A Local Projection Stabilization method with shock capturing and - PowerPoint PPT Presentation

A Local Projection Stabilization method with shock capturing and diagonal mass-matrix for solving non-stationary transport dominated problems Friedhelm Schieweck 1 Piotr Skrzypacz 1 and 1 Institut für Analysis und Numerik Otto-von-Guericke-Universität Magdeburg ”Martin-60 workshop”, Dresden, 16-18th Dec. 2011 LPS-method with shock capturing and diagonal mass-matrix 1

Model Problem 1D model problem : Ω := (0 , 1) , 0 ≤ ε ≪ 1 u t − εu xx + b · u x + cu = f in Ω × (0 , T ) , b.c. u (0 , t ) = g 0 ( t ) , u (1 , t ) = g 1 ( t ) for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω LPS-method with shock capturing and diagonal mass-matrix 2

Model Problem 1D model problem : Ω := (0 , 1) , 0 ≤ ε ≪ 1 u t − εu xx + b · u x + cu = f in Ω × (0 , T ) , b.c. u (0 , t ) = g 0 ( t ) , u (1 , t ) = g 1 ( t ) for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω asymptotic solution in the case b = 1 , c = 0 , f = 0 idea is the transformation: w ( x, t ) := u ( x + bt, t ) then PDE + initial condition are equivalent to w t − εw xx = 0 in R × (0 , T ) , w ( x, 0) = u 0 ( x ) ˜ for x ∈ R ( ˜ u 0 = extension of u 0 by 0 ) � 1 � � − ( x − t − y ) 2 1 ⇒ u ε ( x, t ) = w ( x − t, t ) = √ ˜ u 0 ( y ) exp dy 4 εt 4 πεt 0 LPS-method with shock capturing and diagonal mass-matrix 2

Quality of asymptotic solution u ε problem : Ω := (0 , 1) , 0 ≤ ε ≪ 1 u t − εu xx + u x = 0 in Ω × (0 , T ) , b.c. u (0 , t ) = 0 , u (1 , t ) = 0 for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω LPS-method with shock capturing and diagonal mass-matrix 3

Quality of asymptotic solution u ε problem : Ω := (0 , 1) , 0 ≤ ε ≪ 1 u t − εu xx + u x = 0 in Ω × (0 , T ) , b.c. u (0 , t ) = 0 , u (1 , t ) = 0 for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω u ε satisfies pde + initial condition , but boundary conditions are not satisfied; on the boundary ∂ Ω it holds: | u ε ( x, t ) − 0 | ≤ Cε m ∀ m ∈ N , ∀ t ∈ ( t 1 , t 2 ) LPS-method with shock capturing and diagonal mass-matrix 3

Quality of asymptotic solution u ε problem : Ω := (0 , 1) , 0 ≤ ε ≪ 1 u t − εu xx + u x = 0 in Ω × (0 , T ) , b.c. u (0 , t ) = 0 , u (1 , t ) = 0 for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω u ε satisfies pde + initial condition , but boundary conditions are not satisfied; on the boundary ∂ Ω it holds: | u ε ( x, t ) − 0 | ≤ Cε m ∀ m ∈ N , ∀ t ∈ ( t 1 , t 2 ) from maximum principle we get: | u ε ( x, t ) − u ( x, t ) | ≤ Cε m ∀ m ∈ N , ∀ ( x, t ) ∈ Ω × ( t 1 , t 2 ) LPS-method with shock capturing and diagonal mass-matrix 3

Semi-discretization in space - part 1 1D model problem : Ω := (0 , 1) u t − εu xx + b · u x + cu = f in Ω × (0 , T ) , b.c. u (0 , t ) = g 0 ( t ) , u (1 , t ) = g 1 ( t ) for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω for 0 ≤ ε ≪ 1 LPS-method with shock capturing and diagonal mass-matrix 4

Semi-discretization in space - part 1 1D model problem : Ω := (0 , 1) u t − εu xx + b · u x + cu = f in Ω × (0 , T ) , b.c. u (0 , t ) = g 0 ( t ) , u (1 , t ) = g 1 ( t ) for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω for 0 ≤ ε ≪ 1 Galerkin part a G ( u, v ) := ε ( u x , v x ) + ( b · u x , v ) + ( cu, v ) LPS-method with shock capturing and diagonal mass-matrix 4

