Stochastic multiscale modeling of subsurface and surface flows. Part - PowerPoint PPT Presentation

Stochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows Ivan Yotov Department of Mathematics, University of Pittsburgh KAUST WEP Workshop January 30–February 1, 2010 Joint work with Vivette Girault, Paris VI, and Danail Vassilev, University of Pittsburgh Department of Mathematics, University of Pittsburgh 1

Outline • Mathematical model for the coupled flow problem – Interface conditions – Existence and uniqueness for a global weak formulation – Equivalence to a domain decomposition weak formulation • Finite element discretizations – Conforming Stokes elements and mixed finite elements for Darcy – Mortar finite elements on all subdomain interfaces – Discrete inf-sup condition – Existence and uniqueness of a discrete solution – Convergence analysis • Non-overlapping domain decomposition - reduction to a mortar interface problem • Coupling of Stokes-Darcy flow with transport Department of Mathematics, University of Pittsburgh 2

Applications of coupled Stokes and Darcy flows • Interactions between surface water and groundwater flows • Flows in fractured porous media • Flows through vuggy rocks • Flows through industrial filters • Fuel cells • Blood flows • Glaucoma Department of Mathematics, University of Pittsburgh 3

Model problem Γ 1 Ω = fluid region (Stokes flow) 1 Γ 1 Γ 1 n Γ I 2 n 1 Γ 2 = saturated porous Γ 2 Ω 2 medium (Darcy’s law) Γ 2 u j : Ω j → R d , fluid velocity in Ω j , p j : Ω j → R , fluid pressure in Ω j . Deformation tensor D and stress tensor T in Ω s : D ( u s ) := 1 2( ∇ u s + ( ∇ u s ) T ) , T ( u s , p s ) := − p s I + 2 µ D ( u s ) . Department of Mathematics, University of Pittsburgh 4

Flow equations Stokes flow on Ω s :  − div T ( u s , p s ) ≡ 2 µ div D ( u s ) + ∇ p s = f s in Ω 1 ( conservation of momentum ) ,    div u s = 0 in Ω s (conservation of mass) ,   u s = 0 on Γ s (no slip) .  Darcy flow on Ω d :  µ K − 1 u d + ∇ p d = f d in Ω d (Darcy’s law) ,    div u d = q d in Ω d (conservation of mass) ,  u d · n d = 0 on Γ d (no flow) ,   Solvability condition: � q d dx = 0 Ω d Department of Mathematics, University of Pittsburgh 5

Interface conditions Mass conservation across Γ sd : u s · n s + u d · n d = 0 on Γ sd . Continuity of normal stress on Γ d : − n s · T · n s ≡ p s − 2 µ n s · D ( u s ) · n s = p d on Γ sd . Slip with friction interface condition : (Beavers-Joseph (1967),Saffman (1971), Jones (1973), J¨ ager and Mikeli´ c (2000)) µα − n s · T · τ j ≡ − 2 µ n s · D ( u s ) · τ j = u 1 · τ j , j = 1 , d − 1 , on Γ sd , � K j where K j = τ j · K · τ j . Department of Mathematics, University of Pittsburgh 6

Global variational formulation For u d , v d in L 2 (Ω d ) d and u s , v s in H 1 (Ω s ) d , d − 1 � � � µα � K − 1 u d · v d +2 µ a ( u , v ) = µ D ( u s ) : D ( v s )+ ( u s · τ j )( v s · τ j ) . � K j Ω d Ω s Γ sd j =1 For v d ∈ H (div; Ω d ) , v s ∈ H (div; Ω s ) and q ∈ L 2 (Ω) , � � b ( v , q ) = − q div v d − q div v s . Ω d Ω s X = { v ∈ H (div; Ω) ; v s ∈ H 1 (Ω d ) d , v | Γ s = 0 , ( v · n ) | Γ d = 0 } , � 1 / 2 . � v � 2 H (div;Ω) + | v s | 2 � ∀ v ∈ X , � v � X = H 1 (Ω s ) Find ( u , p ) ∈ X × L 2 0 (Ω) such that � ∀ v ∈ X , a ( u , v ) + b ( v , p ) = f · v , Ω � ∀ r ∈ L 2 0 (Ω) , b ( u , r ) = − r q d . Ω Department of Mathematics, University of Pittsburgh 7

