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Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs inequality andpreconditioning estimates
Janos [email protected]
Department of Applied Analysis and Comp. Math. & Numnet Research GroupELTE University, Budapest, Hungary
3 March 2014
February 25, 2014
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Outline of the talk
Convection-diffusion equations and preconditioning
PreliminariesThe diffusion-dominated caseThe convection-dominated case
Streamline Poincare-Friedrichs inequality
Robust preconditioning estimates
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Convection-diffusion equations
A simple Dirichlet problem:−ε∆u + w · ∇u = g
u|∂Ω = 0.(1)
Assumptions:
(i) Ω ⊂ Rn is a polyhedral domain.
(ii) w ∈ C 1(Ω, Rn), divw = 0.
(iii) g ∈ L2(Ω).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Convection-diffusion equations
A simple Dirichlet problem:−ε∆u + w · ∇u = g
u|∂Ω = 0.(1)
Assumptions:
(i) Ω ⊂ Rn is a polyhedral domain.
(ii) w ∈ C 1(Ω, Rn), divw = 0.
(iii) g ∈ L2(Ω).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Remark. More general mixed BVPs:−ε∆u + w · ∇u + qu = g
u|ΓD= 0, ∂u
∂ν + βu|ΓN= 0.
(2)
Assumptions:
(i) Ω ⊂ Rn is a polyhedral domain; ∂Ω = ΓD ∪ ΓN .
(ii) w ∈ C 1(Ω, Rn), q ∈ L∞(Ω), β ∈ L∞(ΓN).
(iii) q − 12 divw ≥ 0 in Ω, w · ν ≥ 0 on ΓN .
(iv) g ∈ L2(Ω).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Remark. More general mixed BVPs:−ε∆u + w · ∇u + qu = g
u|ΓD= 0, ∂u
∂ν + βu|ΓN= 0.
(2)
Assumptions:
(i) Ω ⊂ Rn is a polyhedral domain; ∂Ω = ΓD ∪ ΓN .
(ii) w ∈ C 1(Ω, Rn), q ∈ L∞(Ω), β ∈ L∞(ΓN).
(iii) q − 12 divw ≥ 0 in Ω, w · ν ≥ 0 on ΓN .
(iv) g ∈ L2(Ω).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Convection-diffusion equations
The Dirichlet problem: weak solution.
Bilinear form on H10 (Ω)× H1
0 (Ω):
a(u, v) :=
∫Ω
(ε∇u · ∇v + (w · ∇u)v
).
Right-hand side functional: `v :=
∫Ω
gv .
Weak formulation:
a(u, v) = `v (∀v ∈ H10 (Ω)).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Convection-diffusion equations
The Dirichlet problem: weak solution.
Bilinear form on H10 (Ω)× H1
0 (Ω):
a(u, v) :=
∫Ω
(ε∇u · ∇v + (w · ∇u)v
).
Right-hand side functional: `v :=
∫Ω
gv .
Weak formulation:
a(u, v) = `v (∀v ∈ H10 (Ω)).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Convection-diffusion equations
The Dirichlet problem: weak solution.
Coercivity and boundedness of the bilinear form
⇒ solvability.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Finite element solution
Type of FEM: depending on the dominating term.
The diffusion-dominated case: ε = O(|w|) → standard FEM
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Finite element solution
Type of FEM: depending on the dominating term.
The diffusion-dominated case: ε = O(|w|) → standard FEM
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Finite element solution: diffusion-dominated case
Standard FEM
FEM subspace: Vh ⊂ H10 (Ω)
Find uh ∈ Vh:
a(uh, vh) = `vh (∀vh ∈ Vh).
Coefficients for uh =∑
ciϕi :
linear algebraic systemAhc = bh. (LAER)
Nonsymmetric but positive definite.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Finite element solution: diffusion-dominated case
Standard FEM
FEM subspace: Vh ⊂ H10 (Ω)
Find uh ∈ Vh:
a(uh, vh) = `vh (∀vh ∈ Vh).
