Goal-oriented adaptive finite element methods for elliptic
PDE
Numerical comparison of methods
Sara Pollock
Collaborators: Michael Holst, Yunrong Zhu
Texas A&M Department of Mathematics
February 28, 2014
S. Pollock
Texas A&M Mathematics
1/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Goal oriented methods
Given a linear or nonlinear second order elliptic problem in weak form: Find u ∈ H01 (Ω)
such that
hA∇u , ∇v i + hb(u , ∇u ) + cu , v i = f (v )
we may be interested in a function of the solution, g (u ), called the quantity of interest.
The function g (·) may be a (weighted) average over a subdomain or a line integral
about its boundary. Goal-oriented methods may be used to estimate a physical
quantity, or be used in pointwise a posteriori error estimation using mollification.
Examples to CFD, see [Giles, Süli, 02] or for structures: [Grätsch, Bathe, 2005].
Goal-oriented methods solve a second PDE called the dual whose solution is referred
to as the influence function, generalized Green’s function or the Lagrange multiplier.
Solving the primal-dual system at each iteration allows us to drive the adaptive
refinement to approximate g (u ) with fewer degrees of freedom than we would require
to approximate u.
Here we compare the standard (DWR) adaptive goal-oriented method to adaptive
residual-based methods that can by shown analytically to converge.
S. Pollock
Texas A&M Mathematics
2/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Analytical results
1
[Holst, Pollock, 2011] Convergence of goal-oriented method for nonsymmetric
linear problems
|g (u ) − g (uh )| ≤ 2|||u − uh ||||||z − zh ||| ≤ |||u − uh |||2 + |||z − zh |||2
with strong contraction shown for each of the quasi-errors, Qk and QkD
|||u − uk |||2 + γp η2h (uh ) ≤ (αk Q0 )2 and |||z − zh |||2 + γd ζ2h (zh ) ≤ αk (Q0D )2 , α < 1.
2
[Holst, Pollock, Zhu, 2012] Convergence of goal-oriented method for semilinear
problems
|g (u ) − g (uh )| ≤ C |||u − uh |||2 + |||ẑ − ẑh |||2
with strong contraction shown for the combined quasi-error, Q̃k
|||ẑ − ẑh |||2 + γ ζ2h (ẑh ) + τ |||u − uh |||2 + τ γp η2h (uh ) ≤ (Q̃k )2 , α < 1.
S. Pollock
Texas A&M Mathematics
3/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Problem class
A : Ω → Rd ×d , Lipschitz, and a.e. symmetric positive-definite and f , g ∈ L2 (Ω).
Semilinear problem: hA∇u , ∇v i + hb(u ), v i = f (v )
I b : Ω × R → R is smooth in the second argument. For simplicity, we write b (u )
instead of b(x , u ). Moreover, we assume that b is monotone (increasing):
b0 (ξ) ≥ 0, for all ξ ∈ R.
Linear problem: hA∇u , ∇v i + hb · ∇u + cu , v i = f (v )
I b : Ω → Rd , with b ∈ L (Ω) , and b divergence-free.
k
∞
I c : Ω → R, with c ∈ L (Ω), and c (x ) ≥ 0 for all x ∈ Ω.
∞
In addition, for the semilinear problem we require the solution to the primal problems
satisfies L∞ bounds. Such bounds have been established (see
[Bank, Holst, Szypowski, and Zhu, 2011]) either conditions on mesh angles or the
growth condition on b, |b(n) (ξ) ≤ K |, ∀ξ ∈ R for some K > 0, and for some
1 ≤ n ≤ (d + 2)/(d − 2) for d ≥ 3 or 1 ≤ n ≤ ∞ for d = 2.
Then there are u− , u+ ∈ L∞ which satisfy
u− (x ) < u (x ), uh (x ) ≤ u+ (x ) for almost every x ∈ Ω,
and we have b0 is Lipschitz on [u− , u+ ] ∩ H01 (Ω) for a.e. x ∈ Ω.
S. Pollock
Texas A&M Mathematics
4/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
The adaptive loop
We will compare numerical results using three different adaptive algorithms
1
HPZ: the algorithm for which we established convergence for linear nonsymmetric
and semilinear problems ([Holst, Pollock, 2011] and [Holst, Pollock, Zhu, 2012] ).
