Nonstandard a posteriori error bounds in Euler-Galerkin
schemes for parabolic problems
with elliptic reconstruction techniques
Omar Lakkis
Mathematics – University of Sussex – Brighton, England UK
15 January 2009
New Directions in Computational Partial Differential Equations
Warwick Mathematical Institute
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Nonstandard a posteriori parabolic estimates
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1
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2
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Outline
Motivation
Introduction
A posteriori analysis
Before reconstruction
Elliptic reconstruction technique
A world without reconstruction
is possible, but. . .
A user’s guide to the elliptic
reconstruction
Remarks
Reconstruction vs. direct
approach
Applications
Spatial L2 estimates via energy
Energy estimates
O Lakkis (Sussex)
4
Higher order estimates
Duality
Duality estimates
Pointwise estimates
Pointwise estimates
Semilinear problems
Allen–Cahn equation
Energy only estimates with
reconstruction
Gradient recovery
Recovery energy estimates
Recovery energy estimates
Recovery energy estimates
Nonconforming methods (dG)
Nonconforming methods (DG)
Closing remarks
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Aim
Derive a posteriori error estimates for
Problem (general linear diffusion)
Find u : Ω × [0, T ] → R satisfying
(linear parabolic diffusion PDE) ∂t u + A u = f in Ω × (0, T ] ⊂ Rd × R,
(with initial Cauchy condition)
u(·, 0) = u0 ,
(and Dirichlet boundary value)
u|∂Ω = 0 on (0, T ].
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
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Aim
Derive a posteriori error estimates for
Problem (general linear diffusion)
Find u : Ω × [0, T ] → R satisfying
(linear parabolic diffusion PDE) ∂t u + A u = f in Ω × (0, T ] ⊂ Rd × R,
(with initial Cauchy condition)
u(·, 0) = u0 ,
(and Dirichlet boundary value)
u|∂Ω = 0 on (0, T ].
with elliptic operator A (t) : H10 (Ω) → H−1 (Ω)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
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Aim
Derive a posteriori error estimates for
Problem (general linear diffusion)
Find u : Ω × [0, T ] → R satisfying
(linear parabolic diffusion PDE) ∂t u + A u = f in Ω × (0, T ] ⊂ Rd × R,
(with initial Cauchy condition)
u(·, 0) = u0 ,
(and Dirichlet boundary value)
u|∂Ω = 0 on (0, T ].
with elliptic operator A (t) : H10 (Ω) → H−1 (Ω) satisfying
Z
hA (t)v | φi := a (v, φ) :=
∇φ(x)| a(x, t)∇v(x) ∀ φ ∈ H10 (Ω),
Ω
α[ (t) |w| ≤ w a(t)w ≤ α] (t) |w|2 ∀ w ∈ Rd
2
|
(e.g., A (t) = −∆
O Lakkis (Sussex)
and 0-Dirichlet BC) .
Nonstandard a posteriori parabolic estimates
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Discrete Model Problem
weak form and spatial semidiscretization
Problem (weak formulation)
Find u : [0, T ] → H10 (Ω) such that
h∂t u, φi + a (u, φ) = hf, φi ∀ φ ∈ H10 (Ω) and u(0) = u0 ∈ L2 (Ω).
O Lakkis (Sussex)
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Discrete Model Problem
weak form and spatial semidiscretization
Problem (weak formulation)
Find u : [0, T ] → H10 (Ω) such that
h∂t u, φi + a (u, φ) = hf, φi ∀ φ ∈ H10 (Ω) and u(0) = u0 ∈ L2 (Ω).
Problem (spatially discrete problem)
Find U : [0, T ] → Vh such that
h∂t U, Φi + B (U, Φ) = hf, Φi ∀ Φ ∈ Vh and U (0) = Πh u0
Πh =L2 -projection
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Discrete Model Problem
weak form and spatial semidiscretization
Problem (weak formulation)
Find u : [0, T ] → H10 (Ω) such that
h∂t u, φi + a (u, φ) = hf, φi ∀ φ ∈ H10 (Ω) and u(0) = u0 ∈ L2 (Ω).
Problem (spatially discrete problem)
Find U : [0, T ] → Vh such that
h∂t U, Φi + B (U, Φ) = hf, Φi ∀ Φ ∈ Vh and U (0) = Πh u0
Πh =L2 -projection
Remarks
conforming method ⇒
O Lakkis (Sussex)
Vh ⊆ H10 (Ω),
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Discrete Model Problem
weak form and spatial semidiscretization
Problem (weak formulation)
Find u : [0, T ] → H10 (Ω) such that
h∂t u, φi + a (u, φ) = hf, φi ∀ φ ∈ H10 (Ω) and u(0) = u0 ∈ L2 (Ω).
Problem (spatially discrete problem)
Find U : [0, T ] → Vh such that
h∂t U, Φi + B (U, Φ) = hf, Φi ∀ Φ ∈ Vh and U (0) = Πh u0
Πh =L2 -projection
Remarks
conforming method ⇒
Vh ⊆ H10 (Ω),
consistent method ⇒ B (V, Φ) = a (V, Φ) ,
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Discrete Model Problem
full discretization
Problem (spatially discrete problem)
Find U : [0, T ] → Vh such that
h∂t U, Φi + B (U, Φ) = hf, Φi ∀ Φ ∈ Vh and U (0) = Πh u0
Πh =L2 -projection
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
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Discrete Model Problem
full discretization
Problem (spatially discrete problem)
Find U : [0, T ] → Vh such that
h∂t U, Φi + B (U, Φ) = hf, Φi ∀ Φ ∈ Vh and U (0) = Πh u0
Πh =L2 -projection
Problem (Fully discrete implicit Euler-Galërkin FEM)
(Vn )n=0,...,N a sequence of FE spaces, find (U n )n=0,...,N such that
U 0 = Π0 u0 and ∀ n ∈ [1 : N ] :
U n − U n−1
, Φ + B (U n , Φ) = hf n , Φi , ∀ Φ ∈ Vn .