Coefficients for uh =∑
ciϕi :
linear algebraic systemAhc = bh. (LAER)
Nonsymmetric but positive definite.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Preliminaries
Finite element solution: diffusion-dominated case
Standard FEM
FEM subspace: Vh ⊂ H10 (Ω)
Find uh ∈ Vh:
a(uh, vh) = `vh (∀vh ∈ Vh).
Coefficients for uh =∑
ciϕi :
linear algebraic systemAhc = bh. (LAER)
Nonsymmetric but positive definite.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - iteration and preconditioning.
Iteration: GCG-LS or CGN.
Convergence for a LAER Ax = b:
depends on the bounds
λ(A) := min〈Ax , x〉‖x‖2
,
Λ(A) := ‖A‖ = max〈Ax , y〉‖x‖‖y‖
.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - iteration and preconditioning.
Iteration: GCG-LS or CGN.
Convergence for a LAER Ax = b:
depends on the bounds
λ(A) := min〈Ax , x〉‖x‖2
,
Λ(A) := ‖A‖ = max〈Ax , y〉‖x‖‖y‖
.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - iteration and preconditioning.
Iteration: GCG-LS or CGN.
Convergence for a LAER Ax = b:
depends on the bounds
λ(A) := min〈Ax , x〉‖x‖2
,
Λ(A) := ‖A‖ = max〈Ax , y〉‖x‖‖y‖
.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - iteration.
Convergence: let λ = λ(A), Λ = Λ(A), then
GCG-LS:‖rk‖‖r0‖
≤(
1−(λ
Λ
)2)k/2
CGN:‖rk‖‖r0‖
≤ 2(Λ− λ
Λ + λ
)k⇒ convergence depends on k(A) :=
Λ
λ.
Problem: k(Ah) = O(h−2)→∞ as h→ 0.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - iteration.
Convergence: let λ = λ(A), Λ = Λ(A), then
GCG-LS:‖rk‖‖r0‖
≤(
1−(λ
Λ
)2)k/2
CGN:‖rk‖‖r0‖
≤ 2(Λ− λ
Λ + λ
)k⇒ convergence depends on k(A) :=
Λ
λ.
Problem: k(Ah) = O(h−2)→∞ as h→ 0.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - iteration.
Convergence: let λ = λ(A), Λ = Λ(A), then
GCG-LS:‖rk‖‖r0‖
≤(
1−(λ
Λ
)2)k/2
CGN:‖rk‖‖r0‖
≤ 2(Λ− λ
Λ + λ
)k⇒ convergence depends on k(A) :=
Λ
λ.
Problem: k(Ah) = O(h−2)→∞ as h→ 0.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - iteration.
Convergence: let λ = λ(A), Λ = Λ(A), then
GCG-LS:‖rk‖‖r0‖
≤(
1−(λ
Λ
)2)k/2
CGN:‖rk‖‖r0‖
≤ 2(Λ− λ
Λ + λ
)k⇒ convergence depends on k(A) :=
Λ
λ.
Problem: k(Ah) = O(h−2)→∞ as h→ 0.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - preconditioning.
Ax = b → S−1Ax = S−1b
S =?
Optimal case: O(N) operations for solving with S ∈ RN×N
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Iteration and preconditioning
Solution of the LAER - preconditioning.
Ax = b → S−1Ax = S−1b
S =?
Optimal case: O(N) operations for solving with S ∈ RN×N
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Return to the FEM solution of the convection-diffusion problem.
Linear algebraic system:
Ahc = bh, (LAER)
where(Ah)ij = a(ϕj , ϕi ).
Preconditioning: Sh =?
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Return to the FEM solution of the convection-diffusion problem.
Linear algebraic system:
Ahc = bh, (LAER)
where(Ah)ij = a(ϕj , ϕi ).
Preconditioning: Sh =?
