Mortar multiscale framework for Stokes-Darcy flows Ivan Yotov - PowerPoint PPT Presentation

Mortar multiscale framework for Stokes-Darcy flows Ivan Yotov Department of Mathematics, University of Pittsburgh Workshop on ”Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration” RICAM, Linz, Austria October 3-7, 2011 Joint work with Vivette Girault, Paris VI and Danail Vassilev, Pitt Acknowledgment: Ben Ganis, UT Austin Department of Mathematics, University of Pittsburgh 1

Coupled Stokes and Darcy flows Γ s Ω s : Stokes region ( u s , p s ) Γ s Γ s Γ sd n d n s Γ d Γ d Ω d : Darcy region ( u d , p d ) Γ d • surface water - groundwater flow • flow in fractured porous media • flow through vuggy rocks • flow through industrial filters • fuel cells • blood flow Department of Mathematics, University of Pittsburgh 2

Outline • Mathematical model for the coupled Stokes-Darcy flow problem – Interface conditions – Existence and uniqueness for a global weak formulation – Equivalence to a domain decomposition weak formulation • Multiscale mortar finite element discretizations h – Fine scale ( h ) conforming Stokes elements and � � � mixed finite elements for Darcy � � � � � � mortar dof – Coarse scale ( H ) mortar finite elements on subdo- � � � � � main interfaces H – Discrete inf-sup condition � � � � � – Convergence analysis � � � • Non-overlapping domain decomposition – Reduction to a mortar interface problem – A multiscale flux basis • Computational results Department of Mathematics, University of Pittsburgh 3

Flow equations 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 ) . Stokes flow in Ω s : −∇ · T ≡ − 2 µ ∇ · D ( u s ) + ∇ p s = in Ω s ( conservation of momentum ) , f s ∇ · u s = 0 in Ω s (conservation of mass) , = 0 on Γ s (no slip) . u s Darcy flow in Ω d : µ K − 1 u d + ∇ p d = in Ω d (Darcy’s law) , f d ∇ · u d = q d in Ω d (conservation of mass) , u d · n d = 0 on Γ d (no flow) . Department of Mathematics, University of Pittsburgh 4

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 sd : − 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 5

Some previous results • Existence and uniqueness of a weak solution – Discacciati, Miglio, Quarteroni 2002 – Layton, Schieweck, Y. 2003 – NSE-Darcy: Girault, Riviere 2009; Discacciati, Quarteroni 2009 • Numerical approximation with Stokes and Darcy elements – Discacciati, Miglio, Quarteroni 2002 – Layton, Schieweck, Y. 2003 – Riviere, Y. 2005 – Galvis, Sarkis 2007 – Babuska, Gatica 2010 – Riviere, Kanschat 2010 • Numerical approximation with unified finite elements (Brinkman model) – Angot 1999 – Mardal, Tai, Winther 2002 – Arbogast, Lehr 2006 – Burman, Hansbo 2005, 2007 – Xie, Xu, Xue 2008 Department of Mathematics, University of Pittsburgh 6

Global variational formulation Stokes: v s ∈ H 1 (Ω s ) n , v s = 0 on Γ s , � � � � 2 µ D ( u s ) : D ( v s ) − p s div v s − Tn s · v s = f s · v s Ω s Ω s Γ sd Ω s Darcy: v d ∈ H (div; Ω d ) , v d · n d = 0 on Γ d � � � � K − 1 u d · v d − µ p d div v d + p d v d · n d = f d · v d Ω d Ω d Γ sd Ω d Interface term: n − 1 � µα � � I = ( u s · τ j )( v s · τ j ) + p d [ v ] · n s � K j Γ sd Γ sd j =1 Department of Mathematics, University of Pittsburgh 7

