一种求解椭圆型微分方程的渐近展开方法及其在辐射热传导方程中的应用.pdf

湘潭大学 硕士学位论文 一种求解椭圆型微分方程的渐近展开方法及其在辐射热传导方 程中的应用 姓名：刘志清 申请学位级别：硕士 专业：计算数学 指导教师：舒适 20080526 ,()PDEs ,, PDEs()PDEs . ,. ,Γ ,,, ?·?0 , .,, ., ,,. ;;;. I Abstract We often encounter some elliptic PDEs with jump ( or multi-scale character- istic) coeffi cients during the research of mathematical physics. The asymptotic expansion method is eff ective to solve such problems, whose main idea is how to decompose the PDEs with jump coeffi cients into several PDEs with smooth (or single-scale characteristic) coeffi cients. In this paper, we study the asymp- totic expansion methods for elliptic PDEs and 2-D radiation heat conduction equations. Firstly, aiming at a kind of jump coeffi cient elliptic scalar equation with the general boundary condition, according to two diff erent interfaces, we propose a linear fi nite element method based on the asymptotic expansion. By using the basic theory of FEM, we obtain the same order of error function as that of the classic linear FEM under L2norm. The numerical experiments verify the correct- ness of theoretical results. Secondly, for a kind of linear radiation heat conduc- tion equations, we design and analyze a linear fi nite element method based on the asymptotic expansion. Furthermore, we give the numerical experiment results for a non-linear radiation heat conduction equation with single-temperature, which show that the asymptotic expansion method is eff ective. Key words:multi-scale characteristic;asymptotic expansion method; linear fi nite element; 2-D radiation heat conduction equations. II ,,, PDEs,, ,,,( )., .,: (IIM)([1,2,3,4,5,6]),([7,8,53,54,55,56]) ([9,10,11,12,13,14, 15,16,17,18]). ,. ,,, . ,, K-BW-K-B. PDEs,PDEs PDEs,PDEs ([19,20,21,22]), ,PDEs. , . ,([16,17,18]), , .,. ,. , .,: (1), (2) ,, , O(ε2).,Klapper,T.shaw([16]), ,,. , 1 PDES (), ,L2 O(h2). PDES, (AMG). ,(GMG)AMG, AMG A.Brandt,S.McCormick,J.Ruge([35 ∼ 47]),GMG , AMG, (), . ,ε = h,, .,, ,, . , . . (ICF)([26 ,27,28,29]), ,, ,. ,,,([23,24,25,32]), ([30,31]). , .,, ,PDEs .,, ,, ,, , . : . ,. , ,. ,. 2 ; AMG;. SobolevW m,p(Ω) . 1.m, SobolevW m,p(Ω) ?v?m,p= ( ? |α|≤m ?Dαv?Lp(Ω)) 1 p, 1 ≤ p 0.,x ? 1 . f(x)= −[t−1ex−t]∞ x − ?∞ x t−2ex−tdt = 1 x − ? ∞ x t−2ex−tdt. f(x)= 1 x + [t−2ex−t]∞ x − 2 ?∞ x t−3ex−tdt = 1 x − 1 x2 + 2 ? ∞ x t−3ex−tdt. n f(x) = n ? m=1 um(x) + Rn(x), ? um(x) = (−1)m−1(m − 1)!x−m, Rn(x) = (−1)nn! ?∞ x t−(n−1)ex−tdt. : |Rn| ≤ n! xn+1, x, ∞ ? m=1 n! xn+1 . (1)n x, n ? m=1 um(x)f(x),. (2)n ? x,,, n ? m=1 um(x)f(x) . 8 , n.xf(x), n ? m=1 umn. , .,,, . 2.