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Optimal preconditioner:
stiffness matrix for a symmetric elliptic problem,
(Sh)ij = b(ϕj , ϕi ) = b(ϕi , ϕj)
→ optimal O(N) solvers are available (multigrid, multilevel)
(or quasi-optimal, O(N log N) like FFT)
E.g.: b(u, v) =
∫Ω∇u · ∇v
(=
∫Ω
(−∆u)v)
→ induces the H10 -norm: b(u, u) =
∫Ω |∇u|2 =: |u|21
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Optimal preconditioner:
stiffness matrix for a symmetric elliptic problem,
(Sh)ij = b(ϕj , ϕi ) = b(ϕi , ϕj)
→ optimal O(N) solvers are available (multigrid, multilevel)
(or quasi-optimal, O(N log N) like FFT)
E.g.: b(u, v) =
∫Ω∇u · ∇v
(=
∫Ω
(−∆u)v)
→ induces the H10 -norm: b(u, u) =
∫Ω |∇u|2 =: |u|21
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Conditioning properties.
Connection between a and b:
coercivity and boundedness of a w.r.t. the | . |1-norm ⇒
|a(u, v)| ≤ M√
b(u, u)b(v , v), a(u, u) ≥ m b(u, u).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Conditioning properties.
Connection between a and Ah, respectively b and Bh:
if uh =∑
ciϕi and vh =∑
diϕi , then
Ahc · d = a(uh, vh), Bhc · d = b(uh, vh).
In particular: |c|2Bh:= Bhc · c = b(uh, uh) = |uh|21.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Conditioning properties.
Connection between a and Ah, respectively b and Bh:
if uh =∑
ciϕi and vh =∑
diϕi , then
Ahc · d = a(uh, vh), Bhc · d = b(uh, vh).
In particular: |c|2Bh:= Bhc · c = b(uh, uh) = |uh|21.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Conditioning properties.
Connection between a and Ah, respectively b and Bh:
if uh =∑
ciϕi and vh =∑
diϕi , then
Ahc · d = a(uh, vh), Bhc · d = b(uh, vh).
In particular: |c|2Bh:= Bhc · c = b(uh, uh) = |uh|21.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Corollary:
using the coercivity and boundedness of a,
|Ahc · d| ≤ M |c|Bh|d|Bh
, Ahc · c ≥ m |c|2Bhon RN .
That is,
|〈B−1h Ahc, d〉Bh
| ≤ M |c|Bh|d|Bh
, 〈B−1h Ahc, c〉Bh
≥ m |c|2Bh
⇒ Λ(B−1h Ah) ≤ M, λ(B−1
h Ah) ≥ m (mesh independent)
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Corollary:
using the coercivity and boundedness of a,
|Ahc · d| ≤ M |c|Bh|d|Bh
, Ahc · c ≥ m |c|2Bhon RN .
That is,
|〈B−1h Ahc, d〉Bh
| ≤ M |c|Bh|d|Bh
, 〈B−1h Ahc, c〉Bh
≥ m |c|2Bh
⇒ Λ(B−1h Ah) ≤ M, λ(B−1
h Ah) ≥ m (mesh independent)
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Corollary:
using the coercivity and boundedness of a,
|Ahc · d| ≤ M |c|Bh|d|Bh
, Ahc · c ≥ m |c|2Bhon RN .
That is,
|〈B−1h Ahc, d〉Bh
| ≤ M |c|Bh|d|Bh
, 〈B−1h Ahc, c〉Bh
≥ m |c|2Bh
⇒ Λ(B−1h Ah) ≤ M, λ(B−1
h Ah) ≥ m (mesh independent)
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Corollary: convergence of the PGCG-LS or PCGN iteration:
‖rk‖‖r0‖
≤(
1−(m
M
)2)k/2
or‖rk‖‖r0‖
≤ 2(M −m
M + m
)k⇒ mesh independent
⇒ optimal O(N) overall algorithm
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Iterative solution
Operator preconditioning
Corollary: convergence of the PGCG-LS or PCGN iteration:
‖rk‖‖r0‖
≤(
1−(m
M
)2)k/2
or‖rk‖‖r0‖
≤ 2(M −m
M + m
)k⇒ mesh independent
⇒ optimal O(N) overall algorithm
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Finite element solution: convection-dominated case
The convection-dominated case: ε O(|w|)
→ standard FEM is problematic.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Finite element solution: convection-dominated case
The convection-dominated case: ε O(|w|)→ standard FEM is problematic.
