Unified A Posteriori Error Control
for all Nonstandard Finite Elements1
Martin Eigel
C. Carstensen, C. Löbhard, R.H.W. Hoppe
Humboldt-Universität zu Berlin
19.05.2010
1
we know of
Guidelines for Applicants
M. Eigel (Humboldt)
Unified ErrorThese
Analysis
EFEF,
Warwick
2010 applying
1 / to
22the
guidelines should help you
as you
think about
Model Example
Flux or stress field p in equilibrium equation g + div p = 0 is approximated
by piecewise constant p` and yields equilibrium residual
Z
Res(v ) := (g · v − p` : D` v ) dx
for v ∈ V := H01 (Ω; Rm )
Ω
Endow V with norm kv kV := kDv kL2 (Ω) s.t. V ∗ ≈ H −1 (Ω) and
Res(v )
v 6=0 kv kV
k div(p − p` )kV ∗ = k Res kV ∗ := sup
Estimation by edge residuals ηE := |E |(p` |T+ · νT+ + p` |T− · νT− ) on each
interior edge E = ∂T+ ∩ ∂T− with outer unit normals νT±
!1/2
p` |T+
X
νT+
2
νT
ηE :=
|ηE |
−
E
E ∈E
M. Eigel (Humboldt)
Unified Error Analysis
p` |T
−
EFEF, Warwick 2010
2 / 22
Estimation of Equilibrium Error
Up to data oscillation osc(g , ·), edge estimator ηE is reliable & efficient
k Res kV ∗ ≈ ηE ± osc(g , elements ∪ edges)
Remark: Edge residual estimator is equivalent to many others
(cf. Ainsworth/Oden, Babuška/Strouboulis, Verfürth)
Example: For triangulation of Ω ⊂ R2 and first-order conforming or
nonconforming FEM
p` := D` u`
M. Eigel (Humboldt)
and V`c := P1 (T` ; Rm ) ∩ V ⊂ ker Res
Unified Error Analysis
EFEF, Warwick 2010
3 / 22
Schemes & Applications
CR
CR
BS
Wilson
KS
KS
Han
Han
Zhang
NR (M)
NR (M)
Ming
NR (A)
NR (A)
LLS
CNR
HMS
DSSY
Laplace
CJY
Stokes
M. Eigel (Humboldt)
Unified Error Analysis
HMS
Elasticity
cf. C. Carstensen,
Hu Jun, A. Orlando
EFEF, Warwick 2010
4 / 22
Unified Analysis of. . .
applications:
Laplace, Stokes, Navier-Lamé, Maxwell equations. . .
schemes:
(all?!) conforming, nonconforming, tri/quad, mixed, mortar elements,
dG, hanging nodes. . .
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
5 / 22
Goals of Unified Analysis
Generalise analysis to cover many different discretisation schemes
and applications in one framework
Reduce repetition of similar mathematical arguments and focus on
specific properties/difficulties
No optimal constants but common point of departure and guiding
principles
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
6 / 22
A Unified Approach
Generic Approach
For each Application: Verify kerrork ≈ kresidualk∗
For each Scheme: Determine discrete space V` ⊂ ker(residual)
and design computable lower/upper bounds of kresidualk∗
Topics
Mixed Setting for Unified A Posteriori Error Control
Unified Equilibrium Estimator
Analysis of Consistency Residual
Applications: Poisson, Lamé, Stokes, . . .
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
7 / 22
Mixed Setting for Unified Analysis
Abstract problem formulation:
Given X` × Y` ⊂ X × Y , A : X × Y → (X × Y )∗
` := `X + `Y ∈ (X × Y )∗ , find (x, y ) ∈ X × Y s.t.
A(x, y )(ξ, η) = `(ξ, η)
(PM)
for all (ξ, η) ∈ X × Y
Given a ∈ (X × X )∗ , Λ : Y → X , b ∈ (X × Y )∗ with b(·, v ) := a(Λv , ·)
c ∈ (Y × Y )∗ , A(x, y ) := a(x, ·) − b(·, y ) + b(x, ·) + c(y , ·)
(PM) then reads
a(x, ξ) − b(ξ, y ) = `X (ξ)
for all ξ ∈ X
b(x, η) + c(y , η) = `Y (η)
for all η ∈ Y
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
8 / 22
Mixed Setting for Unified Analysis
Abstract problem formulation:
Given X` × Y` ⊂ X × Y , A : X × Y → (X × Y )∗
` := `X + `Y ∈ (X × Y )∗ , find (x, y ) ∈ X × Y s.t.
