The role of the error function in three-dimensional singularly perturbed convection-diffusion problems with discontinuous data

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1 The role of he error funcion in hree-dimensional singularl perurbed convecion-diffusion problems wih disconinuous daa José Luis López García, Eser Pérez Sinusía Depo. de Maemáica e Informáica, U. Pública de Navarra jl.lopez@unavarra.es, eser.perez@unavarra.es Nico M. Temme Cenrum Voor Wiskunde en Informáica, Amserdam Nico.Temme@cwi.nl Resumen We consider singularl perurbed convecion-diffusion problems defined in hree-dimensional domains, problems of parabolic pe: ε(u + u ) + u + v u + v u = and of ellipic pe: ε(u + u + u zz ) + v u + v u + v 3 u z =, where v, v and v are real consans. For ever one of hese wo kind of problems we consider several hree-dimensional domains. We also consider for all of hese problems Dirichle daa disconinuous a cerain regions of he boundaries of he domains. For each problem, an asmpoic approimaion of he soluion is obained from an inegral represenaion when he singular parameer ε +. The soluion is approimaed b producs of error funcions, and his approimaion characerizes he effec of he disconinuiies on he small ε behaviour of he soluion and is derivaives in he boundar laers or he inernal laers. Palabras clave: singular perurbaion problem, disconinuous boundar daa, asmpoic epansions, error funcion. Clasificación para el Ceda 5: EDPs.. Inroducion As i has been shown recenl []-[5], he error funcion plas a fundamenal role in he approimaion of D singularl perurbed convecion-diffusion problems wih disconinuous Dirichle daa. In he problems here analzed i is shown ha he soluion is approimaed b a finie combinaion of error funcions as ε +. For eample, in he problem defined on he firs quadran Ω (, ) (, ) and wih a disconinuous Dirichle boundar condiion a he corner poin (, ) (see Figure ) []: { ε U + v U =, (, ) Ω, (P ) U(, ) =, U(, ) =, U C( Ω) D (Ω), where Ω Ω \ {(, )}, he soluion can be approimaed, for < β < /, b wih U β (, ) = U β(, )( + O( ε)) as ε +, r r > Uβ(, ) [ ( )] r φ β erfc sin. () We have used he polar coordinaes = r sin φ, = r cos φ ( r <, φ /). Afer he wo-dimensional sud performed in []-[], we wonder if i is possible o eend our wo-dimensional analsis o hree-dimensional problems wih disconinuous daa and if he error funcion is also useful in he approimaion of hese kind of problems. Then, we analze several 3D parabolic and ellipic problems defined on differen domains. We onl deail here a 3D parabolic problem defined on a half space (,, ) Ω (, ) (, ) (, ) wih a disconinuous iniial condiion over a recangle and a 3D ellipic problem defined on he firs ocan Ω = (, ) (, ) (, ), wih a disconinuous Dirichle daa a he X and Y aes.

2 β Ω Figura : Domain Ω and boundar condiions in problem (P ). U (,) β v Figura : Graph of he firs order approimaion U /4(, ) o he soluion of problem (P ) for ε =, and β = /4. The convecion vecor v drags he disconinui of he boundar condiion a (, ) originaing a parabolic laer along he characerisic defined b v and emanaing from (, )... A 3D parabolic problem in a half space We consider a problem defined in he upper half space: (,, ) Ω (, ) (, ) (, ), wih a recangular source of conaminaion locaed on he recangle (a, b) (c, d) (see Figure 3): U C( Ω ), U, U, U, U, U C(Ω ), U bounded on bounded subses of Ω, ε(u + U ) + v U + v U + U =, in Ω, U(,, ) = χ (a,b) ()χ (c,d) (), for (, ) R. In his formula, a < b, c < d and he posiive numbers b a and d c represen he lengh of he sides of he source of conaminaion. Observe ha he iniial condiion is disconinuous a { = a, =, c < < d}; { = b, =, c < < d}; { = c, =, a < < b} and { = d, =, a < < b}. The se Ω is he closed se Ω wih he regions of disconinui of he boundar daa removed: Ω Ω \ {{(, c, ), (, d, ), a b} {(a,, ), (b,, ), c d}}. () (,c,) (b,,) (a,,) (,d,) Figura 3: Domain Ω and iniial condiions of problem (). The change of variable U(,, ) = e [v +v /]/() F (,, ) in () ields he following pro-