Semi-discretization in space - part 1 1D model problem : Ω := (0 , 1) u t − εu xx + b · u x + cu = f in Ω × (0 , T ) , b.c. u (0 , t ) = g 0 ( t ) , u (1 , t ) = g 1 ( t ) for t ∈ (0 , T ) i.c. u ( x, 0) = u 0 ( x ) for x ∈ Ω for 0 ≤ ε ≪ 1 Galerkin part a G ( u, v ) := ε ( u x , v x ) + ( b · u x , v ) + ( cu, v ) new semi-discretization: Find t �→ u h ( t ) ∈ V h such that u h (0) = π h u 0 , � � � � d t u h ( t ) , v h + a h ˜ u h ( t ) , u h ( t ) , v h = ( f, v h ) ∀ v h ∈ V h where a h (˜ u h , u, v ) := a G ( u, v ) + a LPS ( u, v ) + a SC (˜ u h , u, v ) LPS-method with shock capturing and diagonal mass-matrix 4

Semi-discretization in space - part 2 (time exact) ansatz: N h � u h ( t ) = U j ( t ) ϕ j , u h ( t ) ∈ V h , N h = dim ( V h ) � �� � j =1 ∈ R LPS-method with shock capturing and diagonal mass-matrix 5

Semi-discretization in space - part 2 (time exact) ansatz: N h � u h ( t ) = U j ( t ) ϕ j , u h ( t ) ∈ V h , N h = dim ( V h ) � �� � j =1 ∈ R Find t �→ u h ( t ) ∈ V h such that u h (0) = π h u 0 , ( d t u h ( t ) , v h ) + a h (˜ u h ( t ) , u h ( t ) , v h ) = ( f ( t ) , v h ) ∀ v h ∈ V h LPS-method with shock capturing and diagonal mass-matrix 5

Semi-discretization in space - part 2 (time exact) ansatz: N h � u h ( t ) = U j ( t ) ϕ j , u h ( t ) ∈ V h , N h = dim ( V h ) � �� � j =1 ∈ R Find t �→ u h ( t ) ∈ V h such that u h (0) = π h u 0 , ( d t u h ( t ) , v h ) + a h (˜ u h ( t ) , u h ( t ) , v h ) = ( f ( t ) , v h ) ∀ v h ∈ V h equivalent to nonlinear ODE-system (time exact): � � ∈ V := R N h such that Find t �→ U ( t ) := U j ( t ) � � U (0) = U 0 Md t U ( t ) = F t, U ( t ) , � � �� � � � � u h ( ˜ where F t, U ( t ) i = f ( t ) , ϕ i − a U ( t )) , u h ( U ( t )) , ϕ i LPS-method with shock capturing and diagonal mass-matrix 5

Time discretization by dG(1) I = ∪ N 0 = t 0 < t 1 < . . . < t N = T , n =1 I n , I n := ( t n − 1 , t n ] U ( t ) := ( U j ( t )) ∈ R N h =: V t n, 1 , t n, 2 ∈ ( t n − 1 , t n ] from 2-point Gauss-Radau formula U j n = u τ ( t n,j ) , j = 1 , 2 � I n ( t ) = U 1 n φ n, 1 ( t ) + U 2 dG(1)-method: u τ n φ n, 2 ( t ) � Find U 1 n , U 2 n ∈ V s.t. � 3 � � � � � 1 U 1 MU 0 4 M + τ n 2 A 1 4 M n + τ n 2 f ( t n, 1 ) n = − 9 5 U 2 − MU 0 4 M 4 M + τ n 2 A 2 n + τ n 2 f ( t n, 2 ) n matrix A j ∼ a h ( u h ( ˜ U ( t n,j )) , · , · ) = a h ( u h ( U ( t n − 1 )) , · , · ) LPS-method with shock capturing and diagonal mass-matrix 6

