麻省理工学院：偏微分方程式数字方法（英文版）：lec7_notes

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a251 a77 a15 a6 a233 a248a37a141 a143a73a144a146a145 a251 a153 a23 a14 a29a32a10 a6 a50a113a15a26a235 a25 a234 a42 a234a32a151a119a69 a21 a141 a154 a67 a249 a67a18a69 a21 a141 a141a33a155 a63 a14 a237a28a64a88a234a84a31a16a15 a6 a233a102a29a44a31a126a233a51a50a123a234a33a10a37a156a124a10a26a234a34a237a157a11a76a158a123a234a33a50a109a159a32a11a88a235 a14a18a160a51a6 a23 a6a28a8a13a25a27a161 a21 a60 a38 a225a99a226a137a227a139a228a141a229 a162a112a163 a150 a233 a6a28a8a13a6 a11a24a237a110a235 a14 a233a51a50 a6a28a8a13a6a1a14 a233 a147 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Solution 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Error u ( x ) u ( x ) - u h ( x ) a164a105a165a102a166a44a166a3a167a56a167a13a168a76a167a46a169a157a170a46a171a24a172a18a173a18a166a22a174a121a175a176a168a37a177a46a178a91a179a119a168a113a180a12a168a76a171a181a175a102a168a73a182a99a166a3a182a51a178a100a170a3a183a122a184a13a185a1a186a188a187a100a189a73a190a146a191a99a192a46a179a193a165a112a169a59a180a26a165a5a169a157a170a35a172a24a194a1a168a76a179a106a171a24a168a33a173a32a166a90a168a73a182a5a178a195a165a102a166 a196 a182a105a166a22a171a44a166a54a170a13a165a75a197a181a172a76a182a105a173a27a172a109a194a28a168a73a179a199a198a89a200a3a174a112a178a127a182a99a168a76a178a127a170a3a168a109a194a28a168a73a179a105a201a109a171a24a168a33a173a32a166a24a168a76a182a202a178a195a165a102a166a24a180a12a168a33a172a73a167a12a170a3a166a90a171a44a166a54a170a13a165a102a203a205a204a127a182a206a178a96a165a51a166a24a168a76a178a195a165a102a166a3a167 a165a102a172a73a182a99a173a76a197a193a184a37a185a1a186a188a187a37a207a121a189a76a190a146a191a105a192a81a169a157a170a35a172a24a194a28a168a73a179a106a171a44a168a121a173a18a166a90a168a76a182a48a178a96a165a51a166 a196 a182a99a166a46a171a24a166a54a170a37a165a5a172a76a182a105a173a113a172a22a165a112a169a9a208a121a165a209a171a24a168a33a173a32a166a22a168a73a182a5a178a195a165a102a166a35a180a12a168a121a172a76a167a12a170a3a166 a171a24a166a56a170a13a165a102a203a44a164a105a165a102a166a84a182a105a166a13a210a32a178 a196 a208a73a174a112a167a26a166a56a170a12a197a188a170a13a165a51a168a76a179a148a178a195a165a102a166a81a166a3a211a120a168a73a194a212a174a112a178a91a169a59a168a76a182a209a168a89a177a108a178a195a165a102a166a81a166a3a167a56a167a13a168a76a167a81a172a76a182a105a173a86a178a195a165a102a166a181a170a3a168a76a194a212a174a112a178a91a169a59a168a76a182a205a178a195a165a104a167a13a168a73a174a18a208a121a165 a178a195a165a102a166a35a171a22a174a112a194a212a178a91a169a9a208a18a167a56a169a59a173a84a175a99a167a26a168a33a180a12a166a54a170a26a170a3a203 a213a99a214a75a215a96a216a188a217a206a218a146a219 a220a108a221a96a222a26a223a3a224a7a225a32a226a108a227a41a228 a185 a222a13a223a3a224a12a229a76a222 a185a28a230a32a186a51a184a127a230a18a186 a222a13a231a51a223a81a232 a186 a223a46a233a44a223 a184 a231 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Solution 