A(x, y )(ξ, η) = `(ξ, η)
(PM)
for all (ξ, η) ∈ X × Y
Given a ∈ (X × X )∗ , Λ : Y → X , b ∈ (X × Y )∗ with b(·, v ) := a(Λv , ·)
c ∈ (Y × Y )∗ , A(x, y ) := a(x, ·) − b(·, y ) + b(x, ·) + c(y , ·)
(PM) then reads
a(x, ξ) − b(ξ, y ) = `X (ξ)
for all ξ ∈ X
b(x, η) + c(y , η) = `Y (η)
for all η ∈ Y
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
8 / 22
Errors & Residuals in Unified Analysis
Given approx. (x` , ỹ` ) ∈ X` × Y to (x, y ), define Res := ResX + ResY
(consistency )
ResX := `X − a(x` , ·) + b(·, ỹ` ) = `X − a(x` − Λỹ` )
(equilibrium)
ResY := `Y − b(x` , ·) − c(ỹ` , ·)
Remarks:
ỹ` ∈ Y close to y` , not necessarily discrete
ResX involves piecewise gradient D` (y` − ỹ` ) of y` − ỹ` ∈
/Y
Since A isomorphism, kerrork ≈ kresidualk∗ , i.e.
kx − x` kX + ky − ỹ` kY ≈ k ResX kX ∗ + k ResY kY ∗
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
9 / 22
Errors & Residuals in Unified Analysis
Given approx. (x` , ỹ` ) ∈ X` × Y to (x, y ), define Res := ResX + ResY
(consistency )
ResX := `X − a(x` , ·) + b(·, ỹ` ) = `X − a(x` − Λỹ` )
(equilibrium)
ResY := `Y − b(x` , ·) − c(ỹ` , ·)
Remarks:
ỹ` ∈ Y close to y` , not necessarily discrete
ResX involves piecewise gradient D` (y` − ỹ` ) of y` − ỹ` ∈
/Y
Since A isomorphism, kerrork ≈ kresidualk∗ , i.e.
kx − x` kX + ky − ỹ` kY ≈ k ResX kX ∗ + k ResY kY ∗
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
9 / 22
[Preparation]
Generic Equilibrium Error Analysis
Two discrete spaces V`c ⊂ V and V`nc ⊂ H 1 (T` ; Rm ) with
J
Π
V −→ V`c −→ V`nc
(H1) ∃ H 1 -stable Clément-type operator J : V → V`c into (conforming)
subspace V`c ⊆ V with first-order approximation property
(H2) V`c and V`nc piecewise smooth w.r.t. shape-regular T`
(H3) For p` ∈ L2 (Ω; Rm×n ) ∃ Π : V`c → V`nc s.t. ∀v` ∈ V`c ∀T ∈ T`
kD(Πv` )kL2 (T ) . kDv` kL2 (ωT )
Z
Z
v` dx =
Πv` dx
T
T
Z
Z
p` : D` v` dx =
p` : D` (Πv` ) dx
Ω
M. Eigel (Humboldt)
Ω
Unified Error Analysis
EFEF, Warwick 2010
10 / 22
[Result]
Generic Equilibrium Error Analysis
S
Thm.[CEHL10 + ]: Suppose RT ∈ L2 (T` ), RE ∈ L2 ( E` ) and
Res : V`nc + V → R reads
Z
Z
Res(v ) :=
RT · v dx + S RE · hv i ds
E`
Ω
Suppose (H1)-(H3) and V`nc ⊂ ker Res
Then

1/2
X
1/2
η` := 
hE kRE k2L2 (E )  = khE RE kL2 (S E` )
E ∈E`
is reliable and efficient in the sense
η` − osc(RT , T` ) − osc(RE , E` ) . k Res kV ∗ . η` + osc(RT , {ωz : z ∈ K` })
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
11 / 22
[Result]
Generic Equilibrium Error Analysis
S
Thm.[CEHL10 + ]: Suppose RT ∈ L2 (T` ), RE ∈ L2 ( E` ) and
Res : V`nc + V → R reads
Z
Z
Res(v ) :=
RT · v dx + S RE · hv i ds
E`
Ω
Suppose (H1)-(H3) and V`nc ⊂ ker Res
Then

1/2
X
1/2
η` := 
hE kRE k2L2 (E )  = khE RE kL2 (S E` )
E ∈E`
is reliable and efficient in the sense