Existence and uniqueness of a weak solution Lemma: b ( v , q ) ∀ q ∈ L 2 0 (Ω) , sup ≥ β � q � L 2 (Ω) � v � X v ∈ X Lemma: a ( v , v ) ≥ γ � v � 2 ∀ v ∈ V = { v ∈ X : div = 0 } X Proof: By Korn’s inequality, a ( v , v ) ≥ C ( | v s | 2 H 1 (Ω s ) + � v d � 2 L 2 (Ω d ) ) . Coercivity now follows from � 1 / 2 . � | v s | 2 H 1 (Ω s ) + � v d � 2 � ∀ v ∈ V , � v s � L 2 (Ω s ) ≤ C L 2 (Ω d ) Lemma: The variational problem has a unique solution. Department of Mathematics, University of Pittsburgh 8

Domain decomposition variational formulation Let Ω s = ∪ Ω s,i , Ω d = ∪ Ω d,i . Interface conditions Stokes-Stokes interfaces: [ v s ] = 0 , [ T · n ] = 0 on Γ ss Darcy-Darcy interfaces: [ u d · n ] = 0 , [ p d ] = 0 on Γ dd X = { v ∈ L 2 (Ω) d : v | Ω s,i ∈ H 1 (Ω s,i ) d , v | Ω d,i ∈ H (div , Ω d,i ) , ˜ v · n ∈ L r ( ∂ Ω d,i )( r > 1) , v | Γ s = 0 , ( v · n ) | Γ d = 0 } , W = L 2 Λ = { ξ : ξ | Γ ij ∈ H − 1 / 2 (Γ ij ) ∀ Γ ij ⊂ Γ ss , 0 (Ω) , ξ | Γ ij ∈ H 1 / 2 (Γ ij ) ∀ Γ ij ⊂ Γ dd ∪ Γ sd } . Department of Mathematics, University of Pittsburgh 9

Domain decomposition variational formulation, cont. d − 1 � � µα � a s,i ( u s,i , v s,i ) = 2 µ D ( u s,i ) : D ( v s,i )+ ( u s,i · τ j )( v s,i · τ j ) � K j Ω s,i ∂ Ω s,i ∩ Γ sd j =1 � � K − 1 u d,i · v d,i , a d,i ( u d,i , v d,i ) = µ b i ( v i , w i ) = − w i div v i Ω d,i Ω i � � � a ( · , · ) = a s,i ( · , · ) + a d,i ( · , · ) , b ( · , · ) = b i ( · , · ) � � � b Γ ( v , λ ) = [ v ] λ + [ v · n ] λ + [ v · n ] λ Γ ss Γ dd Γ sd Find u ∈ ˜ X , p ∈ W , λ ∈ Λ : � ∀ v ∈ ˜ X, a ( u , v ) + b ( v , p ) + b Γ ( v , λ ) = f · v , Ω � ∀ w ∈ W, b ( u , w ) = − w q d , Ω ∀ ξ ∈ Λ , b Γ ( u , ξ ) = 0 . Department of Mathematics, University of Pittsburgh 10

Equivalence Lemma: The two variational formulations are equivalent. Department of Mathematics, University of Pittsburgh 11

✟✠ ✡☛ ✝✞ ☎✆ ✂✄ �✁ ☞✌ ✍✎ ✏✑ Finite element discretization Partition T h i on Ω i ; T h i and T h j need not match at Γ ij . X h s,i × W h Stokes elements in Ω s,i : 1 MINI (Arnold-Brezzi-Fortin), Taylor- Hood, Bernardi-Raugel ; contain at least polynomials of degree r s and r s − 1 resp. Velocity Pressure Mixed finite elements X h d,i × W h d,i in Ω d,i : RT, BDM, BDFM, BDDF ; contain at least Lagrange polynomials of degree r d velocity pressure multplier X h := i , W h := � � X h W h i ∩ L 2 0 (Ω) T H ij - partition of Γ ij , possibly different from the traces of T h i and T h j Λ H ij : continuous or discontinuous piecewise polynomials of degree at least m s on Γ ss or m d on Γ dd and Γ sd Λ H := � Λ H ij Nonconforming approximation: Λ h �⊂ Λ Department of Mathematics, University of Pittsburgh 12

