Convergence Analysis For The Residual-Free Bubbles Method Applied To Parabolic Problems

It is well known that when we use normal Galerkin Method to solve an advection-diffusion Problem like this — εΔu + a · ▽u = f as ε << h · |a|, the answer will be unstable. There are various possible ways to stabilize the answer. Except FEM, SUPG etc., the RESIDUAL-FREE BUBBLES method come out in the last 10 years.when we discuss the Parabolic problem u_t — εΔu + a · ▽ = f as ε<< h · |a|, the RESIDUAL-FREE BUBBLES method is also useful to discrete —εΔu + a · ▽u;and then we can use the backward Euler-Galerkin method to discrete u_t. and then,give a priori error analysis for this method.At the end of the paper, give extensions for the same method.This thesis is divided into four chapters. In Chapter 1, we begin with the problems:which the RESIDUAL-FREE BUBBLES method came out 10 years ago. Then, definite some equations and functions, and tell what is the RESIDUAL-FREE BUBBLES method is.In the first chapter, there arc two parts. In part one, tell the reason why equation:is a multiscale problem, and why normal Galerkin method is useless. Then definite the bubble function. So inVh = VL + VB (1.1.3)the scattered equation can be rewrite as:{find uh = uL + uB, s.t.s \ e In VuhVuhdx + Ju(a ■ Vuh)vhdx = Jn fvhdx \/vh G Vh. Just because the particularity of bubble function, the problem equal to:{find ub G Vb, s.t.e JT VuB ■ VvTBdx + JT{a ■ VuB)vTBdx = (/ - a ? VuL)\T JT vTBdx Vu￡ G VB.(1.1.6)where:f -?■ A6f + a ? VbT = 1 in T,{I (1.1.7)\ 6f = 0 e/se.and now we can get the approximate answer more easier.In part two, will tell what people had done, and what I will do.In Chapter 2. we shall work on the parabolic problems. Give several definitions, and then scatter the equation.There are two parts in this chapter.In part one, print out the equation: ut - eAu + a ? Vu = /, x G fI t G [0, TE].u(x,t) = 0, xedfl te[0,TE\, (2.1.1)u(x. 0) = v, x G ft.Give adefinition: imitated ellipse projection Pb:In part two, give the partly scattered equation first:then, the fully scattered: ^^ E a?(eV^-, V&) n = 1, 2, ? ? ? ,N E a ji E a"(a ? V0 0fc) = (/ &) * = 1, 2, ? - - , JVa = 7i, ; = 1.2.-- TiV + l.rewrite as:Aa'(t) + (sB + C)a(f) = /, t > 0, a(0) = 7. (2.2.4)(/T + reB + rCjo" = Aa"-1 + r/(^n). (2.2.7)In Chapter 3, give the convergence analysis. There are two parts in this chapter.In part one, give a remark first: remark the skill is very useful: because dizu = 0,(a-Vf/,,t'h) = -(vh,a-S7vh) = -(a-Vvh,vh) => (a-Vvh,vh) = 0 Vtv, € VL. And then two definitions:definition 2 VxET, if there is axQon the boundary of T Svt.: \a ■ (x — xa}\ = tl°llllx ~ ^ali and a ■ (x — xa)} > 0, then we call xathe upwind of x.definition 3 let/4 = ma* a - fa' ~ ra)/N2then give two lemma:lemmal Vc, Vx € TwhereF € Th, i￡xais the upwind of x, then: 0 < b\ < a ? (x - xa)/|a|2lemma 2 inf {thesmallestangleofT} < ^owhere^o > 0. then there exist a Constance C independent of T, s.t.and then the convergence analysis Theorem:theorem 1 ifufcsatisfied( 1.1.4), u,eLis definite by(3.1.10)(3.1.14). then:￡' ||Ve||0.n = (s||VeL||jj.ntheorem 2 ifuhsatisfied(l. 1.4). u, eL,-)T^ definite by(3.1.10)(3.1.14)(3.1.17), and ifu € i/s(fi)f)f/o(f2)is the accurate answer, then: -l\u\l.T: (3.1-21) ^^Hir- (3-1-22)T Twhere Constance C only depend on#oof7/j.theorem 3 ifuhsatisfied(l. 1.4). u, eL. -yris definite by(3.1.10)(3.1.14)(3.1.17). and ihi € Hs(il) f] Hq(Q.)is the accuiate answer, then:— u/i)||o.i2 < (C 2_^~'t"t \u\s.t) ■ (3.1.23). T Er)^ (3.1.24) r rwhere Constance C only depend on^oofT^.In part two, give the convergence analysis of the parabolic problems: lemma 3 ifu € HS(Q.) f] H^{Q). wherel 0. (3.2.8)Jotheorem 5 iff/"and u is the solution of(2.2.5)and(2.1.1). there is:\\Un-u(tn)\\ < ||i-/l-i;||+C/jr+^--5{||i'||r+ I " \\ut\\rds}+K f " \\utt\\ds, n > 0.Jo Jowhere Constance C only dependent on#0, and constance K is the time step of the backward Euler-Galerkin method.In Chapter 4, give the extension and result.