Proceedings in Applied Mathematics and Mechanics, 31 May 2016
Convergence Analysis of the Dirichlet-Neumann Iteration for Finite
Element Discretizations
Azahar Monge1,∗ and Philipp Birken1
1
Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund
We analyze the convergence rate of the Dirichlet-Neumann iteration for the fully discretized unsteady transmission problem.
Specifically, we consider the coupling of two linear heat equations on two identical non overlapping domains with jumps in
the material coefficients across these. In this context, we derive the iteration matrix of the coupled problem. In the 1D case,
the spectral radius of the iteration matrix tends to the ratio of heat conductivities in the semidiscrete spatial limit, but to the
ratio of the products of density and specific heat capacity in the semidiscrete temporal one. This explains the fast convergence
previously observed for cases with strong jumps in the material coefficients.
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1
Model problem
The unsteady transmission problem is as follows, where we consider a domain Ω ⊂ Rd which is cut into two subdomains
Ω = Ω1 ∪ Ω2 with transmission conditions at the interface Γ = Ω1 ∩ Ω2 :
αm
∂um (x, t)
− ∇ · (λm ∇um (x, t)) = 0, t ∈ [t0 , tf ], x ∈ Ωm ⊂ Rd , m = 1, 2,
∂t
um (x, t) = 0, t ∈ [t0 , tf ] x ∈ ∂Ωm \Γ,
u1 (x, t) = u2 (x, t), x ∈ Γ,
(1)
∂u2 (x, t)
∂u1 (x, t)
λ2
= −λ1
, x ∈ Γ,
∂n2
∂n1
um (x, 0) = u0m (x), x ∈ Ωm .
where nm is the outward normal to Ωm for m = 1, 2, and we consider d = 1, 2.
The constants λ1 and λ2 describe the thermal conductivities of the materials on Ω1 and Ω2 respectively. D1 and D2 represent
with αm = ρm Cm where ρm represents the
the thermal diffusivities of the materials and they are defined by Dm = αλm
m
density and Cm the heat capacity of the material placed in Ωm , m = 1, 2.
1.1
(1)
uI
Discretization
(2)
uI
Let
and
correspond to the unknowns on Ω1 and Ω2 respectively and uΓ correspond to the unknows at the interface
(1)
(2)
Γ, then the compact FEM formulation of (1) for the vector of unknowns u = (uI , uI , uΓ )T will be




(1)
(1)
A1
0
AIΓ
M1
0
MIΓ




(2)
(2)
M̃u̇ − Ãu = 0 where M̃ =  0
 , Ã =  0
 , (2)
M2
MIΓ
A2
AIΓ
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
AΓI AΓI −AΓΓ − AΓΓ
MΓI MΓI MΓΓ + MΓΓ
(m)
(m)
(m)
(m)
(m)
(m)
for the mass matrices Mm , MΓΓ , MIΓ , MΓI and the stiffness matrices Am , AΓΓ , AIΓ , AΓI for m = 1, 2.
Applying the implicit Euler method with time step ∆t to the system (2), we get for the vector of unknowns un+1 =
(1),n+1
(2),n+1
(uI
, uI
, un+1
)T
Γ
Aun+1 = M̃un where A = M̃ − ∆tÃ.
1.2
(3)
Fixed Point Iteration
We now employ a standard Dirichlet-Neumann iteration [3, 4] to solve the discrete system (3), getting in the k-th iteration the
two equation systems
(1),n+1,k+1
(M1 − ∆tA1 )uI
∗
(1)
(1)
(1),n
= −(MIΓ − ∆tAIΓ )un+1,k
+ M1 u I
Γ
(1)
+ MIΓ unΓ ,
(4)
Corresponding author: e-mail azahar.monge@na.lu.se, phone +46 462224757
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2
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Âûk+1 = M̂un − bk ,
(5)
to be solved in succession. Here,
!
(2)
0
M2
MIΓ
and
, M̂ =
 =
(1)
(2)
(1)
(2)
MΓI MΓI MΓΓ + MΓΓ
!
