麻省理工学院：偏微分方程式数字方法（英文版）：lec22_notes

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Approach Galerkin ( ) () ( ) () ( ) 1 , , n ijij japprox approx approx surface surface surface iji x x dS G x x x x dS dS Ab ?α? = ′′ ′ ′′Ψ=!""" #$$$%$$$& #$$$$$$%$$$$$$$& () () () () () ( ) ( ) 1 ,0 n ii ij approx surface x R x dS x x dS x G x x x dS dS?? ?α? = ′′′=Ψ? =!""" njσ njσ a67a68 a68 a68 a69 a70 a108a52a71 a108 a72a57a72a12a72 a72a12a72a19a72 a70 a108a52a71a112 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a75 a70 a112a28a71 a108a76a72a57a72a12a72 a72a12a72a19a72 a70 a112a63a71a112 a77a1a78 a78 a78 a79 a67a68 a68 a68 a69 a36 a112a1a108 a75 a75 a75 a75 a75 a75 a36 a112a85a112 a77a1a78 a78 a78 a79 a131 a67a68 a68 a68 a69 a8 a108 a75 a75 a75 a75 a75 a75 a8 a112 a77a1a78 a78 a78 a79 a9a11a10 a58 a127a129a128 a109a157a128 a59 a130a153a131 a58 a127a129a128 a59 a109a157a128a24a130 a91a69a97a8a76a87a86 a70 a126a40a71 a35 a131 a70 a35 a71a126 a34a13a12 a122a138a161a96a161a15a14a17a16a18a16 a144a23a121a87a145a38a122a138a118 a212a175a213a41a214a216a215a4a19 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a71a70a223a42a28a141a57a28a109a16a72a65a67a33a74a114a60a78a71a106a62a65a103a87a220a169a71a73a62a35a57a82a223a182a71a73a66a113a57a52a67a106a86a104a116 a71a73a66a80a74a80a116a65a71a73a108a73a71a106a62a65a67a70a77a63a60a78a74a125a126a84a86a9a71a73a62a31a64a43a60a233a74a114a60a78a71a106a62a54a220a7a67a73a62a65a64a10a28 a72a35a62a35a79a81a62a35a71a33a221a58a62a65a103a52a220a20a71a106a62a35a57a113a223a182a71a73a66a117a57a28a67a73a86a104a116a154a69a65a67a106a103a105a60a124a103a76a223a182a72a35a62a65a86a9a74a80a60a63a71a73a62a91a221a222a57a87a60a63a108a73a116a16a74a52a227 a219 a77a124a103a80a71a65a220a28a74a80a116a35a57a222a103a105a126a43a103a105a74a80a57a52a59a100a64a35a71a16a57a28a103a20a62a65a71a70a74a7a116a31a67a121a111a73a57a156a74a114a116a35a57a169a112a47a71a70a74a114a57a87a62a16a74a80a60a124a67a70a77a16a57a9a229a43a112a35a77a63a60a63a86a87a60a233a74a114a77a78a126a136a67a106a103a41a74a80a116a65a57a85a66a114a60a78a108a106a116a16a74a20a116a65a67a70a62a31a64a117a103a105a60a124a64a43a57a106a227a20a176a177a62a43a224 a103a105a74a80a57a52a67a106a64a41a220a33a74a114a116a35a57a39a36 a209a94a93 a66a80a60a63a108a73a116a16a74a105a224a122a116a65a67a73a62a65a64a133a103a80a60a124a64a43a57a37a57a87a62a16a74a80a66a114a126a117a60a124a103a92a74a80a116a35a57a68a67a121a111a106a57a87a66a104a67a70a108a106a57a169a71a70a223a31a74a80a116a35a57a58a112a65a66a80a71a43a64a43a72a65a86a9a74a234a71a70a223a31a74a80a116a35a57a58a112a47a71a70a74a114a57a87a62a16a74a80a60a124a67a70a77 a67a70a62a31a64a107a74a114a116a35a57 a36 a209a94a93 a69a65a67a73a103a80a60a63a103a222a223a182a72a65a62a65a86a216a74a114a60a78a71a106a62a54a227 a21 a22 a142a143a10a24a23a117a11a37a135a92a134a117a11a58a10a12a148a85a11a26a25 a10a12a137a117a6a28a27a119a147a20a8a87a147 a29a37a150a105a151 a30a18a31a58a153a33a32 a166a35a34a7a36a38a37a232a168a35a163a40a39a35a34a7a36a41a32 a155 a4a20a172 a99 a172 a99 a99a81a101 