τn
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Finite element spaces V0 , V1 , . . . , VN
. . . and their interaction
O Lakkis (Sussex)
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Finite element spaces V0 , V1 , . . . , VN
. . . and their interaction
for each n, Tn be a partition of (aka mesh on) Ω, into elements
(simplexes/quadrilaterals/. . . ),
O Lakkis (Sussex)
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Finite element spaces V0 , V1 , . . . , VN
. . . and their interaction
for each n, Tn be a partition of (aka mesh on) Ω, into elements
(simplexes/quadrilaterals/. . . ),
denote by hn the meshsize function of Tn ,
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
6 / 53
Finite element spaces V0 , V1 , . . . , VN
. . . and their interaction
for each n, Tn be a partition of (aka mesh on) Ω, into elements
(simplexes/quadrilaterals/. . . ),
denote by hn the meshsize function of Tn ,
denote by Σn the union of sides (edges/faces) of Tn ,
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
6 / 53
Finite element spaces V0 , V1 , . . . , VN
. . . and their interaction
for each n, Tn be a partition of (aka mesh on) Ω, into elements
(simplexes/quadrilaterals/. . . ),
denote by hn the meshsize function of Tn ,
denote by Σn the union of sides (edges/faces) of Tn ,
two successive meshes Tn−1 and Tn are compatible, i.e., one is local
refinement of other:
∀ K ∈ Tn , K 0 ∈ Tn−1 : K ∩ K 0 = ∅ or K ⊆ K 0 or K 0 ⊆ K,
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
6 / 53
Finite element spaces V0 , V1 , . . . , VN
. . . and their interaction
for each n, Tn be a partition of (aka mesh on) Ω, into elements
(simplexes/quadrilaterals/. . . ),
denote by hn the meshsize function of Tn ,
denote by Σn the union of sides (edges/faces) of Tn ,
two successive meshes Tn−1 and Tn are compatible, i.e., one is local
refinement of other:
∀ K ∈ Tn , K 0 ∈ Tn−1 : K ∩ K 0 = ∅ or K ⊆ K 0 or K 0 ⊆ K,
many constants depend on shape-regularity
µ(Tn ) := inf
sup
K∈Tn Bρ (x)∈K
O Lakkis (Sussex)
ρ/ diam K,
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Finite element spaces V0 , V1 , . . . , VN
. . . and their interaction
for each n, Tn be a partition of (aka mesh on) Ω, into elements
(simplexes/quadrilaterals/. . . ),
denote by hn the meshsize function of Tn ,
denote by Σn the union of sides (edges/faces) of Tn ,
two successive meshes Tn−1 and Tn are compatible, i.e., one is local
refinement of other:
∀ K ∈ Tn , K 0 ∈ Tn−1 : K ∩ K 0 = ∅ or K ⊆ K 0 or K 0 ⊆ K,
many constants depend on shape-regularity
µ(Tn ) := inf
sup
K∈Tn Bρ (x)∈K
ρ/ diam K,
example of (conforming) finite elment space
p∈N
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and
Vn := {Φ ∈ C(Ω) :
Φ|K poly of deg p} .
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a posteriori estimates in general
first used in linear algebra 1960’s
Exact problem
Given f find u ∈ V (dim V = ∞) such that λ[u] = f .
O Lakkis (Sussex)
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a posteriori estimates in general
first used in linear algebra 1960’s
Exact problem
Given f find u ∈ V (dim V = ∞) such that λ[u] = f .
Approximate problem
Let F ≈ f, Λ ≈ λ find U ∈ V (dim V < ∞) s.t. Λ[U ] = F .
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
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a posteriori estimates in general
first used in linear algebra 1960’s
Exact problem
Given f find u ∈ V (dim V = ∞) such that λ[u] = f .
Approximate problem
Let F ≈ f, Λ ≈ λ find U ∈ V (dim V < ∞) s.t. Λ[U ] = F .
“Model” theorem
Error bound: There exists a computable estimator functional E such
that
(upper bound)
(optimal order)
kU − uk ≤E [U ; f, F ; λ, Λ]
and E [U ; f, F, λ, Λ] = O(kU − uk).
Main point: “estimator” E is independent of exact solution u.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
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Direct (violent) FEM a posteriori analysis
Error-Residual PDE
1
Subtract exact
h∂t u, φi + a (u, φ) = hf, φi ∀ φ ∈ H10 (Ω)
from residual, i.e., apply exact weak PDO on discrete solution,
h∂t U, φi + a (U, φ) .
O Lakkis (Sussex)
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Direct (violent) FEM a posteriori analysis
Error-Residual PDE
1
Subtract exact
h∂t u, φi + a (u, φ) = hf, φi ∀ φ ∈ H10 (Ω)
from residual, i.e., apply exact weak PDO on discrete solution,
h∂t U, φi + a (U, φ) .
2
Obtain error (e = U − u) relation
h∂t e, φi + a (e, φ) = h∂t U − f, φi + a (U, φ) =: hr | φi .
| {z }
|
{z
}
(weak) PDO on error
residual
for all φ ∈ H10 (Ω).
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Direct (violent) FEM a posteriori analysis
Error-Residual PDE
1
Subtract exact
h∂t u, φi + a (u, φ) = hf, φi ∀ φ ∈ H10 (Ω)
from residual, i.e., apply exact weak PDO on discrete solution,
h∂t U, φi + a (U, φ) .
2
Obtain error (e = U − u) relation
h∂t e, φi + a (e, φ) = h∂t U − f, φi + a (U, φ) =: hr | φi .
| {z }
|
{z
}
residual
(weak) PDO on error
3
for all φ ∈ H10 (Ω).
Briefly, error-residual PDE (generalized sense)
∂t e + A e = r.
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Direct (violent) FEM a posteriori analysis (continued)
Galërkin orthogonality of residual
Remark (Galërkin orthogonality of residual)
Key property in analysis:
hr | Φi = h∂t U − f, Φi + a (U, Φ) = 0
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Nonstandard a posteriori parabolic estimates
∀ Φ ∈ Vh .
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Direct (violent) FEM a posteriori analysis (continued)
Galërkin orthogonality of residual
Remark (Galërkin orthogonality of residual)
Key property in analysis:
hr | Φi = h∂t U − f, Φi + a (U, Φ) = 0
∀ Φ ∈ Vh .
Combined with error-residual PDE (∂t e + A e = r)
h∂t e, φi + a (e, φ) = hr | φ − Πh φi
O Lakkis (Sussex)
∀ φ ∈ H10 (Ω),
Nonstandard a posteriori parabolic estimates
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Direct (violent) FEM a posteriori analysis (continued)
Galërkin orthogonality of residual
Remark (Galërkin orthogonality of residual)
Key property in analysis:
hr | Φi = h∂t U − f, Φi + a (U, Φ) = 0
∀ Φ ∈ Vh .
Combined with error-residual PDE (∂t e + A e = r)
h∂t e, φi + a (e, φ) = hr | φ − Πh φi
∀ φ ∈ H10 (Ω),
Πh : H10 (Ω) → Vh Clément-type interpolant:
k(φ − Πh φ)/hk ≤ C1 |φ|a
O Lakkis (Sussex)
and
√
k(φ − Πh φ)/ hkΣ ≤ C2 |φ|a ,
Nonstandard a posteriori parabolic estimates
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Direct (violent) FEM a posteriori analysis (continued)
Galërkin orthogonality of residual
Remark (Galërkin orthogonality of residual)
Key property in analysis:
hr | Φi = h∂t U − f, Φi + a (U, Φ) = 0
∀ Φ ∈ Vh .
Combined with error-residual PDE (∂t e + A e = r)
h∂t e, φi + a (e, φ) = hr | φ − Πh φi
∀ φ ∈ H10 (Ω),
Πh : H10 (Ω) → Vh Clément-type interpolant:
k(φ − Πh φ)/hk ≤ C1 |φ|a
and
√
k(φ − Πh φ)/ hkΣ ≤ C2 |φ|a ,
Σ geometric mesh (i.e., union of sides)
O Lakkis (Sussex)
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Direct (violent) FEM a posteriori analysis (continued)
Galërkin orthogonality of residual
Remark (Galërkin orthogonality of residual)
Key property in analysis:
hr | Φi = h∂t U − f, Φi + a (U, Φ) = 0
∀ Φ ∈ Vh .
Combined with error-residual PDE (∂t e + A e = r)
h∂t e, φi + a (e, φ) = hr | φ − Πh φi
∀ φ ∈ H10 (Ω),
Πh : H10 (Ω) → Vh Clément-type interpolant:
k(φ − Πh φ)/hk ≤ C1 |φ|a
and
√
k(φ − Πh φ)/ hkΣ ≤ C2 |φ|a ,
Σ geometric mesh (i.e., union of sides)
|φ|a = k∇φk.
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Test with φ = e relation
h∂t e, φi + a (e, φ) = hr | φ − Πh φi , ∀ φ ∈ H10 (Ω).