desirable FE-spaces wish for FE-space: ∃ L 2 -orthogonal local basis ⇒ mass-matrix M is diagonal !!! example: non-conforming Crouzeix-Raviart element ( P 1 / P 0 ) LPS-method with shock capturing and diagonal mass-matrix 7

desirable FE-spaces wish for FE-space: ∃ L 2 -orthogonal local basis ⇒ mass-matrix M is diagonal !!! example: non-conforming Crouzeix-Raviart element ( P 1 / P 0 ) ˆ idea: construct in 1D an L 2 -orthogonal basis on K = [ − 1 , 1] hierarchical basis of P 2 : ϕ 1 ( x ) = 1 ϕ 2 ( x ) = 1 ϕ 3 ( x ) = 1 − x 2 ˜ 2 (1 − x ) , ˜ 2 (1 + x ) , new enriched bubble-function of P 5 : (theory see [Ma/Sk/To ’07]) ϕ 4 ( x ) = (1 − x 2 ) x (1 + cx 2 ) , c ∈ R suitably chosen !!! LPS-method with shock capturing and diagonal mass-matrix 7

desirable FE-spaces wish for FE-space: ∃ L 2 -orthogonal local basis ⇒ mass-matrix M is diagonal !!! example: non-conforming Crouzeix-Raviart element ( P 1 / P 0 ) ˆ idea: construct in 1D an L 2 -orthogonal basis on K = [ − 1 , 1] hierarchical basis of P 2 : ϕ 1 ( x ) = 1 ϕ 2 ( x ) = 1 ϕ 3 ( x ) = 1 − x 2 ˜ 2 (1 − x ) , ˜ 2 (1 + x ) , new enriched bubble-function of P 5 : (theory see [Ma/Sk/To ’07]) ϕ 4 ( x ) = (1 − x 2 ) x (1 + cx 2 ) , c ∈ R suitably chosen !!! ansatz: ϕ i ( x ) = ˜ ϕ i ( x ) + a i ϕ 3 ( x ) + b i ϕ 4 ( x ) i = 1 , 2 determine c, a i , b i such that: ( ϕ j , ϕ i ) = 0 ∀ i � = j LPS-method with shock capturing and diagonal mass-matrix 7

basis functions of the new Q + 2 element in 1D + L2−orthogonal basis for LPS−Q 2 1 φ 1 φ 3 φ 2 φ 1 0.8 φ 3 φ 4 0.6 0.4 φ 4 0.2 φ 2 0 −0.2 −0.4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x LPS-method with shock capturing and diagonal mass-matrix 8

tensor product basis of the new Q + 2 element in 2D φ 3 φ 7 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 1 1 0.5 1 0.5 1 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 y y x x φ 8 φ 16 0.3 0.1 0.2 0.05 0.1 0 0 −0.1 −0.2 −0.05 −0.3 −0.4 −0.1 1 1 0.5 1 0.5 1 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 y y x x LPS-method with shock capturing and diagonal mass-matrix 9

LPS = Local Projection Stabilization Galerkin bilinear form: a G ( u, v ) := ε ( ∇ u, ∇ v ) + (( b · ∇ ) u, v ) + ( cu, v ) LPS stabilization: � � � a LPS ( u, v ) := c 0 h K ∇ u − π h ( ∇ u ) , ∇ v − π h ( ∇ v ) K K π h : L 2 (Ω) → D h := P dc where (local projection space) 1 LPS-method with shock capturing and diagonal mass-matrix 10

LPS = Local Projection Stabilization Galerkin bilinear form: a G ( u, v ) := ε ( ∇ u, ∇ v ) + (( b · ∇ ) u, v ) + ( cu, v ) LPS stabilization: � � � a LPS ( u, v ) := c 0 h K ∇ u − π h ( ∇ u ) , ∇ v − π h ( ∇ v ) K K π h : L 2 (Ω) → D h := P dc where (local projection space) 1 sufficient for theory: local inf-sup-condition ( w L , v h ) K 0 < c 2 ≤ inf sup ∀ K. � w L � 0 ,K � v h � 0 ,K w L ∈ D h v h ∈ B h ( V h ) . . . is fulfilled for our new Q + 2 -element LPS-method with shock capturing and diagonal mass-matrix 10