![](data:image/png;base64,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)
Phöông phaùp phaàn töû höõu haïn cho baøi toaùn elliptic phi tuyeán bieân cong
2
(0.6)
()() ()()()
() ()() ()
.01usinb1'u,1M
,00u
,1x0,0xusinxu,xgx'u,xM
dx
d
=+
=
<<=+−
Ñeå giaûi baøi toaùn (0.6), caùc taùc giaû trong [5] söû duïng phöông phaùp phaàn töû höõu haïn
caáp 1, moät chieàu.
Baøi toaùn (0.1)-(0.3) maø chuùng
toâi khaûo saùt ôû ñaây laø tröôøng hôïp hai
chieàu vôùi mieàn Ω coù bieân ∂Ω goàm ba
caïnh thaúng OA, AB, OC vaø moät phaàn
bieân cong Γ0 = BC , trong ñoù O (0,0),
A(1,0), B(1,ϕ(1)), C(0,ϕ(0)) (xem hình
veõ).
O
Ω
C
B
Vì vaäy ñeå söû duïng phöông
phaùp phaàn töû höõu haïn vôùi caùc ña thöùc
xaáp xæ caáp 1 caàn xaáp xæ bieân cong Γ0
thaønh bieân coù “hình raêng cöa” (ñöôøng
gaáp khuùc).
A
Ngoaøi phaàn môû ñaàu, keát luaän, taøi lieäu tham khaûo, luaän vaên ñöôïc chia thaønh 5
chöông.
Chöông 1 giôùi thieäu moät soá kyù hieäu vaø caùc keát quaû chung chuaån bò ñeå khaûo saùt
trong caùc chöông sau.
Trong chöông 2 chuùng toâi chöùng minh söï toàn taïi duy nhaát lôøi giaûi cuûa baøi toaùn
(0.1)-(0.3). Keát quaû thu ñöôïc ôû chöông naøy ñaõ toång quaùt hoùa töông ñoái caùc keát quaû trong
[5],[6].
Trong chöông 3 chuùng toâi söû duïng phöông phaùp phaàn töû höõu haïn tam giaùc ñeå xaáp
xæ lôøi giaûi chính xaùc baøi toaùn (0.1)-(0.3) trong tröôøng hôïp Ω xaùc ñònh bôûi haøm ϕ lieân tuïc
vaø baäc nhaát töøng khuùc treân [0 , 1], töùc laø bieân ∂Ω laø ña giaùc. Keát quaû thu ñöôïc trong phaàn
naøy laø ñaùnh giaù sai soá giöõa lôøi giaûi xaáp xæ vaø lôøi giaûi chính xaùc theo moät caáp ñoä phuï
thuoäc vaøo tính “trôn” cuûa lôøi giaûi chính xaùc. Cuõng trong phaàn naøy chuùng toâi cho keát quaû
cuï theå öùng vôùi tröôøng hôïp rieâng M1(x,y,z) = M2(x,y,z) = z. Keát quaû trong phaàn naøy toång
quaùt hoùa caùc keát quaû trong [5].
Chöông 4 aùp duïng keát quaû cuûa chöông 3 cho mieàn Ωn, trong ñoù Ωn ⊂ Ω vaø bieân
∂Ωn ñònh bôûi ba caïnh thaúng OA, AB, OC vaø ñöôøng gaáp khuùc xaùc ñònh bôûi haøm ϕn lieân tuïc
vaø baäc nhaát töøng khuùc treân [0 , 1], ϕn “xaáp xæ” ϕ treân [0 , 1]. Keát quaû cuûa chöông naøy laø
caùc ñaùnh giaù sai soá giöõa lôøi giaûi phaàn töû höõu haïn vaø lôøi giaûi chính xaùc trong tröôøng hôïp
Ωn. Ngoaøi ra chuùng toâi cuõng ñaùnh giaù ñöôïc sai soá giöõa lôøi giaûi xaáp xæ baèng phaàn töû höõu
haïn treân Ωn vaø lôøi giaûi chính xaùc treân Ω.