η` − osc(RT , T` ) − osc(RE , E` ) . k Res kV ∗ . η` + osc(RT , {ωz : z ∈ K` })
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
11 / 22
Analysis of Consistency Error
Thm.[CEHL10 + ]:
Given X = L2 (Ω), x` = D` y` ∈ X` = P0 (T` ; Rn ),
Y = H01 (Ω), y` ∈ Y` = CR1 (T` )
and
µ` := minη∈Y kx` − DηkL2 (Ω)
µ` ≈
≈
min
kx` − D` η` kL2 (Ω)
min
khT−1 (y` − η` )kL2 (Ω)
η` ∈P1 (T` )∩Y
η` ∈P1 (T` )∩Y
≈ khT−1 (y` − Sy` )kL2 (Ω)
−1/2
≈ khE
[y` ]kL2 (S E` )
1/2
≈ khE [D` y` · τE ]kL2 (S E` )
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
12 / 22
Analysis of Consistency Error
Thm.[CEHL10 + ]:
Given X = L2 (Ω), x` = D` y` ∈ X` = P0 (T` ; Rn ),
Y = H01 (Ω), y` ∈ Y` = CR1 (T` )
and
µ` := minη∈Y kx` − DηkL2 (Ω)
µ` ≈
≈
min
kx` − D` η` kL2 (Ω)
min
khT−1 (y` − η` )kL2 (Ω)
η` ∈P1 (T` )∩Y
η` ∈P1 (T` )∩Y
≈ khT−1 (y` − Sy` )kL2 (Ω)
−1/2
≈ khE
[y` ]kL2 (S E` )
1/2
≈ khE [D` y` · τE ]kL2 (S E` )
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
12 / 22
[Application]
Laplace Equation
Poisson model problem:
∆u = g in Ω
and
u = 0 on ∂Ω
leads to primal mixed formulation with
Z
Z
a(p, q) :=
p · q dx, Λu := D` u, b(q, u) =
D` u · q dx
Ω
Ω
Given g ∈ L2 (Ω), find (p, u) ∈ X × Y := L2 (Ω; Rn ) × H01 (Ω) s.t.
A(p, u)(q, v ) = (g , v )L2 (Ω)
for all (q, v ) ∈ X × Y
Since A isomorphism
kp − p` kL2 + ku − ũ` kH 1 ≈ k ResX kX ∗ + k ResY kY ∗
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
13 / 22
[Application (cont.)]
Laplace Equation
Treatment of different schemes in the unified framework:
Conforming FEM Discrete solution u` = ũ` ∈ Y defines discrete flux
p` := Λu` = Du` . Consistency error vanishes, equilibrium
residual treated as suggested previously.
Nonconforming FEM Discrete solution u` defines discrete flux p` := D` u` .
Same analysis for equilibrium residual while consistency
residual kp` − D ũ` kL2 (Ω) can be bounded as before.
Mixed FEM Discrete solution is discrete flux p` and u` ∈
/ Y is Lagrange
multiplier. Equilibrium residual reduces to osc(g , elements),
consistency residual minũ` ∈H 1 (Ω) kp` − D ũ` kL2 (Ω) can be
0
bounded as in [Carstensen, Math. Comp. 1997].
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
14 / 22
[Application]
Stokes Equations
R
R
Let a(u, v ) := − Ω uv dx, Λu := div u, c(u, v ) := Ω 2µε(u) : ε(v ) dx
R
`X (p, q) := − Ω pq dx and X := L20 (Ω; Rn ), Y := H01 (Ω; Rn )
With ε(v ) := symDv and deviatoric operator dev σ := σ − n1 (tr(σ))1
the linear operator A : X × Y → (X × Y )∗ reads
A(σ, u)(τ, v ) :=
1
(dev σ, dev τ )L2 (Ω) − (σ, ε(v ))L2 (Ω) − (τ, ε(u))L2 (Ω)
2µ
A isomorphism. . .
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
15 / 22
[Application (cont.)]
Stokes Equations
Stokes problem: Given g ∈ L2 (Ω; Rn ), find (σ, u) ∈ X × Y s.t.