1. Boundary layers:
2. Deteriorating lower bound:
a(u, u) ≥ ε|u|21 (i.e. m = ε ≈ 0 ).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Finite element solution: convection-dominated case
The convection-dominated case: ε O(|w|)→ standard FEM is problematic.
1. Boundary layers:
2. Deteriorating lower bound:
a(u, u) ≥ ε|u|21 (i.e. m = ε ≈ 0 ).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Finite element solution: convection-dominated case
A widespread improvement: streamline diffusion FEM (SDFEM)
Choose paramaters δk > 0 on elements Tk ∈ TReplace test functions: vh → vh + δk w · ∇vh on Tk
⇒ stabilized bilinear form
aSD(uh, vh) :=
∫Ω
(ε∇uh·∇vh+(w·∇uh)vh
)+
N∑k=1
δk
∫Tk
(w · ∇uh) (w · ∇vh)
on Vh × Vh.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Finite element solution: convection-dominated case
A widespread improvement: streamline diffusion FEM (SDFEM)
Choose paramaters δk > 0 on elements Tk ∈ TReplace test functions: vh → vh + δk w · ∇vh on Tk
⇒ stabilized bilinear form
aSD(uh, vh) :=
∫Ω
(ε∇uh·∇vh+(w·∇uh)vh
)+
N∑k=1
δk
∫Tk
(w · ∇uh) (w · ∇vh)
on Vh × Vh.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Finite element solution: convection-dominated case
A widespread improvement: streamline diffusion FEM (SDFEM)
Choose paramaters δk > 0 on elements Tk ∈ TReplace test functions: vh → vh + δk w · ∇vh on Tk
⇒ stabilized bilinear form
aSD(uh, vh) :=
∫Ω
(ε∇uh·∇vh+(w·∇uh)vh
)+
N∑k=1
δk
∫Tk
(w · ∇uh) (w · ∇vh)
on Vh × Vh.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Finite element solution: convection-dominated case
Stabilized problem: find uh ∈ Vh s.t.
aSD(uh, vh) = `SDvh (∀vh ∈ Vh).
Stabilized inner product: SD-inner product
〈uh, vh〉SD :=
∫Ωε∇uh · ∇vh +
N∑k=1
δk
∫Tk
(w · ∇uh) (w · ∇vh) .
⇒ stable lower bound:
aSD(uh, uh) ≥ ‖uh‖2SD (i.e. m = 1 )!
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Preconditioned CG iteration for the LAER:apply operator preconditioning!
Preconditioner = stiffness matrix for the SD-inner product:
(Sh)ij = 〈ϕi , ϕj〉SD
⇒ here 〈uh, vh〉SD =
∫Ω
(Lεuh) vh ,
where Lεu := −div(Aε∇u) with Aε = εI + δw ·wT
⇒ Sh corresponds to symmetric elliptic problems
⇒ optimal O(N) solvers available (multigrid, multilevel)
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Preconditioned CG iteration for the LAER:apply operator preconditioning!
Preconditioner = stiffness matrix for the SD-inner product:
(Sh)ij = 〈ϕi , ϕj〉SD
⇒ here 〈uh, vh〉SD =
∫Ω
(Lεuh) vh ,
where Lεu := −div(Aε∇u) with Aε = εI + δw ·wT
⇒ Sh corresponds to symmetric elliptic problems
⇒ optimal O(N) solvers available (multigrid, multilevel)
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Preconditioned CG iteration for the LAER:apply operator preconditioning!