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Test with φ = e relation
h∂t e, φi + a (e, φ) = hr | φ − Πh φi , ∀ φ ∈ H10 (Ω).
Obtain
1
dt kek2 + |e|2a = hr | e − Πh ei
2
= hR, e − Πh ei + hJ, e − Πh eiΣ ,
√
√
= hRh, (e − Πh e)/hi + hJ h, (e − Πh e)/ hiΣ ,
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Test with φ = e relation
h∂t e, φi + a (e, φ) = hr | φ − Πh φi , ∀ φ ∈ H10 (Ω).
Obtain
1
dt kek2 + |e|2a = hr | e − Πh ei
2
= hR, e − Πh ei + hJ, e − Πh eiΣ ,
√
√
= hRh, (e − Πh e)/hi + hJ h, (e − Πh e)/ hiΣ ,
Residual decomposition r := J|Σ + R where
R = R[U ; f, Πh f, A ] internal (regular) part of distribution r
J = J[U ; A ] jump (singular Σ-concentrated) part of r
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Energy-residual, interpolation and CBS inequalities yield
√
1
dt kek2 + |e|2a ≤ (C1 kRhk + C2 kJ hkΣ ) |e|a .
2
|
{z
}
=: E [U ; f, Πh f, A , Σ] = E [U ]
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Energy-residual, interpolation and CBS inequalities yield
√
1
dt kek2 + |e|2a ≤ (C1 kRhk + C2 kJ hkΣ ) |e|a .
2
|
{z
}
=: E [U ; f, Πh f, A , Σ] = E [U ]
Integrate in time to obtain final a posteriori error estimate
1/2
Z t
Z t
2
2
ke(t)k +
|e|a
≤C
E [U ].
0
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Nonstandard a posteriori parabolic estimates
0
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Energy-residual, interpolation and CBS inequalities yield
√
1
dt kek2 + |e|2a ≤ (C1 kRhk + C2 kJ hkΣ ) |e|a .
2
|
{z
}
=: E [U ; f, Πh f, A , Σ] = E [U ]
Integrate in time to obtain final a posteriori error estimate
1/2
Z t
Z t
2
2
ke(t)k +
|e|a
≤C
E [U ].
0
0
Pros simple, straightfoward, natural, optimal rate for |e|a
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Energy-residual, interpolation and CBS inequalities yield
√
1
dt kek2 + |e|2a ≤ (C1 kRhk + C2 kJ hkΣ ) |e|a .
2
|
{z
}
=: E [U ; f, Πh f, A , Σ] = E [U ]
Integrate in time to obtain final a posteriori error estimate
1/2
Z t
Z t
2
2
ke(t)k +
|e|a
≤C
E [U ].
0
0
Pros simple, straightfoward, natural, optimal rate for |e|a
Cons limited to residual, limited to energy, inflexible, mixes the
norms and the error indicators, suboptimal rate for kek,
reinventing the wheel.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Energy-residual, interpolation and CBS inequalities yield
√
1
dt kek2 + |e|2a ≤ (C1 kRhk + C2 kJ hkΣ ) |e|a .
2
|
{z
}
=: E [U ; f, Πh f, A , Σ] = E [U ]
Integrate in time to obtain final a posteriori error estimate
1/2
Z t
Z t
2
2
ke(t)k +
|e|a
≤C
E [U ].
0
0
Pros simple, straightfoward, natural, optimal rate for |e|a
Cons limited to residual, limited to energy, inflexible, mixes the
norms and the error indicators, suboptimal rate for kek,
reinventing the wheel.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Direct (violent) FEM a posteriori analysis (continued)
Heat energy estimate
Energy-residual, interpolation and CBS inequalities yield
√
1
dt kek2 + |e|2a ≤ (C1 kRhk + C2 kJ hkΣ ) |e|a .
2
|
{z
}
=: E [U ; f, Πh f, A , Σ] = E [U ]
Integrate in time to obtain final a posteriori error estimate
1/2
Z t
Z t
2
2
ke(t)k +
|e|a
≤C
E [U ].
0
0
Pros simple, straightfoward, natural, optimal rate for |e|a
Cons limited to residual, limited to energy, inflexible, mixes the
norms and the error indicators, suboptimal rate for kek,
reinventing the wheel.
Elliptic reconstruction’s purpose:
Get rid of cons (and retain pros :-)
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Explicit a posteriori estimates for the heat equation
A one-slide (incomplete!) history
Duality techniques
[Eriksson and Johnson, 1991] and many others,
optimal order in L∞ (0, T ; L2 (Ω)),
serious mesh restrictions in many cases,
no estimates for gradients.
Energy techniques
[Picasso, 1998], [Chen and Jia, 2004], [Verfürth, 2003],
[Bergam et al., 2005], [Bernardi and Süli, 2005] . . .
optimal order in L2 (0, T ; H10 ),
suboptimal order in L2 (Ω) spaces,
mesh change effects either absent or implicit (i.e., hidden in
constants).
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Other possible techniques
Anything that works for the PDE stability analysis is worth trying
(especially for nonlinear problems) in time:
1
Heat kernel estimates, e.g., to derive pointwise estimates
[Demlow et al., vorg],
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Other possible techniques
Anything that works for the PDE stability analysis is worth trying
(especially for nonlinear problems) in time:
1
Heat kernel estimates, e.g., to derive pointwise estimates
[Demlow et al., vorg],
2
Continuous dependence, useful in nonlinear problems
[Feng and Wu, 2005]
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Other possible techniques
Anything that works for the PDE stability analysis is worth trying
(especially for nonlinear problems) in time:
1
Heat kernel estimates, e.g., to derive pointwise estimates
[Demlow et al., vorg],
2
Continuous dependence, useful in nonlinear problems
[Feng and Wu, 2005]
3
Semigroup techniques, e.g., [Whalbin et al]
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Other possible techniques
Anything that works for the PDE stability analysis is worth trying
(especially for nonlinear problems) in time:
1
Heat kernel estimates, e.g., to derive pointwise estimates
[Demlow et al., vorg],
2
Continuous dependence, useful in nonlinear problems
[Feng and Wu, 2005]
3
Semigroup techniques, e.g., [Whalbin et al]
4
Monotonicity estimates, e.g., [Nochetto et al., 2000]
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Mesh adaption and a posteriori
Ultimate goal of a posteriori error estimates are: error control via
mesh-adaptive methods.
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Mesh adaption and a posteriori
Ultimate goal of a posteriori error estimates are: error control via
mesh-adaptive methods.
Elliptic convergence thoroughly understood for linear
problems [Nochetto, 2008, Binev et al., 2004, ?] and some nonlinear
problems [Veeser 2007].
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Mesh adaption and a posteriori
Ultimate goal of a posteriori error estimates are: error control via
mesh-adaptive methods.
Elliptic convergence thoroughly understood for linear
problems [Nochetto, 2008, Binev et al., 2004, ?] and some nonlinear
problems [Veeser 2007].
Convergence yet to be fully understood. Noted new directions:
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
14 / 53
Mesh adaption and a posteriori
Ultimate goal of a posteriori error estimates are: error control via
mesh-adaptive methods.
Elliptic convergence thoroughly understood for linear
problems [Nochetto, 2008, Binev et al., 2004, ?] and some nonlinear
problems [Veeser 2007].
Convergence yet to be fully understood. Noted new directions:
Tolerance reached under termination assumption [Chen and Jia, 2004].