3 blem for F (,, ): F C( Ω ), F, F, F, F, F C(Ω ), F bounded on bounded subses of Ω, F + F ɛ F =, in Ω, F (,, ) = e [v+v]/() χ (a,b) ()χ (c,d) (), for (, ) R. (3) We add a radiaion condiion o () in order o assure he uniqueness of he soluion. Hence, we consider he following problem: U C( Ω ), U, U, U, U, U C(Ω ), U bounded on bounded subses of Ω, ε(u + U ) + v U + v U + U =, in Ω, U(,, ) = χ (a,b) ()χ (c,d) (), for (, ) R, U, U, U, U, U, U = o (/r), as r, where r +. This problem has a mos one soluion. A soluion of problem (3) ma be derived b using he Fourier ransform in (3) wih respec o and. Therefore, we obain he soluion of (P ) epressed in erms a double inegral: U(,, ) = 4ɛ e[v+v /]/() b a e ( s) /(4ɛ) e vs/() ds Tha ma be evaluaed eacl in erms of error funcions: U(,, ) = { ( ) ( )} v + a erfc 4 v + b erfc ε ε { ( ) ( )} v + c erfc v + d erfc ε. ε d c (P ) e ( u) /(4ɛ) e vu/() du, (4) (5) U(,,.) U(,,) Figura 4: Graphs of he soluion of problem (P ) given in (5) for ε =,, a = b =, c = d =, v =,5 and differen values of..3. A 3D ellipic problem in an ocan We consider he firs ocan of R 3 wih disconinuiies a one-dimensional corners of he boundar of he domain Ω = (, ) (, ) (, ), wih an infinie source of conaminaion locaed a he plane z = (see Figure 5) and wih he convecion vecor v = (,, ): U C( Ω ) D (Ω ), U bounded on bounded subses of Ω, ε U + U z =, in Ω, (6) U(,, ) =, U(,, z) = U(,, z) =, for (,, z) Ω. Observe ha he Dirichle daa are disconinuous a he X and Y aes. The se Ω is he closed se Ω wih he X and Y aes removed: Ω Ω {(,, );, > } {(,, z);, z > } 3

4 z Figura 5: Domain Ω and Dirichle condiions of problem (6). {(,, z);, z > }. Afer he change U(,, z) = e v r /() F (,, z), problem (6) is ransformed ino he Yukawa equaion for F (,, z): F C( Ω ) D (Ω ), F bounded on bounded subses of Ω, F 4ɛ F =, in Ω, (7) F (,, ) =, F (,, z) = F (,, z) =, for (,, z) Ω. This problem ma have no a unique soluion unless we impose a convenien condiion upon U(,, z) (or upon F (,, z)) concerning is growh a infini. Then, we add a radiaion condiion o (6) and consider he following problem: U C( Ω ) D (Ω ), U bounded on bounded subses of Ω, ε U + U z =, in Ω, U(,, ) =, ( U(,, z) = U(,, z) =, for (,, z) Ω, ) (P ) U(,, z) = o e (r k+z)/(), as r k wih k =,, 3, rk where r + z, r + z and r 3 +. As a difference wih problem (P ), he soluion of (P ) canno be evaluaed in erms of known funcions, bu ma be approimaed in erms of error funcions when ɛ +. For his purpose we define he region: Ω Ω \ {{(,, z) Ω, <, < z z } {(,, z) Ω, <, < z z }}. The unique soluion of problem (P ) ma be derived b using he Fourier sine ransform wih respec o and : U(,, z) = ez/() d ds sin(/()) sin(s/()) e z + +s /(). (8) s Afer he change of variable s u defined b s = + u in he s inegral we obain: U(,, z) = ez/() sin(/()) d sin( + u/()) u e z + +u /() du. The inegral in he u variable is jus he soluion of a similar wo-dimensional convecion-diffusion problem defined on he quarer plane (, z) (, ) (, ) where i is shown ha: e z/() sin(u/()) e z +u /() + z du = erf z + u R(, z, ɛ), wih R(, z, ɛ) = O( ɛ) uniforml in, z for + z r > : R(, z, ɛ) C ɛ ( + z ) 3/4 e[z +z ]/(). (9) 4

5 wih C a posiive consan independen of, z and ɛ. Therefore, we can wrie wih and U (,, z) ez/() U(,, z) = U (,, z) + U (,, z), () U (,, z) ez/() sin(/()) e z + + /() erf [ + z z]d sin(/()) e z + /() R(, z, / + )d. The funcion U admis he following bound uniforml valid in Ω wih + z r > : On he oher hand we wrie wih and U (,, z) erf U (,, z) C + z e[z +z ]/(). () U (,, z) = U (,, z) + U (,, z), () z + z e z/() sin(/()) e z + /() d U (,, z) ez/() sin(/()) e z + /() + erf [ z + z + z] erf z d. From here we can obain a bound for U similar o () and use he soluion of a similar D problem defined on a quarer plane o analze U. We finall obain ha, for (,, z) Ω, he soluion U(,, z) of problem (P ) is (3) U(,, z) = erf z + z erf z + z [ + O( ɛ) ]. (4) U(,,) U(,,4) (a) z = (b) z = 4 Bibliografía [] J.L. López and E. Pérez Sinusía, Asmpoic epansions for wo singularl perurbed convecion-diffusion problems wih disconinuous daa: he quarer plane and he infinie srip, Sud. Appl. Mah., 3 (4), [] J.L. López and E. Pérez Sinusía, Asmpoic approimaions for a singularl perurbed convecion-diffusion problem wih disconinuous daa in a secor, JCAM, o be published. [3] S.-D. Shih, A novel uniform epansion for a singularl perurbed parabolic problem wih corner singulari, Meh. Appl. Anal., 3 (996), n., 3-7. [4] N.M. Temme, Analical mehods for a singular perurbaion problem in a secor, SIAM J. Mah. Anal., 5 (974), n. 6, [5] N.M. Temme, Analical mehods for a selecion of ellipic singular perurbaion problems, Recen advances in differenial equaions (Kunming, 997), 3-48, Piman Res. Noes Mah. Ser., 386, Longman, Harlow,