2
MS: the GOAFEM algorithm for which convergence and optimality results are
shown for the scaled Laplacian, shown in [Mommer, Stevenson, 2009].
3
DWR: the standard algorithm for GOAFEM, discussed in, e.g.,
[Becker, Rannacher, 2001, Prudhomme, Oden, 1999, Estep, Holst, Larson, 2002,
Grätsch, Bathe, 2005, Carey, Estep, Tavener, 2009].
The GOAFEM algorithm is based on the standard AFEM iterative loop:
SOLVE → ESTIMATE → MARK → REFINE .
Solve: In practice the linear problem may be solved by a standard iterative solver.
The nonlinear problem may be solved by a standard inexact Newton + multilevel
algorithm. Here we assume the exact solution to each problem
..
.
Refine: In our analysis, the refinement (including the completion) is performed
according to newest vertex bisection.
S. Pollock
Texas A&M Mathematics
5/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Estimate
HPZ, MS: The finite element space for both primal and dual problems:
VT := H01 (Ω) ∩
∏ P1 (T )
and Vk := VTk .
T ∈T
HPZ, MS:
SOLVE →
ESTIMATE → MARK → REFINE .
Estimate: We use an error indicator derived from the strong-form of the residual,
given elementwise as
η2T (v , T ) := hT2 kR (v )k2L2 (T ) + hT kJT (v )k2L2 (∂T ) ,
v ∈ VT ,
with R (v ) = f − L (v ) = f + ∇ · (A∇u ) − b · ∇u − cu, for linear problems. The dual
residuals are defined analogously. The jump residual for primal and dual problems is
JT (v ) := J[A∇v ] · nK∂T and JφK∂T := lim φ(x + tn) − φ(x − tn)
t →0
where n is the the outward normal defined piecewise on ∂T . The error estimator is the
l2 sum of indicators. For the Galerkin solution uk we use the notation
η2k =
∑
η2T (uk , T ).
T ∈Tk
S. Pollock
Texas A&M Mathematics
6/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
DWR: Estimate
DWR: The finite element space
VT := H01 (Ω) ∩
∏ P1 (T )
and Vk := VTk .
T ∈T
is used for the primal problem. The enriched space
V2T := H01 (Ω) ∩
∏ P2 (T )
T ∈T
and V2k := V2Tk .
is used for the dual.
DWR:
SOLVE →
ESTIMATE → MARK → REFINE .
Estimate: The DWR indicator estimates the influence of the dual solution on the
primal residual. Elementwise
1
ηTk (v , T ) := hR (v ), z 2 − Ik z 2 iT + hJT (v ), z 2 − Ik z 2 i∂T ,
2
The error estimator is the absolute value of the sum of indicators.
v ∈ VTk
ηk = ∑ ηT (uk , T ) ≤ ∑ |ηT (uk , T )|
T ∈T
T ∈T
k
S. Pollock
Texas A&M Mathematics
k
7/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Mark
SOLVE → ESTIMATE →
MARK → REFINE .
The mesh is marked at each iteration using the Dörfler stategy with respect to each
indicator: Given θ ∈ (0, 1)
Mark a set M ⊂ Tk such that,
∑
η2k (uk , T ) ≥ θ2 η2k (uk , Tk )
T ∈M
HPZ: Let M = Mp ∪ Md the union of sets found for the primal and dual
problems respectively.
MS: Let M the set of lesser cardinality between Mp and Md . In our
implementation, M = Mp if the cardinalities are equal: #Mp = #Md .
DWR: Let M the marked set using the DWR indicator.
Notice: this strategy does not account for cancellation error with the DWR estimator,
which gives a signed quantity on each element, whereas the residual estimators are
always positive.