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
14 / 53
Mesh adaption and a posteriori
Ultimate goal of a posteriori error estimates are: error control via
mesh-adaptive methods.
Elliptic convergence thoroughly understood for linear
problems [Nochetto, 2008, Binev et al., 2004, ?] and some nonlinear
problems [Veeser 2007].
Convergence yet to be fully understood. Noted new directions:
Tolerance reached under termination assumption [Chen and Jia, 2004].
Proof of termination [Siebert et al].
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
14 / 53
Mesh adaption and a posteriori
Ultimate goal of a posteriori error estimates are: error control via
mesh-adaptive methods.
Elliptic convergence thoroughly understood for linear
problems [Nochetto, 2008, Binev et al., 2004, ?] and some nonlinear
problems [Veeser 2007].
Convergence yet to be fully understood. Noted new directions:
Tolerance reached under termination assumption [Chen and Jia, 2004].
Proof of termination [Siebert et al].
Convergence for wavelets on tensor meshes in space-time
formulation [Schwab and Stevenson, 2008].
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
14 / 53
A world without reconstruction is possible,
but one with it is better.
Most work on parabolic a posteriori estimates is based on elliptic
estimates in one way or another.
Each elliptic technique (mainly residuals) is rederived at each attempt
to derive parabolic estimates.
Estimators are divided into “elliptic”, “time” and “mixed”.
Fully discrete estimates can be very complicated.
Why re-invent the wheel when elliptic a posteriori estimates can be
read off the book, e.g., [Ainsworth and Oden, 2000]?
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
15 / 53
What can Reconstruction buy us,
that the direct approach wouldn’t?
Energy Optimal-rate L2 (Ω)-norm estimates via energy techniques
[Makridakis and Nochetto, 2003],
[Lakkis and Makridakis, 2006]. (Application is the use of L2
estimates for Allen–Cahn simulations.)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
16 / 53
What can Reconstruction buy us,
that the direct approach wouldn’t?
Energy Optimal-rate L2 (Ω)-norm estimates via energy techniques
[Makridakis and Nochetto, 2003],
[Lakkis and Makridakis, 2006]. (Application is the use of L2
estimates for Allen–Cahn simulations.)
Duality A wider range of problems and estimates via duality,
following [Eriksson and Johnson, 1991]
[Lakkis and Makridakis, 2007].
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
16 / 53
What can Reconstruction buy us,
that the direct approach wouldn’t?
Energy Optimal-rate L2 (Ω)-norm estimates via energy techniques
[Makridakis and Nochetto, 2003],
[Lakkis and Makridakis, 2006]. (Application is the use of L2
estimates for Allen–Cahn simulations.)
Duality A wider range of problems and estimates via duality,
following [Eriksson and Johnson, 1991]
[Lakkis and Makridakis, 2007].
ZZ to Parabolic Recovery based estimates [Lakkis and Pryer, 2008].
(Previous work by [Leykekhman and Wahlbin, 2006] but
unrealistic time-constraints in proof).
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
16 / 53
What can Reconstruction buy us,
that the direct approach wouldn’t?
Energy Optimal-rate L2 (Ω)-norm estimates via energy techniques
[Makridakis and Nochetto, 2003],
[Lakkis and Makridakis, 2006]. (Application is the use of L2
estimates for Allen–Cahn simulations.)
Duality A wider range of problems and estimates via duality,
following [Eriksson and Johnson, 1991]
[Lakkis and Makridakis, 2007].
ZZ to Parabolic Recovery based estimates [Lakkis and Pryer, 2008].
(Previous work by [Leykekhman and Wahlbin, 2006] but
unrealistic time-constraints in proof).
Pointwise norms Pointwise (L∞ ([0, T ] × Ω)-norm) estimates in parabolic
problems derived by [Demlow et al., vorg]. (Previous work by
[Boman & Larsson 2000], but very complex and no fully
discrete result.)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
16 / 53
What can Reconstruction buy us,
that the direct approach wouldn’t?
Energy Optimal-rate L2 (Ω)-norm estimates via energy techniques
[Makridakis and Nochetto, 2003],
[Lakkis and Makridakis, 2006]. (Application is the use of L2
estimates for Allen–Cahn simulations.)
Duality A wider range of problems and estimates via duality,
following [Eriksson and Johnson, 1991]
[Lakkis and Makridakis, 2007].
ZZ to Parabolic Recovery based estimates [Lakkis and Pryer, 2008].
(Previous work by [Leykekhman and Wahlbin, 2006] but
unrealistic time-constraints in proof).
Pointwise norms Pointwise (L∞ ([0, T ] × Ω)-norm) estimates in parabolic
problems derived by [Demlow et al., vorg]. (Previous work by
[Boman & Larsson 2000], but very complex and no fully
discrete result.)
Non-conforming FEMs New simple estimates in non-conforming methods,
e.g., [Georgoulis and Lakkis, vorg] for fully discrete
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
16 / 53
A User’s Guide to the Elliptic Reconstruction
[Makridakis and Nochetto, 2003] and [Lakkis and Makridakis, 2006]
u ∈ H10 (Ω) exact solution
u
H10 (Ω)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
17 / 53
A User’s Guide to the Elliptic Reconstruction
[Makridakis and Nochetto, 2003] and [Lakkis and Makridakis, 2006]
u ∈ H10 (Ω) exact solution
U ∈ Vh parabolic Vh -FE
approximation of u.
u
Vh
U
H10 (Ω)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
17 / 53
A User’s Guide to the Elliptic Reconstruction
[Makridakis and Nochetto, 2003] and [Lakkis and Makridakis, 2006]
u ∈ H10 (Ω) exact solution
U ∈ Vh parabolic Vh -FE
approximation of u.
want w intermediate object
between u and U s.t.
u
w?
Vh
U
H10 (Ω)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
17 / 53
A User’s Guide to the Elliptic Reconstruction
[Makridakis and Nochetto, 2003] and [Lakkis and Makridakis, 2006]
u ∈ H10 (Ω) exact solution
U ∈ Vh parabolic Vh -FE
approximation of u.
u
w = RU
Vh
want w intermediate object
between u and U s.t.
U ∈ Vh an elliptic Vh -FE
approximation of w, error
= U − u,
U
H10 (Ω)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
17 / 53
A User’s Guide to the Elliptic Reconstruction
[Makridakis and Nochetto, 2003] and [Lakkis and Makridakis, 2006]
u ∈ H10 (Ω) exact solution
U ∈ Vh parabolic Vh -FE
approximation of u.
u
w = RU
Vh
want w intermediate object
between u and U s.t.
U ∈ Vh an elliptic Vh -FE
approximation of w, error
= U − u,
Galerkin orthogonality ⇒
a posteriori bounds on kk are
available off the shelf,
U
H10 (Ω)
Galerkin ⊥
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
17 / 53
A User’s Guide to the Elliptic Reconstruction
[Makridakis and Nochetto, 2003] and [Lakkis and Makridakis, 2006]
u ∈ H10 (Ω) exact solution
U ∈ Vh parabolic Vh -FE
approximation of u.
u
want w intermediate object
between u and U s.t.