Preconditioner = stiffness matrix for the SD-inner product:
(Sh)ij = 〈ϕi , ϕj〉SD
⇒ here 〈uh, vh〉SD =
∫Ω
(Lεuh) vh ,
where Lεu := −div(Aε∇u) with Aε = εI + δw ·wT
⇒ Sh corresponds to symmetric elliptic problems
⇒ optimal O(N) solvers available (multigrid, multilevel)
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Convergence estimate → we need bounds m and M.
Seen above: m = 1.
M =?
Upper bound needed:
|aSD(uh, vh)| ≤ M ‖uh‖SD‖vh‖SD (∀uh, vh ∈ Vh)
where
‖uh‖2SD =
∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Convergence estimate → we need bounds m and M.
Seen above: m = 1.
M =?
Upper bound needed:
|aSD(uh, vh)| ≤ M ‖uh‖SD‖vh‖SD (∀uh, vh ∈ Vh)
where
‖uh‖2SD =
∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Convergence estimate → we need bounds m and M.
Seen above: m = 1.
M =?
Upper bound needed:
|aSD(uh, vh)| ≤ M ‖uh‖SD‖vh‖SD (∀uh, vh ∈ Vh)
where
‖uh‖2SD =
∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Convergence estimate → we need bounds m and M.
Seen above: m = 1.
M =?
Upper bound needed:
|aSD(uh, vh)| ≤ M ‖uh‖SD‖vh‖SD (∀uh, vh ∈ Vh)
where
‖uh‖2SD =
∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Convection-dominated problems
Iteration and preconditioning
Convergence estimate → we need bounds m and M.
Seen above: m = 1.
M =?
Upper bound needed:
|aSD(uh, vh)| ≤ M ‖uh‖SD‖vh‖SD (∀uh, vh ∈ Vh)
where
‖uh‖2SD =
∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Estimation of the upper bound M.
aSD(uh, vh) :=
∫Ωε∇uh · ∇vh +
N∑k=1
δk
∫Tk
(w · ∇uh) (w · ∇vh)︸ ︷︷ ︸〈uh,vh〉SD
+
∫Ω
(w·∇uh)vh
⇒ |aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
Here
‖w · ∇uh‖2L2(Ω) ≤
1
δ0
N∑k=1
δk
∫Tk
|w · ∇uh|2 ≤1
δ0‖uh‖2
SD
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Estimation of the upper bound M.
aSD(uh, vh) :=
∫Ωε∇uh · ∇vh +
N∑k=1
δk
∫Tk
(w · ∇uh) (w · ∇vh)︸ ︷︷ ︸〈uh,vh〉SD
+
∫Ω
(w·∇uh)vh
⇒ |aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
Here
‖w · ∇uh‖2L2(Ω) ≤
1
δ0
N∑k=1
δk
∫Tk
|w · ∇uh|2 ≤1
δ0‖uh‖2
SD
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Estimation of the upper bound M.
aSD(uh, vh) :=
∫Ωε∇uh · ∇vh +
N∑k=1
δk
∫Tk
(w · ∇uh) (w · ∇vh)︸ ︷︷ ︸〈uh,vh〉SD
+
∫Ω
(w·∇uh)vh
⇒ |aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)︸ ︷︷ ︸↓
‖vh‖L2(Ω).
Here
‖w · ∇uh‖2L2(Ω) ≤
1
δ0
N∑k=1
δk
∫Tk
|w · ∇uh|2 ≤1
δ0‖uh‖2
SD
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Also needed: ‖vh‖L2(Ω) ≤ (?) · ‖vh‖SD . We follow [Kirby, 2010].
Squared:∫Ω|vh|2 ≤ (?)2 ·
(∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2).
Sufficient: ∫Ω|vh|2 ≤ (?)2 · ε
∫Ω|∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Also needed: ‖vh‖L2(Ω) ≤ (?) · ‖vh‖SD . We follow [Kirby, 2010].