S. Pollock
Texas A&M Mathematics
8/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Linear Test Problems
A convection dominated problem on the domain Ω = (0, 1)2 \ (1/3, 2/3)2 .
a(u , v ) :=
1
1000
h∇u , ∇v i + hb · ∇u , v i = f (v ),
with b = (y , 21 − x )T . The goal function g (u ) =
R
Ω g (x , y )u (x , y )
g (x , y ) = 500 exp(−500((x − 0.9)2 + (y − 0.675)2 ))
and f (v ) chosen so the exact solution u is of the form
u = sin(ωπx ) sin(ωπy )((x − x0 )2 + (y − y0 )2 + 10−3 )−1 .
Four sets of problem parameters:
ω = 3,
(x0 , y0 ) = (0.1, 0.1)
(P1)
ω = 3,
(x0 , y0 ) = (0.1, 0.7)
(P2)
ω = 3,
(x0 , y0 ) = (0.7, 0.7)
(P3)
ω = 3,
(x0 , y0 ) = (0.7, 0.1)
(P4)
The initial mesh has 128 elements.
S. Pollock
Texas A&M Mathematics
9/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Numerics: P1
u = sin(3πx ) sin(3πy )((x − 0.1)2 + (y − 0.1)2 + 10−3 )−1
Finite element mesh
Goal Error Reduction
4
Finite element mesh
10
−1
n
HP
MS
DWR
3
10
2
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
2
10
3
10
4
10
5
10
Figure: Goal error after 18 HPZ (blue), 36 MS (red) and 19 DWR (green) iterations, compared
with n−1 .
The mesh on the left is plotted after 13 iterations of HPZ (2728 elements) and the
mesh on the right shows 14 iterations of DWR with (2821 elements).
S. Pollock
Texas A&M Mathematics
10/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Numerics: P2
u = sin(3πx ) sin(3πy )((x − 0.1)2 + (y − 0.7)2 + 10−3 )−1
Finite element mesh
Goal Error Reduction
3
Finite element mesh
10
−1
n
HP
MS
DWR
2
10
1
10
0
10
−1
10
−2
10
−3
10
2
10
3
10
4
10
5
10
Figure: Goal error after 18 HPZ (blue), 36 MS (red) and 18 DWR (green) iterations, compared
with n−1 .
The mesh on the left is plotted after 15 iterations of HPZ (8667 elements) and the
mesh on the right shows 15 iterations of DWR (8621 elements).
S. Pollock
Texas A&M Mathematics
11/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Numerics: P3
u = sin(3πx ) sin(3πy )((x − 0.7)2 + (y − 0.7)2 + 10−3 )−1
Finite element mesh
Goal Error Reduction
1
Finite element mesh
10
−1
n
HP
MS
DWR
0
10
−1
10
−2
10
−3
10
−4
10
2
10
3
10
4
10
5
10
Figure: Goal error after 18 HPZ (blue), 36 MS (red) and 17 DWR (green) iterations, compared
with n−1 .
The mesh on the left is plotted after 14 iterations of HPZ (5332) elements and the
mesh on the right shows 13 iterations of DWR (5292 elements).
S. Pollock
Texas A&M Mathematics
12/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Numerics: P4
u = sin(3πx ) sin(3πy )((x − 0.7)2 + (y − 0.1)2 + 10−3 )−1
Finite element mesh
Goal Error Reduction
1
Finite element mesh
10
0
10
−1
10
−2
10
−3
10
−4
10
2
10
3
10
4
10
5
10
Figure: Goal error after 17 HPZ (blue), 35 MS (red) and 17 DWR (green) iterations, compared
with n−1 .
The mesh on the left is plotted after 10 iterations of HPZ (2845 elements) and the
mesh on the right shows 10 iterations of DWR (2732 elements).
S. Pollock
Texas A&M Mathematics
13/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
Numerical results: linear convection-diffusion
primal spike interacts
with dual solution
primal and dual data spikes
close
primal and dual data
spikes far
(P3) HPZ most efficient,
MS in the middle; (P4)
residual and DWR indicators comparable
(P2) residual indicators
more efficient
primal spike remote
from dual solution
(P1) residual indicators
and DWR comparable
Even where results are comparable, they are achieved for qualitatively different mesh
refinements.
S. Pollock
Texas A&M Mathematics
14/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014
thank you!
thank you!
S. Pollock
Texas A&M Mathematics
15/15
Convergence of GOAFEM
Finite Element Rodeo February 28, 2014