Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA www2.math.umd.edu/ ˜ rhn 7th Z¨ urich Summer School, August 2012 A Posteriori Error Control and Adaptivity
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Adaptive Finite Element MethodsLecture 1: A Posteriori Error Estimation
Ricardo H. Nochetto
Department of Mathematics andInstitute for Physical Science and Technology
University of Maryland, USA
www2.math.umd.edu/˜rhn
7th Zurich Summer School, August 2012A Posteriori Error Control and Adaptivity
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Outline
Piecewise Polynomial Interpolation in Sobolev Spaces
Model Problem and FEM
FEM: A Posteriori Error Analysis
Surveys
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Outline
Piecewise Polynomial Interpolation in Sobolev Spaces
Model Problem and FEM
FEM: A Posteriori Error Analysis
Surveys
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Warm-up: 1d Example
Question: given a continuous function u : [0, 1] → R, a partitionTN = xnN
n=0 with x0 = 0, xN = 1, and a pw constant approximationUN of u over TN , what is the best decay rate of ‖u− UN‖L∞(0,1)?
Answer 1: W 1∞-Regularity. Let u ∈ W 1
∞(0, 1) and TN be quasi-uniform.Then UN (x) = u(xn−1) for xn−1 ≤ x < xn satisfies
|Un(x)− u(x)| = |u(xn−1)− u(x)| ≤∫ xn−1
x
|u′(s)|ds 41N‖u′‖L∞(0,1).
Answer 2: W 11 -Regularity. Let u ∈ W 1
1 (0, 1). If xn is defined by∫ xn
xn−1
|u′(s)|ds =1N‖u′‖L1(0,1),
then
|Un(x)− u(x)| = |u(xn−1)− u(x)| ≤∫ xn−1
x
|u′(s)|ds ≤ 1N‖u′‖L1(0,1).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Warm-up: 1d Example
Question: given a continuous function u : [0, 1] → R, a partitionTN = xnN
n=0 with x0 = 0, xN = 1, and a pw constant approximationUN of u over TN , what is the best decay rate of ‖u− UN‖L∞(0,1)?
Answer 1: W 1∞-Regularity. Let u ∈ W 1
∞(0, 1) and TN be quasi-uniform.Then UN (x) = u(xn−1) for xn−1 ≤ x < xn satisfies
|Un(x)− u(x)| = |u(xn−1)− u(x)| ≤∫ xn−1
x
|u′(s)|ds 41N‖u′‖L∞(0,1).
Answer 2: W 11 -Regularity. Let u ∈ W 1
1 (0, 1). If xn is defined by∫ xn
xn−1
|u′(s)|ds =1N‖u′‖L1(0,1),
then
|Un(x)− u(x)| = |u(xn−1)− u(x)| ≤∫ xn−1
x
|u′(s)|ds ≤ 1N‖u′‖L1(0,1).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Warm-up: 1d Example
Question: given a continuous function u : [0, 1] → R, a partitionTN = xnN
n=0 with x0 = 0, xN = 1, and a pw constant approximationUN of u over TN , what is the best decay rate of ‖u− UN‖L∞(0,1)?
Answer 1: W 1∞-Regularity. Let u ∈ W 1
∞(0, 1) and TN be quasi-uniform.Then UN (x) = u(xn−1) for xn−1 ≤ x < xn satisfies
|Un(x)− u(x)| = |u(xn−1)− u(x)| ≤∫ xn−1
x
|u′(s)|ds 41N‖u′‖L∞(0,1).
Answer 2: W 11 -Regularity. Let u ∈ W 1
1 (0, 1). If xn is defined by∫ xn
xn−1
|u′(s)|ds =1N‖u′‖L1(0,1),
then
|Un(x)− u(x)| = |u(xn−1)− u(x)| ≤∫ xn−1
x
|u′(s)|ds ≤ 1N‖u′‖L1(0,1).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Sobolev Number
Let ω ⊂ Rd be Lipschitz and bounded, k ∈ N, 1 ≤ p ≤ ∞. The Sobolevnumber of W k
p (ω) is
sob(W kp ) := k − d
p.
Remark 1. This number governs the scaling properties of seminorm|v|W k
p (ω): consider x = 1hx which transforms ω into ω and note
|v|W kp (bω) = hsob(W k
p )|v|W kp (ω) ∀v ∈ W k
p (ω).
Remark 2. Let d = 1 and ω = (0, 1). Then W 1∞(ω) is the linear (and
usual) Sobolev scale of L∞(ω), but W 11 (ω) is in the nonlinear scale of
L∞(ω), i.e.
sob(W 11 ) = 1− 1
1= 0− 1
∞= sob(L∞).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Sobolev Number
Let ω ⊂ Rd be Lipschitz and bounded, k ∈ N, 1 ≤ p ≤ ∞. The Sobolevnumber of W k
p (ω) is
sob(W kp ) := k − d
p.
