positive or negative, 673 00:46:51,390 --> 00:46:53,180 I could switch sign if I want to. 674 00:46:56,740 --> 00:46:58,270 So it has to be 0. 675 00:46:58,270 --> 00:47:06,380 So a of u, v has to equal l of v, and that's the weak form. 676 00:47:06,380 --> 00:47:11,560 That's the weak form, that's the form integral of c u prime v 677 00:47:11,560 --> 00:47:17,780 prime dx equals integral of f*v*dx. 678 00:47:17,780 --> 00:47:24,530 That's the l of v, and this is the a of u and v. 679 00:47:24,530 --> 00:47:29,190 This is what you see in engineering papers, 680 00:47:29,190 --> 00:47:32,540 when they're launching into the finite element method, 681 00:47:32,540 --> 00:47:38,860 and they're planning to get some notation. 682 00:47:38,860 --> 00:47:41,540 That's a very familiar notation. 683 00:47:41,540 --> 00:47:49,290 And then the final point is what about this term? 684 00:47:49,290 --> 00:47:53,130 So now we know this is 0. 685 00:47:53,130 --> 00:47:59,090 This is the weak form of the equation, find u, 686 00:47:59,090 --> 00:48:02,210 so that this holds for all v. That's 687 00:48:02,210 --> 00:48:03,590 the weak form of the equation. 688 00:48:06,930 --> 00:48:08,980 Oh yeah, I better not rush by it. 689 00:48:08,980 --> 00:48:12,430 This is the form in which you plug in -- 690 00:48:12,430 --> 00:48:15,310 this is the form finite elements come from. 691 00:48:21,910 --> 00:48:24,530 It doesn't solve this equation exactly. 692 00:48:24,530 --> 00:48:27,210 This is the differential equation. 693 00:48:27,210 --> 00:48:27,970 I'm in here. 694 00:48:27,970 --> 00:48:30,390 It's our Euler-Lagrange equation. 695 00:48:34,840 --> 00:48:40,390 The finite element method says, OK, take the trial functions, 696 00:48:40,390 --> 00:48:44,290 and get it right for those guys. 697 00:48:44,290 --> 00:48:49,830 So that give us our N equations, where 698 00:48:49,830 --> 00:48:52,640 this is a continuous problem. 699 00:48:52,640 --> 00:48:53,540 Yeah. 700 00:48:53,540 --> 00:48:57,570 So this is what just leads you -- 701 00:48:57,570 --> 00:49:01,200 the weak form is what leads you naturally to finite elements. 702 00:49:01,200 --> 00:49:04,490 You take this, and you only make it 703 00:49:04,490 --> 00:49:07,800 true on a finite dimensional space. 704 00:49:07,800 --> 00:49:08,300 OK. 705 00:49:08,300 --> 00:49:12,680 And the final comment is what about this guy? 706 00:49:12,680 --> 00:49:17,110 Well the whole point is that we assume positive definite, 707 00:49:17,110 --> 00:49:19,820 we assume stability, we've made our life easy 708 00:49:19,820 --> 00:49:25,970 by guaranteeing by the fact that this is always greater equal 0. 709 00:49:25,970 --> 00:49:28,690 I don't even have to think about this one. 710 00:49:28,690 --> 00:49:31,100 OK. 711 00:49:31,100 --> 00:49:32,800 Because if u and v are the same, this 712 00:49:32,800 --> 00:49:35,540 is a square, and that material coefficient that 713 00:49:35,540 --> 00:49:38,620 better be not negative, right. 714 00:49:38,620 --> 00:49:41,370 OK. 715 00:49:41,370 --> 00:49:44,350 Thank you for your patience to listen to that. 716 00:49:44,350 --> 00:49:46,810 See this notation for the same thing 717 00:49:46,810 --> 00:49:50,720 that we did last time for the weak form, 718 00:49:50,720 --> 00:49:56,190 and you see how it pays off immediately to give us 719 00:49:56,190 --> 00:49:58,350 the finite element form. 720 00:49:58,350 --> 00:49:59,240 OK. 721 00:49:59,240 --> 00:49:59,840 Good. 722 00:49:59,840 --> 00:50:04,200 So the project ones are all there, 723 00:50:04,200 --> 00:50:06,660 and there's just two or three left here. 724 00:50:06,660 --> 00:50:09,590 And I hope you have a super weekend, 725 00:50:09,590 --> 00:50:13,300 and do give a thought to project two.