Bakhvalov, N. S.

Knyazev, A. V.

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Nikolai Sergeevich Bachvalov; A. V. KnyazevA new iterative algorithm for solving the fictitious fluxes methodproblems for elliptic equationsIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 225--227.Persistent URL: http://dml.cz/dmlcz/702342Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czA NEW ITERATIVE ALGORITHM FOR SOLVING THE FICTITIOUS FLUXES METHOD PROBLEMS FOR ELLIPTIC EQUATIONS BAKHVALOV N.S.,KNYAZEV A.V., MOSKVA, USSR We consider a variant of the fictitious domain method for solving the Dirichlet problem with the diffusion coefficient K * . ^ L ^ C-^X ^ D ^ -Z > ^ in a com-plex shaped domain 5 ) . Let us complement iD by "fictitious" domain/) so, that' in the resulting domain D (in fact O -* J> U J)'L ) the equation of the type A"U. - •? y -u~ e W ^ C Q ) ^ •£- € W ^ C a ) could be efficiently solved. In Q we construct the Dirichlet prob-where iy jT K^Q in «0 , and K. is a large (may be infinitely large) constant. Let %) an(j Q be bounded Lipschitzian domains. The inequalities hold and if *--J is simply connected then I*--**' wiC-D) * C ^ > l l ^ l l ^ C O ^ * C l~L. In the .case |L=.-J-oo one can speak about fictitious fluxes in 2> Which provide, that g-ĉ -rf 1v - O in £> , so that f*uw/it = ss. t̂--3-W Ic^ in ^ 15 Kurzweil, Equadiff 7 225 We introduce the flux vector p "̂  |<£ <VX£ui 1c a n d rewrite the equation (2j in the form t Let now Q^^cuic^LM^ \ IL^ C C-0 ~* 11-^(0) , where £>s cTa* g*cô ?( : W / C Q ' i "* ^ i * " ̂ Q ^ • We can eliminate the unknown function 1/L from (4) and obtain the equations and 1*. could be computed, for example, by the formulatCsA -?lct>- K. p if the (approximate) solution p of (5) had been found before. The main point here is that the operator Q is the orthoprojector in IL^CQ) on the subspace d£ •= <J •t-â  Ofr̂  CQ) ^ ---.CO) . We need (5) exept (2) first of all to treat the case {--.=.-*- oo , when K = 0 in X/ . In this case the uniqueness of p in (5) lost and we mean by p the normal (with minimal VL-^CQ) norm) solution of (5). Theorem 1. Let 4. -- I"- ̂  "*" °"° . The unique solution p of (5) exists for every Q, €• Q and the inequality I P - ^ C Q ) ~ c H - I L ^ C O ) , <-f c f t l ) is true, i.e. problem (5) is well-posed uniformly in l*-»*£-<4-. Theorem 2. Let p/> denote the solution p of (5) with K-> d. and ppo is the normal solutio/i of (5) with fc. --h oo . Then lPt-p'->iL J tCD) « <-*-"i H-I.-.CC-0-The inequalities (3) are the consequences of this estimate. The numerical solution of equations (5) can be obtained by the simple process e ^ f c t a V l p - - ° . - - < » . i . - ,r~f. <«> To study convergence properties of the method (6) we need the following statements. Lemma 1 . Let IfJl » { x ^ I L ^ ( a ) : *fii> X « O , *t-= g-u-W 1> i n «0 , for a T > C N^ 1 ( Q ) ^ Then : (a)llV/ is a subspace in lt-x t ^ ) » (b) IM -. R CQ-'-K-1-) i f £ ~ + < ~ . ^ _ (c)lhll is an invarian t subspace for the operator Q K and Q X K ' 4 •= Q ^ l O ^ Q / - on M , (d) for the so lution p of (5) we have p - Ck, C INI (e) the se l fadjo in t operator Q^K'^Q^ : INI ~* lh/l 226 s a t i s f i e s the i n e q u a l i t i e s 0 < c l ^ Q X « " 1 q > 1 * < | L - ± I on IhJl , i . e . o<c$ i-ci^a * y K ^ I M ^ O * £^1 M ^ a ^e/A//^^o a a a where C r̂  C f tt.) if dL £ K .£ 4-o-=> # Now statements (a)-(c) enable us to prove the important Lemma 2. Let % K'= p ^ - ^ . The the iterative method (6) is equivalent to - h Q ^ K * ̂  X*'- O Vv~ O, tJMi 5 t°s O and x J ^ e thJj . From Lemma 1 (d), (e) and Lemma 2 follows immediately Theorem 3. With appropriate choice of t (for example, with **-•- *£ ) the iterative approximations p from (6) converge to the solution p of (5) as fast as geometric sequence uniformly in TL^i.. In the case l ^ + o o ^ K^ ~ i we t a k e T<Ni £" Ldlv> f C l p** and from (6) we have V U ^ - Í Í C V Г ^ ^ U ^ Ä O , 1 ^ * 0 ^ ^ . . o Here t ^ K ^ W,*" C CI ) .. , so the residuals will belong to W ^ C d ) with support only on 7) SO . There exist generalizations of the results. First of all, K?\ and K might be matrices not functions. Secondly, one can substitute the original space W;J" C Q ) for the space W ^ ( Q ) with other boundary conditions (for instance, with periodic boundary condition when.D is a parallelepiped). Substituting in Wj^ C.C-) vector for scalar functions we get the method for solving elliptic systems. Some boundary value problems of the elasticity theory in displacements could also be solved \f<£coA is changed to the matrix (h , which transforms the displacements to an uppertriangle part of the deforma-tion tensorf^i^ to & and - A to 6 G> with the boundary con-ditions of "rigit contact" on 2 Q : ̂ ^ ^ ^ ~ O , ̂ t a * ^ /dh~~0 (in such a case the equation Q,TG> <-<_ ~ m could be solved by the Fourier Method). r References 1. Kobel'kov G.M. - Sov.J.Num.Anal.Math.Modelling, 1987, v.2, N 6, p.407-420. 2. Bakhvalov N.S., Kobel'kov G.M., Chizhonkov E.V. - Preprint N 190, Dep.Num.Math.USSR Ac.Sci., Moscow, 1988 (In Russian). 227 

A new iterative algorithm for solving the fictitious fluxes method problems for elliptic equations

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A new iterative algorithm for solving the fictitious fluxes method problems for elliptic equations

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Institute of Mathematics AS CR, v. v. i.