Remark 1. This number governs the scaling properties of seminorm|v|W k
p (ω): consider x = 1hx which transforms ω into ω and note
|v|W kp (bω) = hsob(W k
p )|v|W kp (ω) ∀v ∈ W k
p (ω).
Remark 2. Let d = 1 and ω = (0, 1). Then W 1∞(ω) is the linear (and
usual) Sobolev scale of L∞(ω), but W 11 (ω) is in the nonlinear scale of
L∞(ω), i.e.
sob(W 11 ) = 1− 1
1= 0− 1
∞= sob(L∞).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Sobolev Number
Let ω ⊂ Rd be Lipschitz and bounded, k ∈ N, 1 ≤ p ≤ ∞. The Sobolevnumber of W k
p (ω) is
sob(W kp ) := k − d
p.
Remark 1. This number governs the scaling properties of seminorm|v|W k
p (ω): consider x = 1hx which transforms ω into ω and note
|v|W kp (bω) = hsob(W k
p )|v|W kp (ω) ∀v ∈ W k
p (ω).
Remark 2. Let d = 1 and ω = (0, 1). Then W 1∞(ω) is the linear (and
usual) Sobolev scale of L∞(ω), but W 11 (ω) is in the nonlinear scale of
L∞(ω), i.e.
sob(W 11 ) = 1− 1
1= 0− 1
∞= sob(L∞).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Conforming Meshes: The Bisection Method and REFINE
• Labeling of a sequence of conforming refinements T0 ≤ T1 ≤ T2 ford = 2 (similar but much more intricate for d > 2)
0
00 0
0 0
0
0
11
1 1
11
1
1
1
1 1
2
2
2 2
2
2
2
2
2
2
2 2
3
33
3
• Shape regularity: the shape-regularity constant of any T ∈ T solelydepends on the shape-regularity constant of T0.
• Nested spaces: refinement leads to V(T ) ⊂ V(T∗) because T ≤ T∗.• Monotonicity of meshsize function hT : if hT |T := hT := |T |1/d, then
hT∗ ≤ hT for T∗ ≥ T , and reduction property with b ≥ 1 bisections
hT∗ |T ≤ 2−b/dhT |T ∀T ∈ T \ T∗.
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Complexity of REFINE
I Recursive bisection of T3 (sequence of compatible bisection patches)
3
1 2
3
2
2
2 2
1
3
1 2
3
22
2
3
3
3
3
4
4 4
4
3
1 2
3
2
22
2
3
33
3
I Naive estimate is NOT valid with Λ0 independent of refinement level
#T∗ −#T ≤ Λ0 #M
I Complexity of REFINE (Binev, Dahmen, DeVore ’04 (d = 2), andStevenson’ 07 (d > 2)): If T0 has a suitable labeling, then there existsa constant Λ0 > 0 only depending on T0 and d such that for all k ≥ 1
#Tk −#T0 ≤ Λ0
k−1∑j=0
#Mj .
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Complexity of REFINE
I Recursive bisection of T3 (sequence of compatible bisection patches)
3
1 2
3
2
2
2 2
1
3
1 2
3
22
2
3
3
3
3
4
4 4
4
3
1 2
3
2
22
2
3
33
3
I Naive estimate is NOT valid with Λ0 independent of refinement level
#T∗ −#T ≤ Λ0 #M
I Complexity of REFINE (Binev, Dahmen, DeVore ’04 (d = 2), andStevenson’ 07 (d > 2)): If T0 has a suitable labeling, then there existsa constant Λ0 > 0 only depending on T0 and d such that for all k ≥ 1
#Tk −#T0 ≤ Λ0
k−1∑j=0
#Mj .
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Complexity of REFINE
I Recursive bisection of T3 (sequence of compatible bisection patches)
3
1 2
3
2
2
2 2
1
3
1 2
3
22
2
3
3
3
3
4
4 4
4
3
1 2
3
2
22
2
3
33
3
I Naive estimate is NOT valid with Λ0 independent of refinement level
#T∗ −#T ≤ Λ0 #M
I Complexity of REFINE (Binev, Dahmen, DeVore ’04 (d = 2), andStevenson’ 07 (d > 2)): If T0 has a suitable labeling, then there existsa constant Λ0 > 0 only depending on T0 and d such that for all k ≥ 1
#Tk −#T0 ≤ Λ0
k−1∑j=0
#Mj .