a64a58a44a182a30a33a55a52a25a43a42a115a44a98a24a7a53a167a36a225a38a41a40a7a32a35a25a52a44a46a45a47a24 a13 a14a16a15a18a17a20a19a45a44 a179a61a180a46a181a47a183a169a184a194a185 a197 a46 a197a48a47 a181a167a16a175a181a31a188 a47 a189a156a180a46a181a31a188a18a183a125a190a81a191a156a188 a181a50a49a52a51a60a16 a46a73a187a33a46a54a53 Convergence Analysis Example Problems 1-D First Kind Equation () is unknownxσ() 3x xxΨ=? The density must be computedThe potential is given Solution = 3x Ψ x x σ a55 a212a175a213a41a214a216a215a1a0 a176a177a62a119a74a114a116a35a57a39a62a65a57a9a229a81a74a222a103a80a57a87a111a106a57a87a66a104a67a70a77a65a103a80a77a78a60a124a64a43a57a28a103a169a221a222a57a96a221a58a60a63a77a78a77a26a60a63a62a81a111a73a57a52a103a105a74a80a60a63a108a106a67a70a74a80a57a37a74a114a116a35a57a225a86a87a71a73a62a81a111a73a57a52a66a80a108a106a57a87a62a65a86a87a57a58a112a35a66a114a71a73a112a47a57a87a66a80a74a80a60a63a57a52a103a234a71a73a223a41a74a80a116a35a57a28a103a105a57 a64a43a60a124a103a80a86a87a66a80a57a87a74a80a60a21a75a52a67a70a74a80a60a63a71a73a62a12a59a61a57a87a74a80a116a35a71a43a64a35a103a52a227a37a130a68a71a33a221a194a74a80a116a35a57a28a103a105a57a117a59a113a57a87a74a80a116a35a71a43a64a35a103a39a86a9a71a73a62a81a111a106a57a87a66a114a108a73a57a76a64a43a57a87a112a47a57a87a62a65a64a65a103a68a71a73a62a93a221a58a116a65a67a70a74a68a79a81a60a78a62a65a64a93a71a73a223 a60a63a62a106a74a114a57a87a108a106a66a114a67a73a77a156a57a52a109a16a72a65a67a33a74a114a60a78a71a106a62a154a60a63a103a143a69a31a57a52a60a78a62a35a108a232a103a105a71a106a77a78a111a81a60a63a62a35a108a65a227 a79a156a229a35a67a70a59a113a60a78a62a65a60a78a62a35a108a93a74a114a116a35a60a124a103a117a60a124a103a80a103a80a72a35a57a107a221a58a60a63a77a78a77a85a60a78a62a16a74a114a66a80a71a43a64a43a72a65a86a87a57a119a71a73a62a35a57a119a71a73a223 a74a80a116a65a57a117a103a105a72a35a69a35a74a80a77a63a57a136a112a31a71a106a60a78a62a16a74a114a103a58a67a73a69a31a71a106a72a43a74a58a60a78a62a16a74a114a57a87a108a73a66a104a67a70a77a41a57a28a109a16a72a65a67a33a74a114a60a78a71a106a62a65a103a87a227 a228a7a71a119a69a31a57a52a108a73a60a63a62a54a220a31a86a87a71a73a62a65a103a80a60a63a64a35a57a87a66a37a74a80a116a65a57a117a57a9a229a35a67a70a59a113a112a35a77a63a57a136a71a73a62a65a57a9a224a177a64a43a60a78a59a113a57a52a62a65a103a105a60a63a71a73a62a31a67a70a77 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[ ]1,1x ∈? 0 1x =? 1nx =1x 1nx ?2x 1c x 2c x nc x () 11 j ii j xn cjc jx x xxdSσ ?= ′′Ψ= ?!" 1nσ nnσ a179a61a180a182a181a47a208 a137 a183a234a184a194a193 a192 a74 a196a7a197 a189a31a192 a74 a185a8a7 a41 a7 a41a10a9 a80 a47 a181a47a208 a137 a16a154a181a31a188 a47 a190a16a191a156a188 a212a175a213a41a214a216a215a12a11 a228a7a71a84a86a9a71a73a59a113a112a35a72a35a74a80a57a113a74a80a116a65a57a107a62a81a72a35a59a113a57a87a66a114a60a63a86a52a67a70a77a234a103a80a71a73a77a63a72a43a74a80a60a63a71a73a62a175a74a80a71a93a74a114a116a35a60a124a103a117a71a73a62a65a57a9a224a177a64a43a60a78a59a113a57a52a62a65a103a105a60a63a71a73a62a31a67a70a77a156a112a35a66a114a71a73a69a35a77a63a57a87a59a93a220a42a86a9a71a73a62a31a103a105a60a124a64a43a57a87a66 