ρ
w = RU
Vh
U ∈ Vh an elliptic Vh -FE
approximation of w, error
= U − u,
Galerkin orthogonality ⇒
a posteriori bounds on kk are
available off the shelf,
U
H10 (Ω)
Galerkin ⊥
w not computable,
but parabolic error ρ = w − U
satifies parabolic equation with
computable data.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
17 / 53
Crucial parabolic-elliptic ρ- error relation
Lemma (elliptic-parabolic error relation)
For each time-slab In , n ∈ [1 : N ], and φ ∈ H10 (Ω),
h∂t ρ | φi + a (ρ, φ) = h∂t , φi + a ((w − wn ) , φ)
+ hΠn f n − f, φi + τn−1 Πn U n−1 − U n−1 , φ
⇔ ∂t ρ + A ρ =
∂t |{z}
space disc.n
− A (w − wn ) + (Πn f n − f ) +
|
{z
} |
{z
}
time disc.n
data approx.n
Πn U n−1 − U n−1
τn
|
{z
}
mesh change
w := R U elliptic reconstruction, w(t) p.w.linear extension,
(
ρ
:= w − u parabolic error,
e := U − u = total error =
− := U − w elliptic error,
n
n
Πn := L2 (Ω)-projection onto
O Lakkis (Sussex)
V̊n .
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
18 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
Use PDE for ρ with ∂t as data to obtain bound on
kρk < C k∂t k.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
Use PDE for ρ with ∂t as data to obtain bound on
kρk < C k∂t k.
∂t = ∂t w − ∂t U = R∂t U − ∂t U .
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
Use PDE for ρ with ∂t as data to obtain bound on
kρk < C k∂t k.
∂t = ∂t w − ∂t U = R∂t U − ∂t U .
∂t U elliptic Vh -FE solution with exact
R∂t U = ∂t w ∈ H10 (Ω).
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
Use PDE for ρ with ∂t as data to obtain bound on
kρk < C k∂t k.
∂t = ∂t w − ∂t U = R∂t U − ∂t U .
∂t U elliptic Vh -FE solution with exact
R∂t U = ∂t w ∈ H10 (Ω).
⇒ k∂t k elliptic error controlled a posteriori by
estimator E [∂t U, ∂t f, Vh ].
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
Use PDE for ρ with ∂t as data to obtain bound on
kρk < C k∂t k.
∂t = ∂t w − ∂t U = R∂t U − ∂t U .
∂t U elliptic Vh -FE solution with exact
R∂t U = ∂t w ∈ H10 (Ω).
⇒ k∂t k elliptic error controlled a posteriori by
estimator E [∂t U, ∂t f, Vh ].
control of the time error A (w − wn ):
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
Use PDE for ρ with ∂t as data to obtain bound on
kρk < C k∂t k.
∂t = ∂t w − ∂t U = R∂t U − ∂t U .
∂t U elliptic Vh -FE solution with exact
R∂t U = ∂t w ∈ H10 (Ω).
⇒ k∂t k elliptic error controlled a posteriori by
estimator E [∂t U, ∂t f, Vh ].
control of the time error A (w − wn ):Variety of methods depending on
parabolic technique used.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Elliptic reconstruction: a user’s guide (controlling ρ)
∂t ρ + A ρ = ∂t + A (w − wn ) + controlled terms
control of the spatial error ∂t :
Use PDE for ρ with ∂t as data to obtain bound on
kρk < C k∂t k.
∂t = ∂t w − ∂t U = R∂t U − ∂t U .
∂t U elliptic Vh -FE solution with exact
R∂t U = ∂t w ∈ H10 (Ω).
⇒ k∂t k elliptic error controlled a posteriori by
estimator E [∂t U, ∂t f, Vh ].
control of the time error A (w − wn ):Variety of methods depending on
parabolic technique used.Example: use relation
∂t U + A wn = Πn f n
leads to explicit a posteriori representation
A wn := Πn f n − ∂t U.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
19 / 53
Towards an Energy estimate
If interested only in energy estimates.
Semidiscrete case
∂t e + A ρ = 0.
Fully discrete case
n n
∂t e + A ρ = I n U n−1 − U n−1 /τn + f n − f + A
| w −{zA w } .
|
{z
}
data approximation/interpolation error
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
time, operator & mesh
Warwick, 15 Jan 2009
20 / 53
Comparison with direct approach
Error relation via reconstruction
h∂t ρ | φi + a (ρ, φ) = h∂t , φi + a (w − wn , φ)
+ hΠn f n − f, φi + τn−1 Πn U n−1 − U n−1 , φ
Direct (reconstructionless) error relation
h∂t e, φi + a (e, φ) = h∂t U, φi + a (U n , φ) + a (U − U n , φ)
+ hΠn f n − f, φi + τn−1 Πn U n−1 − U n−1 , φ .
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
21 / 53
A useful analogy with the a priori
ER is to a posteriori what Ritz/elliptic projection is to a priori
analysis.
Optimal-order yielding properties of ER can interpreted as an
a posteriori analog of the similar phenomena of superconvergence
observed in the a priori analysis with the Ritz projection.
In spatially discrete (or very small timestep) case ρ converges with a
higher order error than .
However, ρ plays an important role when time error is comparable to
spatial error.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
22 / 53
Energy-residual estimates in L2 (H1 ) and L∞ (L2 )
Lemma (optimal-order elliptic residual a posteriori estimates)
For all V ∈ Vn we have
|RV − V |H1 (Ω) ≤ E [V, H1 (Ω)] = O(hn ) Residual-type estimator
kRV − V kL2 (Ω) ≤ E [V, L2 (Ω)] O(h2n ) Residual-type estimator on convex Ω
Lemma (parabolic energy a posteriori estimate)
There are estimators E1 and E2 such that
max kρ(t)k2L2 (Ω)
[0,tm ]
Z
+2
0
tm
|ρ(t)|2a
1/2
dt
≤ ρ0 + C (E1,m + E2,m ) .
|
{z
}
O(hn )2 (small τn )
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
23 / 53
Remarks on ρ’s order of convergence
The space estimate of ρ derives from estimating the term
h∂t , ρi ≤ k∂t k kρk
or ( k∂t kH−1 (Ω) |ρ|a )
Remark (superconvergence of ρ)
The fact that energy norm |ρ|a = O(h2n ) leads to optimal-order estimates
for norms kekX X = L2 (Ω), L∞ (Ω)
[Makridakis and Nochetto, 2003, Lakkis and Makridakis, 2006,
Lakkis and Makridakis, 2007, Demlow et al., vorg].
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
24 / 53
Higher order energy-residual estimates
Norms of H1 (L2 ) and L∞ (H1 )
Similar elliptic results, but higher norm (test by ∂t ρ) yield
Lemma (higher-order parabolic energy a posteriori estimate)
There are estimators E˜i , i = 1, 2, such that
max |ρ(t)|2a + 2
t∈[0,tm ]
Z
0
tm
k∂t ρk2
1/2
1/2
2
2
≤ ρ0 a + 4 E˜1,m
+ E˜2,m
,
Lead to higher order norms optimal-order energy estimates.
[Lakkis and Makridakis, 2006].
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
25 / 53
Duality estimates
Definition (Error estimators)
Suppose an a posteriori elliptic error estimator function E [·, ·, ·] is
available. Let
εn := E [U n , Vn , L2 (Ω)],
ηn := E [∂t U n , Vn ∩ Vn−1 , L2 (Ω)],
( 1 1 1
Π f − ∂U 1 − A0 U 0 for n = 1,
2
θn := 1 n n
n τ
for n ∈ [2 : N ] ,
n
2 ∂ Π f − ∂U
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
26 / 53
Duality estimates
Definition (Logarithmic factors)
(
bn =
1
4
1
8
log
T −tn−1
T −tn
for n ∈ [1 : N − 1] ,
for n = N ;
 τn+1
λ τn
T −tn − λ − T −tn , for n ∈ [0 : n − 2] ,
an =
λ τN −1 − 1, for n = N − 1,
τN
where
(
(1 + 1/x) log(1 + x)
λ(x) :=
1
O Lakkis (Sussex)
for |x| ∈ (0, 1),
for x = 0.