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Error u ( x ) - u h ( x ) u ( x ) a234 a177a12a178a235a166a3a167a35a178a195a165a104a167a13a166a12a166a22a170a56a171a44a168a121a168a76a178a195a165a104a169a96a182a104a208a205a169a96a178a235a166a54a167a13a172a73a178a100a169a59a168a73a182a102a170a35a178a195a165a102a166a22a165a104a169a9a208a120a165a176a166a54a167a56a167a13a168a73a167a22a171a24a168a121a173a18a166a35a165a51a172a120a170a44a200a12a166a12a166a3a182a202a166a3a194a212a169a96a171a90a169a96a182a99a172a73a178a124a166a12a173a18a203a206a164a105a165a102a166 a194a28a168a73a179a41a166a54a167a56a167a13a168a73a167a46a171a44a168a121a173a18a166a46a165a51a172a120a170a81a170a56a174a54a236a181a166a54a167a26a166a26a173a103a194a212a169a96a178a91a178a100a194a28a166a90a180a12a165a102a172a76a182a104a208a32a166a121a203 a213a99a214a75a215a96a216a188a217a206a218a102a218 a220a108a221a96a222a26a223a3a224a181a237 a230 a229a18a224 a184 a223a7a238a32a224 a185a157a239 a237 a230 a224a26a224a13a223a121a237a56a222 a185a28a230a32a186a48a187a195a240a24a185 a222a13a223a33a224a26a229a73a222 a185a1a230a18a186a51a184a12a192 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Solution 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Error u ( x ) u ( x ) - u h ( x ) a234 a177a12a178a235a166a3a167a103a178a96a165a51a166a205a180a26a168a121a172a76a167a12a170a3a166a24a208a18a167a56a169a59a173a202a180a12a168a76a167a56a167a13a166a12a180a3a178a100a169a59a168a73a182a102a197a46a178a96a165a51a166a109a194a28a168a76a179a199a171a44a168a121a173a18a166a103a165a51a172a120a170a27a200a12a166a12a166a54a182a241a170a56a174a51a200a56a170a56a178a235a172a76a182a16a178a100a169a59a172a73a194a96a194a212a242a48a167a26a166a26a173a73a174a51a180a12a166a26a173a76a197 a200a3a174a112a178a66a182a99a168a73a179a20a197a66a170a3a168a76a171a24a166a103a182a105a166a54a179a41a165a104a169a9a208a120a165a243a171a44a168a121a173a18a166a54a170a86a165a51a172a76a211a76a166a205a200a26a166a12a166a3a182a243a169a96a182a51a178a91a167a13a168a121a173a76a174a51a180a12a166a12a173a205a178a195a165a104a167a13a168a73a174a18a208a121a165a206a178a195a165a102a166a113a169a96a182a51a178a235a166a3a167a91a175a102a168a73a194a28a172a76a178a91a169a59a168a76a182 a175a99a167a26a168a33a180a12a166a54a170a26a170a3a203 a207a33a244 a245a99a246a75a247a96a248a188a249a206a250a102a251 a252a108a253a96a254a26a255a1a0a3a2a5a4a7a6a9a8a11a10a13a12a15a14a37a254a7a14a17a16a18a12a19a12a73a254a21a20a23a22a25a24a27a26a28a22a28a254a13a255a29a0a21a30a73a254a31a22a32a12a33a24a27a14a35a34a59a255a1a24a27a36a37a12a73a253a39a38a41a40a42a38a1a43a28a255a18a44a46a45 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Solution 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Error u ( x ) u ( x ) - u h ( x ) a47a49a48a23a50a35a48a19a51a53a52a54a48a56a55a18a57a46a58a33a50a41a59a3a48a23a60a62a61a62a50a11a63a13a57a5a64a9a65a21a50a66a50a1a63a68a67a31a50a66a58a62a69a27a70a66a50a66a58a33a71 a245a99a246a75a247a96a248a188a249a206a250a19a72 