workshop in honor of Martin Stynes November 16–18, 2011 remarks on asymptotic expansions for singularly perturbed second order ODEs with multiple scales J.M. Melenk joint work with L. Oberbroeckling B. Pichler C. Xenophontos TU Wien Institut f¨ ur Analysis und Scientific Computing
Intro weakly coupled strongly coupled DG conclusions Introduction 1 weakly coupled elliptic-elliptic case 2 strongly coupled systems 3 DG 4 Conclusions 5 singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions Introduction 1 weakly coupled elliptic-elliptic case 2 strongly coupled systems 3 DG 4 Conclusions 5 singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions setting of the talk − Eu ′′ + Bu ′ + Au = f in I = (0 , 1) , u (0) = u (1) = 0 . � ε 1 � E = , 0 < ε 1 ≤ ε 2 ≤ 1 ε 2 questions: identification of the correct limit problem for “small” E layer structure of the solution asymptotic expansions and their regularity continuous dependence of solution on data for small E singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions “ E small” =? � ε 1 � E = ε 2 key ingredient of asymptotic analysis: identify the length scales assume scale separation singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions “ E small” =? � ε 1 � E = ε 2 key ingredient of asymptotic analysis: identify the length scales assume scale separation expect here: solution features on scales O ( ε 1 ) , O ( ε 2 ) , O (1) singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions “ E small” =? � ε 1 � E = ε 2 key ingredient of asymptotic analysis: identify the length scales assume scale separation expect here: solution features on scales O ( ε 1 ) , O ( ε 2 ) , O (1) = ⇒ scale separation governed by ε 1 ε 2 and ε 2 1 singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions “ E small” =? � ε 1 � E = ε 2 key ingredient of asymptotic analysis: identify the length scales assume scale separation expect here: solution features on scales O ( ε 1 ) , O ( ε 2 ) , O (1) = ⇒ scale separation governed by ε 1 ε 2 and ε 2 1 = ⇒ for the phrase “ E small” to be meaningful, have to specify which quantities ε 1 /ε 2 and/or ε 2 / 1 are small ( → different asymptotic regimes!) example singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions � � � 1 � 3 − 2 − Eu ′′ + u ′ = , u (0) = u (1) = 0 . − 2 0 1 < 1 ε 1 < < ε 2 < limit boundary conditions u 1 (0) − 2 ε 1 /ε 2 → 0 and ε 2 / 1 → 0 : 3 u 2 (0) = 0 , u 2 (1) = 0 ε 1 = 10 − 9 , ε 2 = 100 ε 1 more details singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions � � � 1 � 3 − 2 − Eu ′′ + u ′ = , u (0) = u (1) = 0 . − 2 0 1 < 1 ε 1 = ε 2 < < 1 ε 1 < < ε 2 < limit boundary conditions u 1 (0) − 2 ε 1 /ε 2 → 0 and ε 2 / 1 → 0 : 3 u 2 (0) = 0 , u 2 (1) = 0 ε 1 = ε 2 → 0 : 2 u 1 (0) − u 2 (0) = 0 , u 1 (1) + 2 u 2 (1) = 0 ε 1 = 10 − 9 , ε 2 = 100 ε 1 more details singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions Introduction 1 weakly coupled elliptic-elliptic case 2 strongly coupled systems 3 DG 4 Conclusions 5 singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions weakly coupled system − Eu ′′ + A ( x ) u = f in I = (0 , 1) , u (0) = u (1) = 0 Assumptions A , f analytic on I , A uniformly positive definite on I . � ε 1 � E = , 0 < ε 1 ≤ ε 2 ≤ 1 . ε 2 features: well-posedness and continuous dependence of solution on data (e.g., in energy norm) asymptotic expansions are designed to yield small residual → truncated expansion yields small error (in appropriate norms) solution u has features on 3 scales: O ( ε 1 ) , O ( ε 2 ) , O (1) singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions expansions correct asymptotic expansion ansatz hinges on the presence of scale separation, i.e., whether ε 2 ε 1 and/or is small 1 ε 2 → 4 different cases! ε 1 ε 2 focus here: ε 2 and 1 small singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions expansions make the ansatz � ε 2 � i � ε 1 � j � � � u ( x ) ∼ + + + 1 ε 2 i,j singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions expansions make the ansatz � ε 2 � i � ε 1 � j � � � u ( x ) ∼ u ij ( x ) + + + 1 ε 2 i,j u ij : functions on the O (1) scale singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions expansions make the ansatz � ε 2 � i � ε 1 � j � � � u ( x ) ∼ u ij ( x ) + � u ij ( � x ) + + 1 ε 2 i,j u ij : functions on the O (1) scale � x = x/ε 2 � u ij : functions on (0 , ∞ ) for the O ( ε 2 ) scale exponentially decaying singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions expansions make the ansatz � ε 2 � i � ε 1 � j � � � u ( x ) ∼ u ij ( x ) + � u ij ( � x ) + � u ij ( � x ) + 1 ε 2 i,j u ij : functions on the O (1) scale � x = x/ε 2 � u ij : functions on (0 , ∞ ) for the O ( ε 2 ) scale exponentially decaying � x = x/ε 1 � u ij : functions on (0 , ∞ ) for the O ( ε 1 ) scale exponentially decaying singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions expansions make the ansatz � ε 2 � i � ε 1 � j � � � u R u ( x ) ∼ u ij ( x ) + � u ij ( � x ) + � u ij ( � x ) + � u R x R ) + � x R ) ij ( � ij ( � 1 ε 2 i,j u ij : functions on the O (1) scale � x = x/ε 2 � u ij : functions on (0 , ∞ ) for the O ( ε 2 ) scale exponentially decaying � x = x/ε 1 � u ij : functions on (0 , ∞ ) for the O ( ε 1 ) scale exponentially decaying u R u R � ij , � ij : analogous boundary layer fct at x = 1 singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions computation of asymptotic expansion insert ansatz into the differential equation separate scales, i.e., view variables x , � x , � x as independent variables equate like powers of ε 1 /ε 2 and ε 2 / 1 this yields a recurrence relation of DAEs for the functions u ij , � u ij , � u ij insertion of the b.c. at x = 0 and x = 1 closes the DAEs singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions � u ij � � � � � � � u ij u ij u ij = , u ij = � , u ij = � , v ij v ij � � v ij convention: functions with negative subscript vanish recursions for A = constant � u ′′ � i − 2 ,j − 2 − + Au ij = F ij = δ ( i,j ) , (0 , 0) f (1a) v ′′ i − 2 ,j � ( � � u i,j − 2 ) ′′ − + A � u i,j = 0 , (1b) v i,j ) ′′ ( � � � u i,j ) ′′ ( � − + A � u i,j = 0 , (1c) v i,j +2 ) ′′ ( � boundary conditions: u ij (0) + � u ij (0) + � u ij (0) = 0 , (1d) decay conditions for � u ij , � u ij at + ∞ (1e) singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions properties of the functions u ij , � u ij , � u ij in general, the equations are systems of DAEs all arising DAEs are successively solvable the regularity of the functions u ij , � u ij , � u ij can be controlled explicitly in terms of i , j . Define approximation to u by truncated expansion: � ε 2 � i � ε 1 � j � � � u M 1 ,M 2 := u R u R x R ) + � x R ) u ij ( x ) + � u ij ( � x ) + � u ij ( � x ) + � ij ( � ij ( � 1 ε 2 i ≤ M 1 j ≤ M 2 u R ( � x R ) + � x R ) =: w ( x ) + � u ( � x ) + � u ( � x ) + � u ( � � �� � � �� � � �� � smooth part layer parts layer parts singularly perturbed coupled systems J.M. Melenk
Intro weakly coupled strongly coupled DG conclusions � ε 2 � i � ε 1 � j � � � u M 1 ,M 2 := u R u ij ( x ) + � u ij ( � x ) + � u ij ( � x ) + � u R ij ( � x R ) + � ij ( � x R ) 1 ε 2 i ≤ M 1 j ≤ M 2 u R ( � x R ) + � x R ) =: w ( x ) + � u ( � x ) + � u ( � x ) + � u ( � Theorem (optimal truncation) Select, for implied constants depending on f and A : M 1 ∼ 1 M 2 ∼ ε 2 , ε 2 ε 1 Then: � e − b/ε 2 + e − bε 2 /ε 1 � � u − u M 1 ,M 2 � E,I ≤ C w is analytic (with control uniformly in ε 1 , ε 2 ) � u is typical boundary layer function on ε 2 -scale � u is typical boundary layer function on ε 1 -scale singularly perturbed coupled systems J.M. Melenk
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