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Piecewise Polynomial Interpolation
Quasi-local error estimate: if 0 ≤ t ≤ s ≤ n + 1 (n ≥ 1 polynomialdegree) and 1 ≤ p, q ≤ ∞ satisfy sob(W s
p ) > sob(W tq ), then for all
T ∈ T
‖Dt(v − IT v)‖Lq(T ) . hsob(W s
p )−sob(W tq )
T ‖Dsv‖Lp(NT (T )),
where NT (T ) is a discrete neighborhood of T and IT is a quasiinterpolation operator (Clement or Scott-Zhang). If sob(W s
p ) > 0, then vis Holder continuous, IT can be replaced by the Lagrange interpolationoperator, and NT (T ) = T .
• Quasi-uniform meshes: if 1 ≤ s ≤ n + 1 and u ∈ Hs(Ω), then
‖∇(v − IT v)‖L2(Ω) 4 |v|Hs(Ω)(#T )−s−1
d .
• Optimal error decay: If s = n + 1 (linear Sobolev scale), then
‖∇(v − IT v)‖L2(Ω) 4 |v|Hn+1(Ω)(#T )−nd .
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Piecewise Polynomial Interpolation
Quasi-local error estimate: if 0 ≤ t ≤ s ≤ n + 1 (n ≥ 1 polynomialdegree) and 1 ≤ p, q ≤ ∞ satisfy sob(W s
p ) > sob(W tq ), then for all
T ∈ T
‖Dt(v − IT v)‖Lq(T ) . hsob(W s
p )−sob(W tq )
T ‖Dsv‖Lp(NT (T )),
where NT (T ) is a discrete neighborhood of T and IT is a quasiinterpolation operator (Clement or Scott-Zhang). If sob(W s
p ) > 0, then vis Holder continuous, IT can be replaced by the Lagrange interpolationoperator, and NT (T ) = T .
• Quasi-uniform meshes: if 1 ≤ s ≤ n + 1 and u ∈ Hs(Ω), then
‖∇(v − IT v)‖L2(Ω) 4 |v|Hs(Ω)(#T )−s−1
d .
• Optimal error decay: If s = n + 1 (linear Sobolev scale), then
‖∇(v − IT v)‖L2(Ω) 4 |v|Hn+1(Ω)(#T )−nd .
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Lower A Posteriori Bound (Continued)
• Oscillation of r: hT ‖r − rT ‖L2(T ) with meanvalue rT . Then
hT ‖r‖L2(T ) 4 ‖R‖H−1(T ) + hT ‖r − rT ‖L2(T )
• Data oscillation: if A is pw constant, then r = f and
hT ‖r − rT ‖L2(T ) = hT ‖f − fT ‖L2(T ) = oscT (f, T )
• Oscillation of j: likewise hS‖j − jS‖L2(S) with meanvalue jS and
h1/2S ‖j‖L2(S) 4 ‖R‖H−1(ωS) + h
1/2S ‖j − jS‖L2(S) + hS‖r‖L2(ωS)
where ωS = T1 ∪ T2 with T1 ∩ T2 = S and T1, T2 ∈ T .
• Local lower bound: let ωT = ∪S∈∂T ωS and the local oscillation beoscT (U, ωT ) := ‖h(r − r)‖L2(ωT ) + ‖h1/2(j − j)‖L2(∂T ). Then
ET (U, T ) 4 α2‖∇(u− U)‖L2(ωT ) + oscT (U, ωT ).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Lower A Posteriori Bound (Continued)
• Oscillation of r: hT ‖r − rT ‖L2(T ) with meanvalue rT . Then
hT ‖r‖L2(T ) 4 ‖R‖H−1(T ) + hT ‖r − rT ‖L2(T )
• Data oscillation: if A is pw constant, then r = f and
hT ‖r − rT ‖L2(T ) = hT ‖f − fT ‖L2(T ) = oscT (f, T )
• Oscillation of j: likewise hS‖j − jS‖L2(S) with meanvalue jS and
h1/2S ‖j‖L2(S) 4 ‖R‖H−1(ωS) + h
1/2S ‖j − jS‖L2(S) + hS‖r‖L2(ωS)
where ωS = T1 ∪ T2 with T1 ∩ T2 = S and T1, T2 ∈ T .
• Local lower bound: let ωT = ∪S∈∂T ωS and the local oscillation beoscT (U, ωT ) := ‖h(r − r)‖L2(ωT ) + ‖h1/2(j − j)‖L2(∂T ). Then
ET (U, T ) 4 α2‖∇(u− U)‖L2(ωT ) + oscT (U, ωT ).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Lower A Posteriori Bound (Continued)
• Oscillation of r: hT ‖r − rT ‖L2(T ) with meanvalue rT . Then
hT ‖r‖L2(T ) 4 ‖R‖H−1(T ) + hT ‖r − rT ‖L2(T )
• Data oscillation: if A is pw constant, then r = f and
hT ‖r − rT ‖L2(T ) = hT ‖f − fT ‖L2(T ) = oscT (f, T )
• Oscillation of j: likewise hS‖j − jS‖L2(S) with meanvalue jS and
h1/2S ‖j‖L2(S) 4 ‖R‖H−1(ωS) + h
1/2S ‖j − jS‖L2(S) + hS‖r‖L2(ωS)
where ωS = T1 ∪ T2 with T1 ∩ T2 = S and T1, T2 ∈ T .