a252a108a253a96a254a26a255a1a0a3a2a15a73a74a6a9a8a11a22a28a254a13a255a1a0a66a30a76a254a21a22a25a12a15a24a27a14 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Solution 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Error u ( x ) - u h ( x ) u ( x ) a245a99a246a75a247a96a248a188a249a206a250a23a75 a252a108a253a96a254a26a255a1a0a7a38a1a12a15a30a33a0a21a14a13a255a3a26a15a0a31a22a76a36a77a38a1a12a33a0a21a0a13a255a46a38a56a254a21a22a25a12a15a24 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Solution 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Error u ( x ) u ( x ) - u h ( x ) a245a99a246a75a247a96a248a188a249a206a250a23a78 a252a108a253a96a254a26a255a1a0a3a2a5a4a7a6a9a8a11a22a28a254a13a255a1a0a66a30a76a254a21a22a25a12a15a24a27a14a79a34a195a255a29a24a27a36a37a12a73a253a80a38a41a40a42a38a41a43a1a255a81a8a33a45 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Solution 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 0.5 1 1.5 Error u ( x ) - u h ( x ) u ( x ) a44a33a44 a82a84a83a35a85a1a83a66a83a35a86a88a87a27a89a62a86a90a86a88a87a27a83a35a85a1a83a66a91a21a92a5a93a13a94a95a91a1a96a33a91a1a97a25a83a81a98a76a85a81a89a62a93a49a89a62a97a25a92a21a99a100a92a62a101a19a85a79a86a102a92a28a86a103a87a23a83a105a104a107a106a66a85a108a86a90a91a1a96a33a91a41a97a32a83a108a109a90a89a5a93a13a94a110a86a103a87a23a89a5a86a111a109a105a89a17a112a66a86a102a83a1a106a81a92a62a93a27a97a53a96 a86a114a113a105a92a77a91a1a96a33a91a41a97a25a83a41a85a66a109a105a86a103a87a23a83a81a85a41a92a5a97a115a101a42a86a114a98a111a92a62a93a56a98a76a85a116a89a5a97a115a117a18a92a62a85a108a86a118a91a66a92a62a93a120a119a54a83a1a106a121a99a100a83a66a94a33a122 a123a13a124a100a125a88a126a128a127a130a129a120a131 a132a84a133a42a134a21a135a25a136a32a137a33a138a21a136a76a139a141a140a1a142a33a143a19a144a33a145a29a138a31a137a15a145a1a143a27a140a1a145a7a144a42a146a1a147a107a146a31a136a32a143a23a137a33a135a32a145a35a137a33a138a21a136a32a139 0 5 10 15 20 25 14 12 10 8 6 4 2 0 2 4 # Iterations l o g 1 0 | | r | | L 2 Weighted Jacobi, w =2/3 2level multigrid: n 1 =2, n 2 =2 a148a49a87a19a98a76a85a105a104a149a99a5a101a42a106a31a83a81a98a88a97a88a97a115a101a19a85a108a86a121a106a21a89a62a86a150a83a108a85a81a86a103a87a23a83a116a117a11a101a27a91a66a87a79a112a41a89a62a85a108a86a102a83a1a106a11a91a66a92a62a93a120a119a54a83a1a106a121a99a100a83a41a93a49a91a21a83a28a92a33a151a1a86a150a89a62a98a88a93a13a83a66a94a77a101a19a85a108a98a88a93a100a99a77a86a88a87a27a83a81a117a28a101a42a97a115a86a114a98a53a99a5a106a108a98a111a94 a152 a106a21a92a29a91a66a83a66a94a62a101a42a106a31a83a46a122a153a148a49a87a23a83a154a91a66a92a62a93a120a119a54a83a1a106a121a99a100a83a41a93a49a91a21a83a77a97a25a83a1a119a54a83a1a97a105a98a76a85a110a85a31a87a27a92a62a113a39a93a155a119a54a83a1a106a66a85a108a101a19a85a18a86a103a87a23a83a77a93a27a101a42a117a110a151a66a83a41a106a95a92a17a112a110a98a88a86a102a83a1a106a31a89a5a86a121a98a111a92a5a93a23a85a141a92a62a93 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