Mortar finite element method Find u h ∈ X h , p h ∈ W h , λ H ∈ Λ H : � ∀ v h ∈ X h , a ( u h , v h ) + b ( v h , p h ) + b Γ ( v h , λ H ) = f · v h , Ω � ∀ w h ∈ W h , b ( u h , w h ) = − w h q d , Ω ∀ ξ H ∈ Λ H , b Γ ( u h , ξ H ) = 0 . Equivalently, letting V h = { v h ∈ X h : b Γ ( v h , ξ H ) = 0 ∀ ξ H ∈ Λ H } , Find u h ∈ V h , p h ∈ W h : � ∀ v h ∈ V h , a ( u h , v h ) + b ( v h , p h ) = f · v h , Ω � ∀ w h ∈ W h , b ( u h , w h ) = − w h q d . Ω Department of Mathematics, University of Pittsburgh 13

Mortar compatibility conditions On Γ ij ⊂ Γ dd ∪ Γ sd , i < j (assume that Ω i is a Darcy domain), � ϕ h i , ξ H � Γ ij ∀ ξ H ∈ Λ H , ≥ β d � ξ H � L 2 (Γ ij ) . sup � ϕ h i � L 2 (Γ ij ) ϕ h i ∈ V h i · n On Γ ij ⊂ Γ ss , i < j , i , ξ H � Γ ij � ϕ h ∀ ξ H ∈ Λ H , ≥ β s � ξ H � H − 1 / 2 (Γ ij ) . sup � ϕ h i � H 1 / 2 (Γ ij ) i | Γ ij ∩ H 1 / 2 ϕ h i ∈ V h 00 (Γ ij ) d These are much more general than the mortar conditions used in [Bernardi-Maday- Patera], [Ben Belgacem] for Laplace and Stokes, and in [Layton-Schieweck-Y.], [Riviere-Y.], [Galvis-Sarkis] for Stokes-Darcy. The above condition on Γ dd is similar to the one used in [Arbogast-Cowsar-Wheeler- Y.] and [Arbogast-Pencheva-Wheeler-Y.]. Department of Mathematics, University of Pittsburgh 14

Weakly continuous interpolants s = { v h ∈ X h s : � [ v h ] , ξ H � Γ ss = 0 ∀ ξ H ∈ Λ H } V h d = { v h ∈ X h d : � [ v h · n ] , ξ H � Γ dd = 0 ∀ ξ H ∈ Λ H } V h There exisit Π h s : H 1 (Ω s ) → V h d such that s v ) , w h ) Ω s,i = 0 ∀ w h ∈ W h , ∀ Ω s,i ⊂ Ω s , (div( v − Π h � � � Π h s v � H 1 (Ω s,i ) ≤ C � v � H 1 (Ω s,i ) There exist Π h d : H 1 (Ω d ) → V h s such that d v ) , w h ) Ω d,i = 0 ∀ w h ∈ W h , ∀ Ω d,i ⊂ Ω d , (div( v − Π h � � � Π h d v � H (div;Ω s,i ) ≤ C � v � H 1 (Ω s,i ) Department of Mathematics, University of Pittsburgh 15

Discrete inf-sup condition Lemma: b ( v h , w h ) ∀ w h ∈ W h , sup ≥ β � w � W � v � X v h ∈ V h Proof: Given w h ∈ W h , construct v ∈ H 1 (Ω) : div v = − w h , � v � H 1 (Ω) ≤ C � w h � L 2 (Ω) . Then construct an interpolant Π h : H 1 (Ω) → V h such that b (Π h v − v , q h ) = 0 ∀ q h ∈ W h , � Π h v � X ≤ C � v � H 1 (Ω) . Department of Mathematics, University of Pittsburgh 16