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a160 a13 a14a16a15a18a17a20a19a22a21a35a21 n = 10 n = 20 n = 40 Answers Are Getting Worse!!! σ x a212a175a213a41a214a216a215a218a217a47a217 a43 a62a35a57a136a72a65a103a80a72a65a67a73a77a78a77a63a126a5a69a47a57a87a77a63a60a78a57a52a111a73a57a28a103a37a74a80a116a65a67a70a74a39a67a119a64a43a60a124a103a80a86a87a66a80a57a87a74a80a60a21a75a52a67a70a74a80a60a63a71a73a62a93a103a80a86a104a116a35a57a52a59a113a57a117a103a105a116a35a71a106a72a35a77a124a64a82a112a65a66a80a71a43a64a43a72a65a86a87a57a136a112a35a66a114a71a73a108a73a66a114a57a52a103a114a103a80a60a78a111a106a57a87a77a63a126 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a26a65a62a65a60a233a74a125a126a93a67a73a103a37a74a114a116a35a57a143a64a43a60a124a103a114a86a9a66a114a57a9a74a80a60a21a75a52a67a70a74a80a60a63a71a73a62a93a60a124a103a58a66a80a57a20a26a65a62a35a57a28a64a41a227a222a228a37a116a65a57a117a66a80a57a28a103a105a72a65a77a233a74a104a103a68a67a70a66a114a57a117a86a9a57a52a66a105a74a104a67a70a60a63a62a35a77a78a126a82a62a35a71a70a74 a86a9a71a106a62a81a111a73a57a87a66a114a108a73a60a63a62a35a108a31a227 a18 a73a49a116a81a126a115a60a63a103a39a74a114a116a35a60a63a103a76a116a31a67a70a112a35a112a47a57a87a62a35a60a63a62a35a108a9a61a167a176a125a103a225a74a80a116a35a57a61a62a81a72a35a59a113a57a87a66a114a60a63a86a52a67a70a77a7a74a80a57a28a86a104a116a35a62a35a60a124a109a106a72a65a57a113a67a70a74a39a223a46a67a73a72a35a77a78a74a52a220a41a71a73a66a76a60a124a103a39a74a80a116a35a57a61a60a78a62a16a74a114a57a87a108a73a66a104a67a70a77 a57a52a109a16a72a65a67a70a74a80a60a63a71a73a62a93a67a61a112a35a66a80a71a106a69a35a77a63a57a87a59a4a61 a29a37a150a11a29 a30a18a31a58a153a33a32 a166a35a34a7a36a38a37a232a168a35a163a40a39a35a34a7a36a41a32 a155 a4a20a172a5a4a20a172 a99 a99a81a101 a3a54a27a81a95a106a45a31a24a42a53 a42a115a44a46a24a42a53a167a36a39a38a41a40a42a32a43a25a52a44a98a45a31a24 a13 a14a16a15a18a17a20a19a22a21a73a174 a179a61a180a182a181a26a183a169a184a141a189a156a180a46a181a47a183 a13 a185 a197 a46 a197 a47 a181a167a16a175a181a31a188 a47 a189a156a180a182a181a31a188a182a183a105a190a16a191a234a188 a181a50a49a4a51a81a16 a46a106a187a20a46 a53 Convergence Analysis Example Problems [ ]1,1x ∈? () is unknownxσ() 3x xxΨ=? The density must be computedThe potential is given Solution = 3x () () () 1 1 x xxxxdSσσ ? ′′′Ψ= +?" a212a175a213a41a214a216a215a218a217 a15 a73a175a57a37a67a73a66a80a57a222a108a73a71a106a60a78a62a65a108a96a74a114a71a39a112a47a71a106a103a105a74a80a112a47a71a73a62a35a57a37a57a87a229a43a67a73a59a113a60a78a62a35a60a63a62a35a108a76a221a58a116a65a67a70a74a42a221a222a57a87a62a16a74a156a221a58a66a80a71a106a62a35a108a225a60a78a62a133a74a80a116a35a57a78a26a65a66a104a103a105a74a105a224a122a79a16a60a63a62a65a64a133a57a9a229a35a67a70a59a113a112a35a77a63a57a73a220 a67a70a62a31a64a119a60a78a62a31a103a125a74a114a57a52a67a73a64a107a77a78a71a81a71a73a79a113a67a33a74a58a67 a71a81a57a28a86a9a71a106a62a65a64a1a0a76a60a63a62a65a64a107a57a28a109a106a72a31a67a33a74a80a60a63a71a73a62a20a227a42a102a65a71a73a66a85a74a80a116a35a60a124a103a169a57a28a109a16a72a65a67a33a74a114a60a78a71a106a62a54a220a106a221a222a57a39a67a106a103a80a103a80a72a35a59a113a57a68a74a114a116a65a67a33a74 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[ ]1,1x ∈? 