Squared:∫Ω|vh|2 ≤ (?)2 ·
(∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2).
Sufficient: ∫Ω|vh|2 ≤ (?)2 · ε
∫Ω|∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Also needed: ‖vh‖L2(Ω) ≤ (?) · ‖vh‖SD . We follow [Kirby, 2010].
Squared:∫Ω|vh|2 ≤ (?)2 ·
(∫Ωε |∇uh|2 +
N∑k=1
δk
∫Tk
|w · ∇uh|2).
Sufficient: ∫Ω|vh|2 ≤ (?)2 · ε
∫Ω|∇uh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Also needed: ‖vh‖L2(Ω) ≤ (?) · ‖vh‖SD . We follow [Kirby, 2010].
Squared:
∫Ω|vh|2 ≤ (?)2 ·
∫
Ωε |∇vh|2 +
N∑k=1
δk
∫Tk
|w · ∇vh|2︸ ︷︷ ︸≥0
.
Sufficient: ∫Ω|vh|2 ≤ (?)2 · ε
∫Ω|∇vh|2 .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Usual Poincare-Friedrichs inequality:∫Ω|vh|2 ≤ C 2
Ω
∫Ω|∇vh|2 .
⇒ ∫Ω|vh|2 ≤
C 2Ω
ε· ε∫
Ω|∇vh|2 ≤
C 2Ω
ε‖vh‖2
SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Usual Poincare-Friedrichs inequality:∫Ω|vh|2 ≤ C 2
Ω
∫Ω|∇vh|2 .
⇒ ∫Ω|vh|2 ≤
C 2Ω
ε· ε∫
Ω|∇vh|2 ≤
C 2Ω
ε‖vh‖2
SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Return to the estimate
|aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
We have seen
‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .
From Poincare-Friedrichs:
‖vh‖L2(Ω) ≤CΩ√ε‖vh‖SD .
Altogether:
|aSD(uh, vh)| ≤(
1 +CΩ√δ0ε
)‖uh‖SD‖vh‖SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Return to the estimate
|aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
We have seen
‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .
From Poincare-Friedrichs:
‖vh‖L2(Ω) ≤CΩ√ε‖vh‖SD .
Altogether:
|aSD(uh, vh)| ≤(
1 +CΩ√δ0ε
)‖uh‖SD‖vh‖SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Return to the estimate
|aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
We have seen
‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .
From Poincare-Friedrichs:
‖vh‖L2(Ω) ≤CΩ√ε‖vh‖SD .
Altogether:
|aSD(uh, vh)| ≤(
1 +CΩ√δ0ε
)‖uh‖SD‖vh‖SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Upper bound M:
|aSD(uh, vh)| ≤(
1 +CΩ√δ0ε
)︸ ︷︷ ︸
M
‖uh‖SD‖vh‖SD .
Problem:
M =(
1 +CΩ√δ0ε
)≈ ∞ if ε ≈ 0.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Standard Poincare-Friedrichs
Estimations
Upper bound M:
|aSD(uh, vh)| ≤(
1 +CΩ√δ0ε
)︸ ︷︷ ︸
M
‖uh‖SD‖vh‖SD .
Problem:
M =(
1 +CΩ√δ0ε
)≈ ∞ if ε ≈ 0.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs
Estimations
Can it be improved?
New estimate: compare ‖uh‖L2(Ω) and ‖w · ∇uh‖L2(Ω)
→ streamline Poincare-Friedrichs inequality.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs
Estimations
Can it be improved?