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
27 / 53
Duality estimates
Theorem (General explicit duality-based a posteriori error estimates)
kρ(T )k ≤ ke(0)k +
N
−1
X
!1/2
an ε2n
+ ηN
n=0
+
N
X
!1/2
bn
n−1
2
U
− U n + ηn
n=1
+
N
−1
X
!1/2
2
bn δn h2n n=1
+
N
X
r
+
τN
kδN hN k
2
τn k∂t f kL1 (In ;L2 (Ω))
n=1
[Lakkis and Makridakis, 2007]
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Warwick, 15 Jan 2009
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Estimates in the L∞ (Ω) norm
[Demlow et al., vorg]
We take A = −∆ thus study:
∂t u − ∆u = f, u(0) = u0
and
u|Ω = 0.
Direct approach is messy and hard, especially for fully discrete case
[Boman 2000, cf.].
Our result based on elliptic a posteriori estimates
[Nochetto, 1995, Nochetto et al., 2006]
kV − RV k ≤ C (log hn )2 E∞,0 [V, g, Vn ] = O (log hn hn )2
for residual-based a posteriori estimator functional E∞,0 .
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L∞ estimates
Main idea [Demlow et al., vorg]
Theorem (semidiscrete estimates)
Recalling e = U − u, ρ = RU − u and = RU − U :
Z t
ke(t)kL∞ (Ω) ≤ k(0)kL∞ (Ω) + k(t)kL∞ (Ω) +
k∂t (s)k ds,
0
ke(t)kL∞ (Ω) ≤ k(0)kL∞ (Ω) + k(t)kL∞ (Ω) + Cp,q (t) k∂t kLq (0,t;W−1
p (Ω))
for p, q ∈ (2, ∞] and d/p + 2/q < 1.
Proof’s idea.
Use heat kernel estimates to bound ρ(= e + ) in terms of knowing
∂t ρ − ∆ρ = ∂t .
Negative Sobolev useful for convex domains and
O Lakkis (Sussex)
P2 or higher elements.
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
30 / 53
L∞ estimates
[Demlow et al., vorg]
Theorem (fully discrete estimates)
ke(tn )kL∞ (Ω) ≤ ke(0)kL∞ (Ω) + data +
N
X
τn g n − g n−1 L∞ (Ω)
n=1
2
+ C log(tn /τn ) (log hn ) max E∞,0 [U n , g n , Vn ]
1≤n≤N
n
n
n−1
n
g := U − U
/τn − f = An − An−1 + f n − Πn f n .
Elliptic error estimators [Nochetto, 1995]
E∞,0 [U n , g n , Vn ] := max h2n (g n − ∆U n )L∞ (K) + khn J∇U n KkL∞ (∂K)
K∈Tn
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31 / 53
Application to the Allen–Cahn equation
Known results by [Kessler et al., 2004] and [Feng and Wu, 2005] on the
Allen-Cahn equation
∂t u − ∆u +
1
f (u) = 0,
ε2
yield a posteriori error estimates in energy norm with a polynomial (instead
of exponential!) dependence on 1/ε exploiting special spectral properties
of the linearised operator. (Un)fortunately rate is optimal only for energy
norm in both results. Using the ER we provide optimal-order estimates for
the L∞ (0, T ; L2 (Ω)) norm based on identity
1
1
dt kρk2 − λ0 kρk2 = h∂t , ρi + 2 hf (w) − f (U ), ρi
2
ε
Obtain an ε-free error.
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Nonstandard a posteriori parabolic estimates
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Towards an Energy estimate
If interested only in energy estimates.
Semidiscrete case
∂t e + A ρ = 0.
Fully discrete case
n n
∂t e + A ρ = I n U n−1 − U n−1 /τn + f n − f + A
| w −{zA w } .
|
{z
}
data approximation/interpolation error
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
time, operator & mesh
Warwick, 15 Jan 2009
33 / 53
General energy estimates
Semidiscrete setting
1. The parabolic-elliptic error relation ∂t ρ + A ρ = ∂t is sometime more
useful written as
∂t e + A ρ = 0.
2. Testing with e we obtain
1
kek2 + kρk2a = a (ρ, ) .
2
Continuity of a and integration in time yield estimate
Z t
Z t
2
kρka ≤ Ca
kk2a .
0
0
3. Note that superconvergence properties of kρka are lost (but not
needed) in this context.
4. These simple observations and some elliptic technical work, lead to
interesting nonstandard results: recovery estimators and
nonconforming methods.
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Nonstandard a posteriori parabolic estimates
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Recovery-based estimates
[Lakkis and Pryer, 2008]
Theorem (Zienkiewicz-Zhou)
Let G be the Zienkiewicz–Zhou gradient recovery operator. Then
kV − RV kH1 (Ω) ≤ C k∇V − GV k .
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Recovery-based estimates
[Lakkis and Pryer, 2008]
Theorem (Zienkiewicz-Zhou)
Let G be the Zienkiewicz–Zhou gradient recovery operator. Then
kV − RV kH1 (Ω) ≤ C k∇V − GV k .
Lemma ((semidiscrete) energy norm parabolic-elliptic estimate)
2
t
Z
kek +
0
O Lakkis (Sussex)
kρk2a
≤C
ke(0)k2a
Z
+
0
Nonstandard a posteriori parabolic estimates
t
kk2a
.
Warwick, 15 Jan 2009
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Recovery-based estimates
[Lakkis and Pryer, 2008]
Theorem (Zienkiewicz-Zhou)
Let G be the Zienkiewicz–Zhou gradient recovery operator. Then
kV − RV kH1 (Ω) ≤ C k∇V − GV k .
Lemma ((semidiscrete) energy norm parabolic-elliptic estimate)
2
t
Z
kek +
0
kρk2a
≤C
ke(0)k2a
Z
+
0
t
kk2a
.
In combination lead to recovery based estimates for parabolic equations.
(Related results by [Leykekhman and Wahlbin, 2006].)
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Recovery estimates
Fully discrete estimators
elliptic (gradient recovery error) estimator εn := k∇U n − Gn [U n ]k ,
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Recovery estimates
Fully discrete estimators
elliptic (gradient recovery error) estimator εn := k∇U n − Gn [U n ]k ,
mesh-change
(coarsening) error
indicator
γn := τn−1 I n U n−1 − U n−1 ,
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
36 / 53
Recovery estimates
Fully discrete estimators
elliptic (gradient recovery error) estimator εn := k∇U n − Gn [U n ]k ,
mesh-change
(coarsening) error
indicator
γn := τn−1 I n U n−1 − U n−1 ,
time error indicator θn := Cτn I n U n−1 − U n a
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
36 / 53
Recovery estimates
Fully discrete estimators
elliptic (gradient recovery error) estimator εn := k∇U n − Gn [U n ]k ,
mesh-change
(coarsening) error
indicator
γn := τn−1 I n U n−1 − U n−1 ,
time error indicator θn := Cτn I n U n−1 − U n a
R tn
kI n f n − f k .
