Multiscale mortar mixed finite element methods for flow in porous - PowerPoint PPT Presentation

Multiscale mortar mixed finite element methods for flow in porous media Ivan Yotov Department of Mathematics, University of Pittsburgh High Demensional Approximation Seminar University of South Carolina, March 17, 2008 Joint work with Todd Arbogast , Gergina Pencheva , and Mary F. Wheeler , The University of Texas at Austin Department of Mathematics, University of Pittsburgh 1

Outline • Motivation • A multiscale mortar mixed finite element method • A priori error estimates • A domain decomposition algorithm • A posteriori error estimates • Mortar and subdomain adaptivity • A relationship between multiscale mortar MFE methods and subgrid upscaling methods • Extension to two-phase flow • Extension to stochastic PDEs Department of Mathematics, University of Pittsburgh 2

Reservoir rock Department of Mathematics, University of Pittsburgh 3

Motivation: flow in heterogeneous porous media Heterogeneous permeability varies on a fine scale. Full fine scale grid resolution ⇒ large, highly coupled system of equations ⇒ solution is computationally intractable • Variational Multiscale Method – Hughes et al; Brezzi – Mixed FEM: Arbogast et al • Multiscale Finite Elements – Hou, Wu, Cai, Efendiev et al – Mixed FEM: Chen and Hou; Aarnes et al New approach: based on domain decomposition and mortar finite elements More flexible - easy to improve global accuracy by refining the local mortar grid where needed Department of Mathematics, University of Pittsburgh 4

Multiscale finite element/subgrid upscaling methods L ǫ u = f ⇒ u ∈ V : a ( u, v ) = ( f, v ) ∀ v ∈ V. Multiscale approximation: H - coarse grid, h ≈ ǫ - fine grid (subgrid) h � � � V H,h = V H + V ′ h � � � � � h ( E ) : φ E Basis for V ′ h,i , i = 1 , . . . , N E , � � � � � H a E ( φ E H,i + φ E h,i , v h ) = 0 ∀ v h ∈ V h ( E ) � � � � � � � � Multiscale solution: u H,h ∈ V H,h , a ( u H,h , v H,h ) = ( f, v H,h ) ∀ v H,h ∈ V H,h Department of Mathematics, University of Pittsburgh 5

Multiblock formulation for single phase flow ¯ i =1 ¯ Ω = ∪ n Ω i ; Γ ij = ∂ Ω i ∩ ∂ Ω j On each block Ω i : u = − K ∇ p in Ω i ∇ · u = q in Ω i u · ν = 0 on ∂ Ω i ∩ ∂ Ω On each interface Γ ij : p i = p j on Γ ij [ u · ν ] ij = 0 on Γ ij where p i = p | ∂ Ω i [ u · ν ] ij ≡ u | Ω i · ν − u | Ω j · ν Department of Mathematics, University of Pittsburgh 6

Multiblock discretization spaces n n � � � V h = V h,i , W h = W h,i , M h = M h,ij i =1 i =1 0 ≤ i<j ≤ n � λ h | Γ ij ∈ M h,ij , [ u h · ν ] ij µ = 0 , µ ∈ M h,ij . Γ ij Subdomain grids do not need to match. Department of Mathematics, University of Pittsburgh 7

The mortar mixed finite element method Find u h ∈ V h , p h ∈ W h , λ h ∈ M h s.t. for 1 ≤ i ≤ n ( K − 1 u h , v ) Ω i − ( p h , ∇ · v ) Ω i + � λ h , v · ν i � Γ i = 0 , v ∈ V h,i , ( ∇ · u h , w ) Ω i = ( q, w ) Ω i , w ∈ W h,i , n � � u h · ν i , µ � Γ i = 0 , µ ∈ M h . i =1 Stability, optimal convergence, superconvergence: Y.(1996,97), Arbogast, Cowsar, Wheeler, Y. (2000) Department of Mathematics, University of Pittsburgh 8