A(σ, u)(τ, v ) = (g , v )L2 (Ω)
for all (τ, v ) ∈ X × Y
Conforming or nonconforming FEM yield u` and p` with
σ = 2µε(u) − p1 and σ` = 2µε` (u` ) − p` 1
Error control: Given FE solution (σ` , u` ) to (σ, u), then
kσ − σ` kL2 (Ω) . min 2µkε` (u` − ũ` )kL2 (Ω) + k ResY kY ∗ + k div` u` kL2 (Ω)
ũ` ∈Y
Examples: all known conforming and nonconforming FEM
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
16 / 22
[Application]
Linear Elasticity
Fourth-order elasticity tensor for λ, µ > 0,
Cτ := λ tr(τ )1 + 2µτ
for all τ ∈ Rn×n
R
Let a(σ, τ ) := Ω (C−1 σ) : τ dx, Λu := Cε(u) and
1
n
X := L2 (Ω; Rn×n
sym ), Y := H0 (Ω; R )
The linear operator A : X × Y → (X × Y )∗ then reads
A(σ, u)(τ, v ) := (C−1 σ, τ )L2 (Ω) − (σ, ε(v ))L2 (Ω) − (τ, ε(u))L2 (Ω)
A isomorphism, λ-independent operator norms of A and A−1
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
17 / 22
[Application (cont.)]
Linear Elasticity
Navier-Lamé problem: Given g ∈ L2 (Ω; Rn ), find (σ, u) ∈ X × Y s.t.
A(σ, u)(τ, v ) = (g , v )L2 (Ω)
for (τ, v ) ∈ X × Y
For conforming or nonconforming FEM, σ = Cε(u) and σ` = Cε` (u` )
Error control: Given FE solution (σ` , u` ) to (σ, u),
kσ − σ` kL2 (Ω) . min kε` (u` − ũ` )kL2 (Ω) + k ResY kY ∗
ũ` ∈Y
is robust for λ → ∞
Examples: Conforming FEM, Kouhia&Stenberg, PEERS
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
18 / 22
Conclusions for Unified Analysis
Framework for unified a posteriori analysis covers large class of
applications & discretisation schemes
Similarities are exposed and encountered obstacles/challenges guide
development of specific error estimators
Approach doesn’t strive for most accurate/efficient estimators but
rather provides an initial line of attack for as many problems as
possible
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
19 / 22
Wrap-Up of Unified A Posteriori Analysis
C. Carstensen: A unifying theory of a posteriori finite element error
control (Numer. Math. 2005)
C. Carstensen, Jun Hu, A. Orlando: Framework for the a posteriori
error analysis of nonconforming finite elements (SINUM 2007)
C. Carstensen, Jun Hu: A Unifying Theory of A Posteriori Error
Control for Nonconforming FEM (Numer. Math. 2007)
C. Carstensen, ME, R.H.W. Hoppe, C. Löbhard: A Unifying Theory of
A Posteriori Error Control (2010 + )
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
20 / 22
Extensions of Unified Analysis
Unified Analysis has also been applied/extended to
Maxwell Equations [C. Carstensen, R.H.W. Hoppe 2009]
Mortar FEM and higher order FEM
dG FEM [C. Carstensen, T. Gudi, M. Jensen 2009]
Hanging nodes [C. Carstensen, Jun Hu 2008]
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
21 / 22
References
C. Carstensen, ME, R.H.W. Hoppe, C. Löbhard: Unified A Posteriori
Error Control, in preparation
C. Carstensen: A unifying theory of a posteriori finite element error
control, Numer. Math. 100 (2005) 617–637
C. Carstensen, Jun Hu, A. Orlando: Framework for the a posteriori
error analysis of nonconforming finite elements, SINUM 45, 1 (2007)
C. Carstensen, Jun Hu: A Unifying Theory of A Posteriori Error
Control for Nonconforming Finite Element Methods, Numer. Math.
(2007), DOI 10.1007/s00211-007-0068-z
C. Carstensen, T. Gudi, M. Jensen: A unifying theory of a posteriori
error control for discontinuous Galerkin FEM
C. Carstensen, R.H.W. Hoppe: Unified framework for an a posteriori
analysis of non-standard finite element approximations of
H(curl)-elliptic problems
M. Eigel (Humboldt)
Unified Error Analysis
EFEF, Warwick 2010
22 / 22