data approximation error indicator βn := τn−1 tn−1
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
36 / 53
Recovery estimates
Fully discrete estimators
elliptic (gradient recovery error) estimator εn := k∇U n − Gn [U n ]k ,
mesh-change
(coarsening) error
indicator
γn := τn−1 I n U n−1 − U n−1 ,
time error indicator θn := Cτn I n U n−1 − U n a
R tn
kI n f n − f k .
data approximation error indicator βn := τn−1 tn−1
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
36 / 53
Recovery estimates
Fully discrete estimators
elliptic (gradient recovery error) estimator εn := k∇U n − Gn [U n ]k ,
mesh-change
(coarsening) error
indicator
γn := τn−1 I n U n−1 − U n−1 ,
time error indicator θn := Cτn I n U n−1 − U n a
R tn
kI n f n − f k .
data approximation error indicator βn := τn−1 tn−1
Lemma (gradient recovery a posteriori error estimates)
2
Z
T
max ke(t)k + 2
t∈[0,T ]
≤ ke(0)k C
N
X
0
Z
(βn τn + θn τn + γn τn ) + 8
n=1
O Lakkis (Sussex)
kρ(t)k2a
Nonstandard a posteriori parabolic estimates
0
T
1/2
dt
!1/2
kk2a
.
Warwick, 15 Jan 2009
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Convergence tests for ZZ-type estimators
[Lakkis and Pryer, 2008]
P1 elements τ
=h
2
2
E O C [ | | e | | L 2 ( 0 , t m ; H 01 (]Ω ) E
) 1 / 2]
) O C[ τ ( Σ² n + ² n − 1
E O C [ τ Σ nθ n]
E ff e c t i v i t y I n d e x
6
6
7
7
5
5
6
6
4
4
5
5
3
3
4
4
3
3
2
2
2
2
1
1
0
0
0.5
0
1
1
0
| | e | | L 2 ( 0 , t m ; H 01 ( Ω ) )
0.5
1
0
1
0
τ ( Σ ² 2n + ² 2n − 1) 1 / 2
0.5
1
0
τ Σ nθ n
0
0
0
0
−2
−2
−2
−4
−4
−4
−4
−6
−6
−6
−6
−8
−8
−8
0
0.5
O Lakkis (Sussex)
1
−10
0
0.5
1
−10
0.5
1
C om b i n e d E s t i m at or
−2
−10
0
−8
0
0.5
1
−10
0
0.5
1
Nonstandard a posteriori parabolic estimates
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Convergence tests for ZZ-type estimators
[Lakkis and Pryer, 2008]
P1 elements τ
= h2
2
2
E O C [ | | e | | L 2 ( 0 , t m ; H 01 (]Ω ) E
) 1 / 2]
) O C[ τ ( Σ² n + ² n − 1
E O C [ τ Σ nθ n]
E ff e c t i v i t y I n d e x
6
6
7
7
5
5
6
6
4
4
5
5
3
3
4
4
3
3
2
2
2
2
1
1
0
0
0.5
0
1
1
0
| | e | | L 2 ( 0 , t m ; H 01 ( Ω ) )
0.5
1
0
1
0
τ ( Σ ² 2n + ² 2n − 1) 1 / 2
0.5
1
0
τ Σ nθ n
0
0
0
0
−2
−2
−2
−4
−4
−4
−4
−6
−6
−6
−6
−8
−8
−8
0
0.5
O Lakkis (Sussex)
1
−10
0
0.5
1
−10
0.5
1
C om b i n e d E s t i m at or
−2
−10
0
−8
0
0.5
1
−10
0
0.5
1
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Convergence tests for ZZ-type estimators
[Lakkis and Pryer, 2008]
P2 elements τ
= h3
2
2
E O C [ | | e | | L 2 ( 0 , t m ; H 01 (]Ω ) E
) 1 / 2]
) O C[ τ ( Σ² n + ² n − 1
E O C [ τ Σ nθ n]
E ff e c t i v i t y I n d e x
6
6
7
7
5
5
6
6
4
4
5
5
3
3
4
4
3
3
2
2
2
2
1
1
0
0
0.5
0
1
1
0
| | e | | L 2 ( 0 , t m ; H 01 ( Ω ) )
0.5
1
0
1
0
τ ( Σ ² 2n + ² 2n − 1) 1 / 2
0.5
1
0
τ Σ nθ n
0
0
0
0
−2
−2
−2
−4
−4
−4
−4
−6
−6
−6
−6
−8
−8
−8
0
0.5
O Lakkis (Sussex)
1
−10
0
0.5
1
−10
0.5
1
C om b i n e d E s t i m at or
−2
−10
0
−8
0
0.5
1
−10
0
0.5
1
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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ZZ Adaptive Mesh Refinement
[Lakkis and Pryer, 2008]
+1
+1
+1
+1
+1
+0.9
+0.9
+0.9
+0.9
+0.9
+0.8
+0.8
+0.8
+0.8
+0.8
+0.7
+0.7
+0.7
+0.7
+0.7
+0.6
+0.6
+0.6
+0.6
+0.6
+0.5
+0.5
+0.5
+0.5
+0.5
+0.4
+0.4
+0.4
+0.4
+0.4
+0.3
+0.3
+0.3
+0.3
+0.3
+0.2
+0.2
+0.2
+0.2
+0.2
+0.1
+0.1
+0.1
+0.1
+0.1
+0
+0
+0
+0
+0
u_h
u_h
u_h
u_h
u_h
+1
+1
+1
+1
+1
+0.8
+0.8
+0.8
+0.8
+0.8
+0.6
+0.6
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0.2
+0.2
-5.55e-17
-5.55e-17
-5.55e-17
-5.55e-17
-5.55e-17
-0.2
-0.2
-0.2
-0.2
-0.2
-0.4
-0.4
-0.4
-0.4
-0.4
-0.6
-0.6
-0.6
-0.6
-0.6
-0.8
-0.8
-0.8
-0.8
-0.8
-1
-1
-1
-1
-1
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
40 / 53
Recovery adaptive method
‘Easy” (regular) problem
Uniform
Tolerance
0.573
0.295
0.149
0.0625
DOF’s
232,290
3,489,090
54,097,020
OOM
CPU
3
49
598
OOM
refine % = 0.65
DOF’s
CPU
24,080
4
42,042
8
82,172
15
206,709
39
Adaptive
refine % = 0.70
DOF’s
CPU
22,792
5
39,414
8
77,932
15
195,810
37
refine % = 0.75
DOF’s
CPU
22,240
4
38,630
6
76,452
16
191,650
37
Adaptive
refine % = 0.7
DOF’s
CPU
11,430
5
16,140
8
100,058
29
460,637
118
refine % = 0.75
DOF’s
CPU
11,498
5
16,201
7
22,597
10
449,568
115
Space-error dominated problem
Uniform
Tolerance
0.296
0.21
0.104
0.03125
DOF’s
3,489,090
13,940,289
54,097,020
OOM
O Lakkis (Sussex)
CPU
47
196
602
OOM
refine % = 0.65
DOF’s
CPU
12,092
5
17,038
7
106,188
32
513,694
120
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Recovery adaptive method
Time-error dominated problem (implicit strategy)
Uniform
Tolerance
1.000
0.569
0.295
0.149
DOF’s
925,809
3,489,090
54,097,020
OOM
CPU
12
49
605
OOM
Adaptive
refine % = 0.7
refine % = 0.75
DOF’s
CPU
DOF’s
CPU
159,070
43
127,610
58
237,960
142
204,376
180
471,733
755
471,542
920
940,618
1410
940,138
1850
Time-error dominated problem (explicit strategy)
Uniform
Tolerance
1.000
0.569
0.295
0.149
DOF’s
925,809
3,489,090
54,097,026
OOM
O Lakkis (Sussex)
CPU
12
49
605
OOM
Adaptive
refine % = 0.75
refine % = 0.7
DOF’s
CPU
DOF’s
CPU
135,788
5
127,004
4
198,628
7
194,311
8
397,716
15
395,876
16
2,177,666
79
2,079,081
76
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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A nonconforming method (DG)
[Georgoulis and Lakkis, vorg]
Discontinuous Galerkin (DG) bilinear form discretizing ∆ is given by
X Z
B(w, v) :=
∇w∇v
K
K∈T
Z
+
Γ
(θ{∇v} JwK − {∇w} JvK + σ JwK JvK) ds
+
− −
where JφK := φ+
S n + φ n is the jump of the scalar φ, and
+
−
{φ} := (φ + φ )/2 is the average of field φ, across the triangulations
sides; θ, σ are method, penalty parameters, respectively.