0 1x =? 1nx =1x 1nx ?2x 1c x 2c x nc x () 11 j ii j xn cjc jx x xxdSσ ?= ′′Ψ= ?!" 1nσ nnσ a179a61a180a182a181a26a208 a137 a183a234a184a141a189a31a192 a195 a13 a193 a192 a74 a196a42a197 a189a65a192 a74 a185 a7 a41 a7 a41a10a9 a80 a47 a181a47a208 a137 a16a175a181a31a188 a47 a190a81a191a156a188 a3 a212a175a213a41a214a216a215a218a217 a81 a228a7a71a225a86a87a71a73a59a113a112a35a72a43a74a114a57a85a74a114a116a35a57a37a62a81a72a35a59a113a57a87a66a114a60a63a86a52a67a70a77a43a103a105a71a106a77a78a72a35a74a80a60a63a71a73a62a117a74a114a71a225a74a80a116a35a57a37a71a73a62a65a57a9a224a177a64a43a60a78a59a113a57a52a62a65a103a105a60a63a71a73a62a31a67a70a77a43a103a105a57a28a86a9a71a106a62a65a64a81a224a122a79a16a60a63a62a65a64a143a57a52a109a16a72a65a67a33a74a114a60a78a71a106a62 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a83a85a71a73a77a63a77a78a71a43a86a87a67a70a74a80a60a63a71a73a62a12a108a73a57a52a62a35a57a87a66a104a67a33a74a114a57a52a103a37a67a113a64a43a60a63a103a114a86a9a66a114a57a9a74a114a57a225a223a182a71a73a66a114a59a3a71a70a223a8a4 a4a218a189a115a184a230a179 a0 a4a107a192a43a189a31a192a107a184a230a179a225a192 Convergence Analysis Example Problems () () 1 11 101 1 01 1 n n n nn xx cc cxx xx nc cc xx xxdS xxdS x xxxdS xxdS σ σ ? ? $%′′ ′′?? &’$%Ψ $%&’ &’= &’&’ &’()()′′ ′′ ??&’ () "" "" ’ ()( (( ’ nσ nnσ nK a228a37a116a65a57a225a59a119a67a70a74a80a66a114a60a233a229 a4 a192 a60a124a103a222a74a80a116a65a57a136a62a35a71a70a74a58a74a114a116a35a57a136a71a73a112a47a57a87a66a104a67a33a74a114a71a73a66 a4 a192a2a1 a212a175a213a41a214a216a215a218a217a41a19 a219 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= 20 n = 40 Eigenvalues accumulating at zero. a212a175a213a41a214a216a215a218a217a18a0 a176a122a223a47a74a80a116a35a57a58a71a106a112a31a57a52a66a114a67a70a74a80a71a73a66a8a4a165a60a124a103a156a103a105a60a63a62a35a108a73a72a65a77a63a67a73a66a52a220a70a71a106a62a35a57a37a59a113a60a78a108a106a116a106a74a156a57a87a229a81a112a47a57a52a86a9a74a156a74a80a71a117a103a80a57a87a57a222a74a80a116a31a67a33a74a234a66a114a57a7a6a65a57a28a86a216a74a80a57a28a64a143a60a63a62a61a74a80a116a35a57a58a57a52a60a78a108a106a57a87a62a43a224 a111a33a67a70a77a63a72a35a57a52a103a85a71a70a223a20a67a143a59a119a67a33a74a80a66a114a60a78a229a113a108a106a57a87a62a35a57a52a66a114a67a70a74a80a57a28a64a113a69a16a126a119a64a43a60a124a103a114a86a9a66a114a57a9a74a80a60a21a75a87a60a63a62a35a108 a4 a227a92a176a177a62a5a112a65a67a73a66a105a74a114a60a63a86a87a72a35a77a63a67a73a66a52a220a106a71a73a62a35a57a39a221a85a71a106a72a35a77a124a64a119a57a9a229a43a112a47a57a52a86a216a74 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Nystrom Piecewise-Constant Centroid Collocation 1-D Second Kind Example Nystrom Method Convergence Comparison