(2),n+1,k+1
0
uI
k+1
k
b =
, û
=
(1)
(1) (1),n+1,k+1
(1)
(1)
(MΓI − ∆tAΓI )uI
+ (MΓΓ + ∆tAΓΓ )un+1,k
un+1,k+1
Γ
Γ
M2 − ∆tA2
(2)
(2)
MΓI − ∆tAΓI
(2)
(2)
MIΓ − ∆tAIΓ
(2)
(2)
MΓΓ + ∆tAΓΓ
!
with some initial condition, here un+1,0
= unΓ . The iteration is terminated according to the standard criterion kuk+1
−ukΓ k ≤ τ
Γ
Γ
where τ is a user defined tolerance [1].
(1),n+1,k+1
(2),n+1,k+1
We now rewrite (4)-(5) as an iteration for un+1
. To this end, we isolate the term uI
from (4) and uI
Γ
from the first equation in (5) and insert them into the second equation in (5). By this, one obtains the iteration un+1,k+1
=
Γ
ΣuΓn+1,k + ψ n , with iteration matrix
Σ = −S(2)
−1 (1)
S
(m)
(m)
(m)
(m)
(m)
(m)
, where S(m) = (MΓΓ +∆tAΓΓ )−(MΓI −∆tAΓI )(Mm −∆tAm )−1 (MIΓ −∆tAIΓ ), (6)
for m = 1, 2 and ψ n contains terms that depend only on the solutions at the previous time step.
Thus, the Dirichlet-Neumann iteration is a linear iteration and the rate of convergence is described by the spectral radius of
the iteration matrix Σ. In the 1D case, Σ can be computed exactly decomposing the matrices (Mm − ∆tAm )−1 by their
eigendecomposition. Then, it can be shown that the limit of the convergence rates when ∆t → 0 is γ := α1 /α2 and the limit
when ∆x → 0 is δ := λ1 /λ2 .
2
Numerical Results
We consider here the thermal interaction between air at 273K with steel at 900K. Physical properties of the materials are
shown in table 1. γ = 3.7434e − 4 and δ = 4.9693e − 4 for the air-steel interaction. Figure 1 shows the 1D rates for the
Table 1: Physical properties of the materials. λ is the thermal conductivity, ρ the density, C the specific heat capacity and α = ρC.
Material
Air
Steel
λ (W/mK)
0.0243
48.9
ρ (kg/m3 )
1.293
7836
α (J/K m3 )
1299.5
3471348
C (J/kgK)
1005
443
air-steel interaction. On the left we plot the rates with respect to the variation of ∆t. On the right we plot the rates for a fixed
∆t and varying ∆x. From figure 1 we can observe that the convergence rates are really fast (factor of ∼ 1e − 4) when there
−3.38
−3
|Σ|
Conv. Rate
γ
−3.39
|Σ|
Conv. Rate
δ
−3.2
log
−3.4
log
−3.4
−3.41
−3.6
−3.42
−3.8
−3.43
−1
−0.5
0
0.5
1
−4
−1.6
−1.4
log(∆t)
(a) The curves are restricted to the discrete values ∆t =
10/40, 2 · 10/40, ..., 40 · 10/40 and ∆x = 1/20.
−1.2
−1
−0.8
−0.6
log(∆x)
(b) The curves are restricted to the discrete values ∆x =
1/3, 1/4, ..., 1/30 and ∆t = 1e8.
Fig. 1: Air-Steel thermal interaction with respect ∆t on the left and ∆x on the right.
exist strong jumps in the coefficient of the materials as observed previously [2].
References
[1] P. Birken, Termination criteria for inexact fixed point methods. Numer. Linear Algebra Appl. 22(4), 702-716 (2015).
[2] P. Birken, T. Gleim, D. Kuhl, and A. Meister. Fast Solvers for Unsteady Thermal Fluid Structure Interaction. Int. J. Numer. Meth. Fluids.
79(1), 16-29 (2015).
[3] A. Quarteroni and A. Valli. Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999).
[4] A. Toselli and O. Widlund. Domain Decomposition Methods - Algorithms and Theory. Springer (2005).
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