εu(x,ε). u(x,ε),ε ()εu(x,ε) u(x,ε) = n ? m=0 δm(ε)um(x) + Rn(x,ε), (0 0. (2.1)u ∈ C(Ω),(Flux)β∇u ∈ C(Ω),u [u] = 0,x ∈ Γ,(2.3) [β un] = 0,x ∈ Γ,(2.4) [u] = u+− u−,[β un] = α+u+ n − ε−1α−u− n, u+,u−u+ n,u − n Ω+Ω− Γ. (2.1) ? −∇ · (β(x)∇u) = f(x),x ∈ Ω, u = 0 ,x ∈ ∂Ω, (2.5) u ∈ H1 0(Ω), a(u,v) = (f,v),∀v ∈ H1 0(Ω), (2.6) a(u,v)= ? Ωβ(x)∇u · ∇vdx, (f,v)= ? Ωfvdx. Th= {Ek,1 ≤ k ≤ M}Ω,Ek,M k, h = max 1≤k≤M hk( hkEk); {Xi,i = 1,··· ,N}ThDirichlet, N . P11,Th V h 0 = {u : u ∈ H1 0(Ω),u|Ek ∈ P1,1 ≤ k ≤ M} (2.6), uh∈ V h 0 , a(uh,vh) = (f,vh),∀vh∈ V h 0 .(2.7) 11 (2.7), AhUh= Fh, ,NAh, UhN, FhN . : (1)(2(a)), (2) (2(b)) 2 (a)(b) 1(2.5) Ω = [0,1] × [0,1],3: 3 β(x,y) = ? 1,y ≥ 1 2, ε−1, otherwise, f(x,y) = 5π2sin(πx)sin(2πy) , u(x,y) = ? sin(2πy)sin(πx),y ≥ 1 2, εsin(2πy)sin(πx), otherwise. 12 1,,Ω,xy nxny,nx = ny = n,h = 1 nx, (2.5) ,CG AMG,: 1.n,(2(a)). CG,. 1? · ?0 nε−1iter?u − uh?0 4811076.06e-4 4842441.76e-3 48164306.87e-3 48647312.74e-2 4825612161.09e-1 1,,, ,, . ,AMG, 2? · ?0 nε−1iter?u − uh?0 48146.02e-4 48451.76e-3 481656.87e-3 486452.74e-2 4825651.09e-1 2AMG. 13 ε−1?u − uh?0?u − uh?0rate 2448 12.42e-36.02e-44.01 47.06e-31.76e-34.01 162.74e-26.87e-33.98 641.09e-12.74e-23.98 2564.38e-11.09e-14.01 3,? · ?0 O(h2). 2.n,(2(b)). 4? · ?0 nε−1iter?u − uh?0 49145.78e-4 49442.25e-2 491651.98e-1 496459.69e-1 4925654.07 4,,, ,,, . 5 ε−1?u − uh?0?u − uh?0rate 2549 12.23e-35.78e-43.85 44.16e-22.25e-21.84 163.86e-11.98e-11.94 641.9019.69e-11.96 2568.014.071.96 5,? · ?0 O(h),. ,, ,(([53,54,55,56]))IIM. ,,. 14 3 ,ε−1→ ∞,(2.1) ,ε, ,, . β(x)(2.2)(2.1), ? −∇ · (α+(x)∇u+) + c0u+= f+(x),x ∈ Ω+, −∇ · (ε−1α−(x)∇u−) + c0u−= f−(x),x ∈ Ω−. (2.8) (2.1)uε, u(x; ε) = ? u+ 0(x) + εu + 1(x) + O(ε 2), x ∈ Ω+, u− 0(x) + εu − 1(x) + ε 2u− 2(x) + O(ε 3), x ∈ Ω−, u+,u−O(ε2)O(ε3), u(x; ε) ? ? u+ 0(x) + εu + 1(x), x ∈ Ω+, u− 0(x) + εu − 1(x) + ε 2u− 2(x), x ∈ Ω−. (2.9) (2.9)(2.8), ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −∇ · (α+(x)∇u+ 0) − ε∇ · (α +(x)∇u+ 1) + c0u + 0 + c0εu+ 1 = f+(x),x ∈ Ω+, −ε−1∇ · (α−(x)∇u− 0) − ∇ · (α −(x)∇u− 1) −ε∇ · (α−(x)∇u− 2) + c0u − 0 + c0εu− 1 + c0ε2u− 2 = f−(x),x ∈ Ω−, (2.10) (2.10),ε,, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −∇ · (α+(x)∇u+ 0) + c0u + 0 = f+(x),x ∈ Ω+, −∇ · (α+(x)∇u+ 1) + c0u + 1 = 0,x ∈ Ω+, −∇ · (α−(x)∇u− 0) = 0, x ∈ Ω−, −∇ · (α−(x)∇u− 1) + c0u − 0 = f−(x),x ∈ Ω−, −∇ · (α−(x)∇u− 2) + c0u − 1 = 0,x ∈ Ω−. (2.11) (2.9)(2.3), u+ 0 + εu+ 1 − u− 0 − εu− 1 − ε2u− 2 = 0,x ∈ Γ, 15 ,ε, ? u+ 0 = u− 0, u+ 1 = u− 1, x ∈ Γ .(2.12) (2.9)(2.4), α+(u+ 0(x) + εu + 1(x))n− ε −1α−(u− 0(x) + εu −。