Two-scale formulation: mortar upscaling Two-scale problem: 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: Effective solution, fine scale on subdomains, coarse scale flux matching Department of Mathematics, University of Pittsburgh 9

Multiscale mortar mixed finite element method Allow for different scales and polynomial approximations on interfaces and subdo- mains. Assume P k ⊂ V h,i , P l ⊂ W h,i , P m ⊂ M H , m ≥ k + 1 Find u h ∈ V h , p h ∈ W h , λ H ∈ M H s.t. for 1 ≤ i ≤ n ( K − 1 u h , v ) Ω i − ( p h , ∇ · v ) Ω i + � λ H , v · ν i � Γ i = 0 , v ∈ V h,i , ( ∇ · u h , w ) Ω i = ( q, w ) Ω i , w ∈ W h,i , n � � u h · ν i , µ � Γ i = 0 , µ ∈ M H . i =1 Stability assumption: � µ � 0 , Γ i,j ≤ C ( �Q h,i µ � 0 , Γ i,j + �Q h,j µ � 0 , Γ i,j ) , µ ∈ M H , 1 ≤ i < j ≤ n. Department of Mathematics, University of Pittsburgh 10

An approximation result Weakly continuous velocities: � n � � V h, 0 = v ∈ V h : � v | Ω i · ν i , µ � Γ i = 0 ∀ µ ∈ M H . i =1 Equivalent formulation: find u h ∈ V h, 0 and p h ∈ W h such that ( K − 1 u h , v ) − ( p h , ∇ · v ) = 0 , v ∈ V h, 0 , ( ∇ · u h , w ) = ( q, w ) , w ∈ W h Interpolation operator Π 0 : V → V h, 0 such that ( ∇ · (Π 0 q − q ) , w ) Ω = 0 , w ∈ W h . n � q � r, Ω i h r + � q � r +1 / 2 , Ω i h r H 1 / 2 � � � � Π 0 q − q � 0 ≤ C , 1 ≤ r ≤ k + 1 i =1 Department of Mathematics, University of Pittsburgh 11

A priori error estimates Theorem: � u − u h � ≤ C ( H m +1 / 2 + h k +1 ) , �∇ · ( u − u h ) � ≤ Ch l +1 ||| u − u h ||| ≤ C ( H m +1 / 2 + h k +1 H 1 / 2 ) � p − p h � ≤ C ( H m +3 / 2 + h k +1 H + h l +1 ) ||| p − p h ||| ≤ CH � u − u h � H ( div ) Balance � u − u h � or ||| p − p h ||| error terms (with l = k ): k +1 k +1 k +1+ m +1 / 2 ⇒ � u − u h � ≤ Ch k +1 , H = h ||| p − p h ||| ≤ Ch m +1 / 2 For example, RT 0 , k = 0 , and quadratic mortars, m = 2 , H = h 2 / 5 : � u − u h � ≤ Ch, ||| p − p h ||| ≤ Ch 1+2 / 5 � p − p h � ≤ Ch, Department of Mathematics, University of Pittsburgh 12

Parallel domain decomposition Two types of subdomain problems: BC BC n BC q � g H ( µ ) = � ¯ u h,i · ν i , µ � Γ i 1 q2 0 BC i =1 BC BC 0 0 a H,i ( λ, µ ) = −� u ∗ h,i ( λ ) · ν i , µ � Γ i 0 0 λ 0 0 n � a H ( λ, µ ) = a H,i ( λ, µ ) 0 0 i =1 The solution ( u h , p h , λ H ) to the original problem satisfies A H λ H = g H or a H ( λ H , µ ) = g ( µ ) , ∀ µ ∈ M H , with u h = u ∗ p h = p ∗ h ( λ H ) + ¯ u h , h ( λ H ) + ¯ p h . Department of Mathematics, University of Pittsburgh 13