• Local lower bound: let ωT = ∪S∈∂T ωS and the local oscillation beoscT (U, ωT ) := ‖h(r − r)‖L2(ωT ) + ‖h1/2(j − j)‖L2(∂T ). Then
ET (U, T ) 4 α2‖∇(u− U)‖L2(ωT ) + oscT (U, ωT ).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Lower A Posteriori Bound (Continued)
• Higher order: we expect oscT (U, ωT ) ‖∇(u−U)‖L2(ωT ) as hT → 0.
• Marking: if ET (U, T ) 4 ‖∇(u− U)‖L2(ωT ) and ET (U, T ) is largerelative to ET (U), then T contains a large portion of the error. Toimprove the solution U effectively, such T must be split giving rise toa procedure that tries to equidistribute errors.
• Global lower bound: we have ET (U) 4 α2‖u−U‖V + oscT (U) where
oscT (U) = ‖h(r − r)‖L2(Ω) + ‖h1/2(j − j)‖L2(Γ).
• Discrete local lower bound (Dorfler’96, Morin, N, Siebert’00):
ET (U, T ) 4 α2‖∇(U∗ − U)‖L2(ωT ) + oscT (U, ωT ).
provided the interior of T and each of its sides contain a node ofT∗ ≥ T (interior node property).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Lower A Posteriori Bound (Continued)
• Higher order: we expect oscT (U, ωT ) ‖∇(u−U)‖L2(ωT ) as hT → 0.
• Marking: if ET (U, T ) 4 ‖∇(u− U)‖L2(ωT ) and ET (U, T ) is largerelative to ET (U), then T contains a large portion of the error. Toimprove the solution U effectively, such T must be split giving rise toa procedure that tries to equidistribute errors.
• Global lower bound: we have ET (U) 4 α2‖u−U‖V + oscT (U) where
oscT (U) = ‖h(r − r)‖L2(Ω) + ‖h1/2(j − j)‖L2(Γ).
• Discrete local lower bound (Dorfler’96, Morin, N, Siebert’00):
ET (U, T ) 4 α2‖∇(U∗ − U)‖L2(ωT ) + oscT (U, ωT ).
provided the interior of T and each of its sides contain a node ofT∗ ≥ T (interior node property).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Lower A Posteriori Bound (Continued)
• Higher order: we expect oscT (U, ωT ) ‖∇(u−U)‖L2(ωT ) as hT → 0.
• Marking: if ET (U, T ) 4 ‖∇(u− U)‖L2(ωT ) and ET (U, T ) is largerelative to ET (U), then T contains a large portion of the error. Toimprove the solution U effectively, such T must be split giving rise toa procedure that tries to equidistribute errors.
• Global lower bound: we have ET (U) 4 α2‖u−U‖V + oscT (U) where
oscT (U) = ‖h(r − r)‖L2(Ω) + ‖h1/2(j − j)‖L2(Γ).
• Discrete local lower bound (Dorfler’96, Morin, N, Siebert’00):
ET (U, T ) 4 α2‖∇(U∗ − U)‖L2(ωT ) + oscT (U, ωT ).
provided the interior of T and each of its sides contain a node ofT∗ ≥ T (interior node property).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Outline
Piecewise Polynomial Interpolation in Sobolev Spaces
Model Problem and FEM
FEM: A Posteriori Error Analysis
Surveys
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto
Outline Polynomial Interpolation Model Problem A Posteriori Error Analysis Surveys
Surveys
• R.H. Nochetto Adaptive FEM: Theory and Applications toGeometric PDE, Lipschitz Lectures, Haussdorff Center forMathematics, University of Bonn (Germany), February 2009 (seewww.hausdorff-center.uni-bonn.de/event/2009/lipschitz-nochetto/).
• R.H. Nochetto, K.G. Siebert and A. Veeser, Theory ofadaptive finite element methods: an introduction, in Multiscale,Nonlinear and Adaptive Approximation, R. DeVore and A. Kunoth eds,Springer (2009), 409-542.
• R.H. Nochetto and A. Veeser, Primer of adaptive finite elementmethods, in Multiscale and Adaptivity: Modeling, Numerics andApplications, CIME Lectures, eds R. Naldi and G. Russo, Springer (toappear).
Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Ricardo H. Nochetto