Global variational formulation, cont. ˜ X = { v ∈ H (div; Ω) ; v s ∈ H 1 (Ω s ) n , v | Γ s = 0 , ( v · n ) | Γ d = 0 } , W = L 2 0 (Ω) � 1 / 2 � v � 2 H (div;Ω) + | v s | 2 � � v � ˜ X = H 1 (Ω s ) Find ( u , p ) ∈ ˜ X × W such that � � � ∀ v ∈ ˜ K − 1 u d · v d + 2 µ X, µ D ( u s ) : D ( v s ) − p div v Ω d Ω s Ω n − 1 � µα � � + ( u s · τ j )( v s · τ j ) = f · v , � K j Γ sd Ω j =1 � � ∀ w ∈ W, w div u = w q d . Ω Ω d Department of Mathematics, University of Pittsburgh 8

Global variational formulation, cont. 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 � ˜ b ( v , w ) = − w div v Ω Find ( u , p ) ∈ ˜ X × W such that � a ( u , v ) + ˜ ∀ v ∈ ˜ X, ˜ b ( v , p ) = f · v , Ω � ˜ ∀ w ∈ W, b ( u , w ) = − w q d . Ω Lemma: The variational formulation is equivalent to the PDE system. Department of Mathematics, University of Pittsburgh 9

Existence and uniqueness of a weak solution Lemma: ˜ b ( v , w ) ∀ w ∈ W, sup ≥ β � w � W � v � ˜ v ∈ ˜ X X Lemma: ∀ v ∈ ˜ X 0 = { v ∈ ˜ a ( v , v ) ≥ γ � v � 2 ˜ X : div = 0 } ˜ X Proof: Korn’s inequality: a ( v , v ) ≥ C ( | v s | 2 H 1 (Ω s ) + � v d � 2 ˜ L 2 (Ω d ) ) Poincare-type inequality: � 1 / 2 ∀ v ∈ ˜ | v s | 2 H 1 (Ω s ) + � v d � 2 � X 0 , � 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 10

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 | Ω s,i ∈ H 1 (Ω s,i ) n , v | Ω d,i ∈ H (div , Ω d,i ) + BCs , v · n | Γ ij ∈ H − 1 / 2 (Γ ij ) ∀ Γ ij ⊂ Γ dd ∪ Γ sd } Λ = { λ | Γ ij ∈ H − 1 / 2 (Γ ij ) n ∀ Γ ij ⊂ Γ ss , λ | Γ ij ∈ H 1 / 2 (Γ ij ) ∀ Γ ij ⊂ Γ dd ∪ Γ sd } . Department of Mathematics, University of Pittsburgh 11

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 , ˜ µ ) = [ v ] µ + [ v · n ] µ + [ v · n ] µ Γ ss Γ dd Γ sd Department of Mathematics, University of Pittsburgh 12

Domain decomposition variational formulation, cont. 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 ∀ ˜ µ ∈ Λ , b Γ ( u , ˜ µ ) = 0 . Lemma: The two variational formulations are equivalent. ˜ ˜ λ = − T · n on Γ ss , λ = p d on Γ sd ∪ Γ dd Department of Mathematics, University of Pittsburgh 13

Porous media scales Full fine scale grid resolution ⇒ large, highly coupled system of equations ⇒ solution is computationally intractable Department of Mathematics, University of Pittsburgh 14

Multiscale methods • Variational Multiscale Method – Galerkin FEM: Hughes et al; Brezzi – Mixed FEM: Arbogast et al • Multiscale Finite Elements – Galerkin FEM: Hou, Wu, Cai, Efendiev et al – Mixed FEM: Chen and Hou; Aarnes et al • Multiscale Mortar Methods: based on domain decomposition and mortar finite elements – Mixed FEM: Arbogast, Pencheva, Wheeler, Y. – DG-Mixed: Girault, Sun, Wheeler, Y. More flexible - easy to improve global accuracy by adapting the local mortar grids Allows for multiphysics subdomain models Department of Mathematics, University of Pittsburgh 15

Multiscale mortar approximation h � � � � � � � � � mortar dof � � � � � H � � � � � � � � • Each block is an element of the coarse grid. • Each block is discretized on the fine scale. • A coarse mortar space on each interface. • Result: Multiscale solution, fine scale on subdomains, coarse scale flux matching Department of Mathematics, University of Pittsburgh 16

✟✠ ✡☛ ✝✞ ☎✆ ✂✄ �✁ ☞✌ ✍✎ ✏✑ 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 17