DG space with respect to triangulation T , with no hanging nodes
restriction:
Vh = {v ∈ L2 (Ω) : v|K ∈ Pp }
where p is a fixed polynomial degree.
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Nonconforming Elliptic Reconstruction
[Georgoulis and Lakkis, vorg]
Definition (the DG elliptic reconstruction)
Let UDG (t) be the (semidiscrete) DG solution at time t ∈ [0, T ], define
elliptic reconstruction w(t) ∈ H01 (Ω), of UDG (t) solves elliptic problem
A w(t) = g(t) ∀ t ∈ [0, T ],
where
g(t) := AUDG (t) + f − Πf,
and A : Vh → Vh is the discrete DG-operator defined by for V ∈ Vh by
hAV, Φi = B (V, Φ) ∀ Φ ∈ Vh .
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Nonconforming Elliptic Reconstruction
[Georgoulis and Lakkis, vorg]
Here w ∈ H10 (Ω) and u ∈ H10 (Ω) ⇒ ρ ∈ H10 (Ω). Define discontinuous
part:
ed := Ud := d ,
and its continous part:
ec := e − ed = e − Ud = ρ + c .
Lemma (basic nonconforming energy estimate)
1
dt kec k2 + kρk2a = B (c , ρ) + h∂t Ud , ec i + ln−1 An−1 U n−1 − An U n , ec
| {z } | {z } |
2
{z
}
elliptic
+
|
O Lakkis (Sussex)
n
I U
nonconforming
n−1
−U
n−1
time & mesh-change
n
/τn + f − f, ec .
{z
}
data & mesh-change
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Closing remarks and outlook
Conclusions
Elliptic reconstruction unifies known a posteriori analysis (with
Makridakis, Nochetto).
Leads to new optimal-order estimates (with Demlow, Makridakis).
Rigorous justification the use of recovery techniques (with Pryer).
Nonconforming methods (with Georgoulis).
Current developments (new directions?)
Wave equation (with Georgoulis & Makridakis).
Semilinear equations, e.g., Allen–Cahn (with Georgoulis &
Makridakis), with applications to stochastic Monte-Carlo simulations
(with Katsoulakis, Kossioris & Romito).
Quasilinear equations, e.g, MCF of function graphs.
Taxpayer’s support
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Wiley-Interscience [John Wiley & Sons], New York.
Bergam, A., Bernardi, C., and Mghazli, Z. (2005).
A posteriori analysis of the finite element discretization of some
parabolic equations.
Math. Comp., 74(251):1117–1138 (electronic).
Bernardi, C. and Süli, E. (2005).
Time and space adaptivity for the second-order wave equation.
Math. Models Methods Appl. Sci., 15(2):199–225.
Binev, P., Dahmen, W., and DeVore, R. (2004).
Adaptive finite element methods with convergence rates.
Numer. Math., 97(2):219–268.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Bibliography II
Chen, Z. and Jia, F. (2004).
An adaptive finite element algorithm with reliable and efficient error
control for linear parabolic problems.
Math. Comp., 73(247):1167–1193 (electronic).
Demlow, A., Lakkis, O., and Makridakis, C. (to appear, preprint
0711-3928@arXiv.org).
A posteriori error estimates in the maximum norm for parabolic
problems.
SIAM J. Numer. Anal.
Eriksson, K. and Johnson, C. (1991).
Adaptive finite element methods for parabolic problems. I. A linear
model problem.
SIAM J. Numer. Anal., 28(1):43–77.
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Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Bibliography III
Feng, X. and Wu, H.-j. (2005).
A posteriori error estimates and an adaptive finite element method for
the Allen-Cahn equation and the mean curvature flow.
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Georgoulis, E. and Lakkis, O. (to appear, preprint
0804.4262@arXiv.org).
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SIAM J. Numer. Anal.
Kessler, D., Nochetto, R. H., and Schmidt, A. (2004).
A posteriori error control for the Allen-Cahn problem: Circumventing
Gronwall’s inequality.
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O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Bibliography IV
Lakkis, O. and Makridakis, C. (2006).
Elliptic reconstruction and a posteriori error estimates for fully discrete
linear parabolic problems.
Math. Comp., 75(256):1627–1658 (electronic).
Lakkis, O. and Makridakis, C. (2007).
A posteriori error control for parabolic problems via elliptic
reconstruction and duality.
Technical Report 0709.0916, arXiv.
Lakkis, O. and Pryer, T. (2008).
Gradient recovery in adaptive methods for parabolic equations.
Technical Report To be submitted, University of Sussex, Brighton, UK.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Bibliography V
Leykekhman, D. and Wahlbin, L. (2006).
A posteriori error estimates by recovered gradients in parabolic finite
element equations.
Technical report, University of Texas, Austin.
Preprint (submitted to Math. Comp.).
Makridakis, C. and Nochetto, R. H. (2003).
Elliptic reconstruction and a posteriori error estimates for parabolic
problems.
SIAM J. Numer. Anal., 41(4):1585–1594 (electronic).
Nochetto, R. H. (1995).
Pointwise a posteriori error estimates for elliptic problems on highly
graded meshes.
Math. Comp., 64(209):1–22.
O Lakkis (Sussex)
Nonstandard a posteriori parabolic estimates
Warwick, 15 Jan 2009
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Bibliography VI
Nochetto, R. H. (2008).
Adaptive finite element methods for elliptic pde.
Lecture notes, University of Maryland.
Nochetto, R. H., Savaré, G., and Verdi, C. (2000).
A posteriori error estimates for variable time-step discretizations of
nonlinear evolution equations.
Comm. Pure Appl. Math., 53(5):525–589.
Nochetto, R. H., Schmidt, A., Siebert, K. G., and Veeser, A. (2006).
Pointwise a posteriori error estimates for monotone semi-linear
equations.
Numer. Math., 104(4):515–538.
Picasso, M. (1998).
Adaptive finite elements for a linear parabolic problem.
Comput. Methods Appl. Mech. Engrg., 167(3-4):223–237.
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Bibliography VII
Schwab, C. and Stevenson, R. (2008).
Space-time adaptive wavelet methods for parabolic evolution
problems.
Report 01, Seminar für Angewandte Mathematik ETH, Eidgenössische
Technische Hochschule CH-8092 Zürich.
Verfürth, R. (2003).
A posteriori error estimates for finite element discretizations of the
heat equation.
Calcolo, 40(3):195–212.
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