Interface iteration Lemma The interface operator A H : M H → M H is symmetric and positive semi- definite. a H,i ( λ, µ ) = ( K − 1 u ∗ h,i ( λ ) , u ∗ h,i ( µ )) . A H : λ H → [ u ∗ h ( λ H ) · ν ] is a Steklov-Poincare operator. Apply the Conjugate Gradient method for A H λ H = g H . Computing the action of the operator (needed at each CG iteration): • Given mortar data λ H ∈ M H , project onto subdomain grids: λ H → Q h i λ H • Solve local problems in parallel with boundary data Q h i λ H • Project fluxes onto the mortar space and compute the jump: u h,i · ν i → Q T A H λ H = Q T h 1 u h, 1 · ν 1 + Q T h i u h,i · ν i , h 2 u h, 2 · ν 2 Department of Mathematics, University of Pittsburgh 14

Numerical experiments m H || p − p h || || u − u h || ||| p − p h ||| ||| u − u h ||| ||| p − λ H ||| full K diag K h 1 / 2 2 1 1 1.5 1.25 1.25 1.5 1 2 h 1 1 2 1.5 1.5 2 Table 1: Theoretical convergence rates for quadratic and linear mortars. Example 1: ( x + 1) 2 + y 2 � � sin ( xy ) p ( x, y ) = x 3 y 4 + x 2 + sin ( xy ) cos ( y ) , K = . ( x + 1) 2 sin ( xy ) Example 2: � x 2 y 3 + cos ( xy ) � I, 0 ≤ x ≤ 1 / 2 , � , p ( x, y ) = K = � 2 y 3 + cos � 2 x +9 � 2 x +9 10 ∗ I, 1 / 2 ≤ x ≤ 1 20 y 20 Department of Mathematics, University of Pittsburgh 15

Computed solution for Example 1 A. Discontinuous quadratic mortars B. Discontinuous linear mortars Computed pressure (shade) and velocity (arrows). Department of Mathematics, University of Pittsburgh 16

Convergence rates for Example 1 1 /h || p − p h || || u − u h || ||| p − p h ||| ||| u − u h ||| ||| p − λ H ||| iter. 4 8 2.64E-1 2.03E-1 4.62E-2 2.13E-2 4.45E-2 16 13 6.37E-2 4.86E-2 2.83E-3 1.81E-3 2.72E-3 64 15 1.59E-2 1.21E-2 1.75E-4 1.60E-4 1.69E-4 256 16 3.98E-3 3.03E-3 1.09E-5 1.77E-5 1.08E-5 O ( h 1 . 01 ) O ( h 1 . 01 ) O ( h 2 . 01 ) O ( h 1 . 71 ) O ( h 2 . 00 ) rate Continuous quadratic mortars on non-matching grids 1 /h || p − p h || || u − u h || ||| p − p h ||| ||| u − u h ||| ||| p − λ H ||| iter. 4 4 2.63E-1 2.04E-1 4.54E-2 2.35E-2 4.55E-2 8 7 1.28E-1 9.82E-2 1.14E-2 7.32E-3 1.13E-2 16 13 6.37E-2 4.86E-2 2.82E-3 2.23E-3 2.83E-3 32 18 3.18E-2 2.43E-2 7.01E-4 6.95E-4 7.05E-4 64 23 1.59E-2 1.21E-2 1.75E-4 2.24E-4 1.76E-4 128 23 7.95E-3 6.06E-3 4.36E-5 7.47E-5 4.40E-5 256 24 3.98E-3 3.03E-3 1.09E-5 2.54E-5 1.09E-5 O ( h 1 . 01 ) O ( h 1 . 01 ) O ( h 2 . 00 ) O ( h 1 . 65 ) O ( h 2 . 00 ) rate Continuous linear mortars on non-matching grids Department of Mathematics, University of Pittsburgh 17

Error in the computed solution for Example 1 A. Discontinuous quadratic mortars B. Discontinuous linear mortars Error in computed pressure (shade) and velocity (arrows). Department of Mathematics, University of Pittsburgh 18