a59a7a172 a99 a172a1a0 a120 a45a47a24a26a97a31a27a16a30a33a29a31a27a81a24a42a95a73a27 a120 a32a106a97a31a27a81a32a43a25 a13 a14a16a15a18a17a20a19a7a6 a44 a176a122a223a48a20a68a126a43a103a125a74a114a66a80a71a106a59a3a59a113a57a9a74a114a116a35a71a43a64a12a86a52a67a70a62a12a116a65a67a121a111a73a57a225a57a9a229a43a112a47a71a73a62a35a57a52a62a16a74a80a60a124a67a70a77a54a86a9a71a106a62a81a111a73a57a87a66a114a108a73a57a52a62a65a86a9a57a106a220a106a221a58a116a81a126a82a72a65a103a105a57a136a67a73a62a16a126a16a74a114a116a35a60a78a62a65a108a61a57a52a77a63a103a80a57a68a61 a75a76a67a73a72a65a103a114a103a105a60a124a67a70a62a12a109a106a72a31a67a73a64a43a66a104a67a33a74a114a72a35a66a80a57a225a116a65a67a106a103a37a57a9a229a43a112a31a71a106a62a35a57a87a62a16a74a114a60a63a67a73a77a54a86a9a71a73a62a81a111a106a57a87a66a114a108a73a57a87a62a31a86a9a57a68a223a182a71a106a66a68a24a7a45a47a24a42a55a52a44a98a24a7a29a47a40a7a34a98a32a43a30a117a79a106a57a87a66a114a62a35a57a87a77a124a103 a231 a174 a77a115a71a16a103a125a74a92a112a35a116a81a126a43a103a105a60a124a86a87a67a73a77a81a112a35a66a80a71a106a69a35a77a63a57a87a59a119a103a42a71a70a223a65a60a63a62a16a74a80a57a87a66a114a57a52a103a105a74a92a116a65a67a121a111a73a57a222a55a52a44a98a24a7a29a47a40a7a34a98a32a43a30a225a123a31a27a16a30a33a24a7a27a81a34a46a55a12a22 a46a1a0a29a52a81a220 a37a38a28a117a22 a36a3a2a117a52a4a0a94a52a81a220a70a57a9a74a104a86a33a24 a5 a6a119a2 a2 a2 a137a136a135 a27 a13 a14a16a15a18a17a20a19a7a6 a5 a23a9a24a26a25a28a27a16a29a31a30a33a32a35a34a37a36a39a38a41a40a42a32a43a25a28a44a46a45a47a24a49a48a50a27a16a25a52a51a42a45a54a53a7a55 a56a58a57a52a111a16a60a63a57a87a221a222a57a52a64a93a75a76a67a70a77a63a57a87a66a114a79a81a60a78a62a93a67a70a62a31a64a93a83a85a71a106a77a78a77a63a71a43a86a87a67a33a74a114a60a78a71a106a62 a36a39a88a54a32a35a89a91a90a92a34a46a27a93a45a31a94 a120 a45a47a24a26a97a31a27a16a30a33a29a47a27a16a24a42a95a73a27a218a23a9a55a52a55a28a40a42a27a16a55a113a44a98a24a194a99a81a101 a235 a103a105a74a68a67a70a62a65a64a93a231a33a62a31a64a82a79a81a60a63a62a65a64a82a60a78a62a16a74a80a57a52a108a73a66a104a67a70a77a41a57a28a109a106a72a31a67a33a74a80a60a63a71a73a62a31a103a87a220a81a62a81a72a35a77a63a77a20a103a105a112a31a67a73a86a9a57a28a103 a120 a45a31a24a26a97a47a27a16a30a121a29a47a27a81a24a7a95a106a27a84a94a122a45a65a30a82a55a52a27a81a95a73a45a47a24a42a53a22a123a54a44a98a24a7a53a49a27a81a38a41a40a42a32a43a25a52a44a98a45a31a24a42a55 a71a81a116a35a71a33a221 a86a9a71a106a62a65a103a105a60a124a103a105a74a80a57a87a62a31a86a9a126a107a67a70a62a31a64a12a103a105a74a114a67a73a69a35a60a63a77a78a60a78a74a125a126a107a60a124a103a80a103a80a72a35a57a52a103 a127a93a128a54a55a52a25a87a30a33a45a31a89a129a89a139a27a16a25a52a51a42a45a54a53a7a55 a130a68a60a63a108a73a116a12a71a73a66a104a64a43a57a52a66a68a86a9a71a73a62a81a111a106a57a87a66a114a108a73a57a87a62a31a86a9a57 a101a12a44a46a53a49a24a42a45a65a25a113a32a35a53a42a53a20a30a33a27a81a55a52a55a119a55a28a44a46a24a42a29a47a40a7a34a98a32a43a30a107a44a46a24a26a25a28a27a16a29a31a30a33a32a35a24a42a53a7a55 a231 a95