New estimate: compare ‖uh‖L2(Ω) and ‖w · ∇uh‖L2(Ω)
→ streamline Poincare-Friedrichs inequality.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs
Streamline Poincare-Friedrichs inequality
Assumption on the vector field w: global rectifiability(sufficient but not necessary for the results)
Definition. The vector field w ∈ C 1(Ω,Rn) is globally rectifiable onΩ if there exists a diffeomorphism h : K → Ω on a compact set Ksuch that
h(s1, . . . , sn−1, t) := γs1,...,sn−1(t)((s1, . . . , sn−1, t) ∈ K
)where t 7→ γs1,...,sn−1(t) are the family of characteristic curvescovering Ω.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs
Streamline Poincare-Friedrichs inequality
Assumption on the vector field w: global rectifiability(sufficient but not necessary for the results)
Definition. The vector field w ∈ C 1(Ω,Rn) is globally rectifiable onΩ if there exists a diffeomorphism h : K → Ω on a compact set Ksuch that
h(s1, . . . , sn−1, t) := γs1,...,sn−1(t)((s1, . . . , sn−1, t) ∈ K
)where t 7→ γs1,...,sn−1(t) are the family of characteristic curvescovering Ω.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs
Streamline Poincare-Friedrichs inequality
Figure : A globally rectifiable vector field.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs
Streamline Poincare-Friedrichs inequality
Theorem. (Streamline Poincare-Friedrichs inequality). Letw ∈ C 1(Ω,Rn), for which w(x) 6= 0 (x ∈ Ω), be a globallyrectifiable vector field on Ω. Then there exists a constant Cw > 0(depending on w but independent of v) such that
‖v‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) (v ∈ H10 (Ω)).
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Streamline Poincare-Friedrichs
Streamline Poincare-Friedrichs inequality
Essence of proof:
Usual Poincare-Friedrichs:
‖v‖L2(Ω) ≤ c · ‖∂1v‖L2(Ω)︸ ︷︷ ︸Newton–Leibniz
≤ c · ‖∇v‖L2(Ω)
Streamline Poincare-Friedrichs:
‖v‖L2(Ω) ≤ Cw · ‖∂wv‖L2(Ω)︸ ︷︷ ︸local change of variables + streamline Newton–Leibniz
= Cw · ‖w · ∇v‖L2(Ω)
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Theoretical estimates
Robust estimations
Return to the estimate
|aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
We have seen
‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .
From streamline Poincare-Friedrichs:
‖vh‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) ≤Cw
δ0‖vh‖SD .
Altogether:
|aSD(uh, vh)| ≤(
1 +Cw
δ0
)‖uh‖SD‖vh‖SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Theoretical estimates
Robust estimations
Return to the estimate
|aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
We have seen
‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .
From streamline Poincare-Friedrichs:
‖vh‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) ≤Cw
δ0‖vh‖SD .
Altogether:
|aSD(uh, vh)| ≤(
1 +Cw
δ0
)‖uh‖SD‖vh‖SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Theoretical estimates
Robust estimations
Return to the estimate
|aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).
We have seen
‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .
From streamline Poincare-Friedrichs:
‖vh‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) ≤Cw
δ0‖vh‖SD .
Altogether:
|aSD(uh, vh)| ≤(
1 +Cw
δ0
)‖uh‖SD‖vh‖SD .
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Theoretical estimates
Robust estimations
That is: the upper bound of aSD satisfies
M ≤ 1 +Cw
δ0
independently of ε.
Consequence: the PCG iterations converge with rate independentlyof ε → robustness.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Theoretical estimates
Robust estimations
That is: the upper bound of aSD satisfies
M ≤ 1 +Cw
δ0
independently of ε.
Consequence: the PCG iterations converge with rate independentlyof ε → robustness.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Estimates and experiments
Numerical experiments
A simple Dirichlet problem:−ε∆u + w · ∇u = g
u|∂Ω = 0.
domain Ω := [0, 1]2 in R2.
w := (1, 0) a fixed constant vector
exact solution u(x , y) =(
x − ex/ε−1e1/ε−1
)4y(1− y)
→ boundary layer near the segment x = 1.
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Estimates and experiments
Numerical experiments
J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning
Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates
Estimates and experiments
Numerical experiments
Figure : result for ε = 10−10 – no unphysical oscillations.J. Karatson Department of Applied Analysis, ELTE
Streamline diffusion preconditioning