PAMM · Proc. Appl. Math. Mech. 14, 851 – 852 (2014) / DOI 10.1002/pamm.201410406
Domain decomposition methods for nonlinear transmission conditions Matthias A. F. Gsell1,∗ and Olaf Steinbach1 1
Institut für Numerische Mathematik, TU Graz, Steyrergasse 30, 8010 Graz, Austria
We consider the transformation of quasilinear partial differential equations to a coupled system of linear equations, but with nonlinear transmission conditions on the interfaces. After deriving a variational formulation, we will discuss Mortar finite element discretization strategies and present a numerical example. c 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Model problem
As a model problem we consider the following quasilinear boundary value problem in a bounded domain Ω ⊂ Rd (d = 2, 3) with Lipschitz boundary Γ = ∂Ω = ΓD ∪ ΓN and ΓD ∩ ΓN = ∅ to find p such that −∇ · kr θ(p) ∇p = f
in Ω,
p=0
on ΓD ,
kr θ(p) ∇p · n = gN
on ΓN .
(1)
This model problem can be seen as a simplification of the stationary Richards equation without gravity, see [1]. If kr only depends on θ, we can introduce Z
p(x)
u(x) := κ(p(x)) :=
kr (θ(s)) ds 0
as the Kirchhoff transformation of p. Here we assume kr ◦ θ to be Lipschitz and bounded with 0 < cl ≤ kr (θ(·)) ≤ cu < ∞. Therefore we obtain ∇u = κ0 (p) ∇p = kr (θ(p)) ∇p and problem (1) can be transformed to the following linear problem to find u such that −∆u = f
in Ω,
u=0
on ΓD ,
∇u · n = gN
on ΓN .
Obviously, such a simplification can not be achieved if kr depends in addition explicitly on x ∈ Ω. However, if this dependency
p2 , kr,2 ◦ θ2 Ω1
Γ2
Th,2
p1 , kr,1 ◦ θ1 Γ1
Th,1
Ω2
ΓI
Fig. 1: several domains
Ih ΓI
Ih′ Fig. 2: dual interface mesh
is piecewise within non–overlapping subdomains Ωi , we can introduce local Kirchhoff transformations ui (x) := κi (pi (x)) for x ∈ Ωi . So we can reformulate problem (1) and obtain the following nonlinear transmission problem, in the case of two subdomains, see Figure 1, to find ui , i = 1, 2, such that −1 −∆ui = f in Ωi , ui = 0 on ΓD,i , ∇ui · ni = gN.i on ΓN,i , ∇u1 · n = ∇u2 · n, κ−1 1 (u1 ) = κ2 (u2 ) on ΓI , (2)
where ΓI = ∂Ω1 ∩ ∂Ω2 , ΓD,i = ∂Ωi ∩ ΓD , ΓN,i = ∂Ωi ∩ ΓN and gN,i = gN |Γ . The second, nonlinear interface condition N,i ensures the continuity of the solution p of the original problem (1), where we started from. Next we will derive a variational formulation for the nonlinear transmission problem (2). ∗
Corresponding author, e-mail
[email protected]
c 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
852
Section 18: Numerical methods of differential equations
2
Primal–hybrid formulation
In [2, 3], an equivalent primal–hybrid formulation for the Poisson equation is presented. The idea is to introduce Lagrange multipliers acting on interfaces which ensure global continuity of the solution. We use this approach for the nonlinear transmission problem (2) and obtain the following variational problem to find (u, λ) ∈ X × M , such that 2 Z X i=1
Z ∇ui · ∇vi dx +
Ωi
ΓI
λ [v]ΓI dsx =
2 Z X i=1
fi vi dx +
Ωi
2 Z X i=1
gN,i vi dsx
for all v ∈ X,
ΓN,i
(3)
Z ΓI
for all µ ∈ M
µ (u)ΓI dsx = 0
1 1 2 (ΓI )0 , and with the jump terms with the ansatz spaces X := {v ∈ L2 (Ω) vi = v|Ωi ∈ H0,Γ (Ωi ), i = 1, 2}, M := H00 D,i −1 −1 0 0 [v]ΓI := (v1 − v2 )|ΓI ∈ M and (u)ΓI := (κ1 (u1 ) − κ2 (u2 ))|ΓI ∈ M .
3
Discretization
For the discretization of (3) we introduce the finite dimensional subspaces Xh := {v ∈ L2 (Ω) vi ∈ Sh1 (Th,i ), i = 1, 2} ⊂ X and Mh := {µ ∈ L2 (ΓI ) µ ∈ Sh0 (Ih0 )} ⊂ M , where Th,i is an admissible triangulation of Ωi , and Ih0 is a modified dual mesh along the interface ΓI as depicted in Figure 2, see also [4]. For the solution of the nonlinear discrete problem we apply Newton’s method by solving a sequence of the following linear problems. For given data wh ∈ Xh , find (uh , λh ) ∈ Xh ×Mh , such that Z 2 Z 2 Z 2 Z X X X gN,i vh,i dsx ∀vh ∈ Xh , fi vh,i dx + λh [vh ]ΓI dsx = ∇uh,i · ∇vh,i dx + ΓI i=1 ΓN,i i=1 Ωi i=1 Ωi (4) Z Z Z 0
ΓI
µh (wh , uh )ΓI dsx =
0
0
ΓI
−1 0 0 where (wh , uh )ΓI := (κ−1 1 ) (wh,1 ) uh,1 − (κ2 ) (wh,2 ) uh,2 corresponds to the previous Newton iteration.
4
µh (wh , wh )ΓI dsx −
|ΓI
ΓI
µh (wh )ΓI dsx
∀µ ∈ Mh
is the linearization of the nonlinear jump (uh )ΓI and wh
Numerical example
Consider R2 ⊃ Ω = [0, 1]2 = [0, 0.5] × [0, 1] ∪ [0.5, 1] × [0, 1] =: Ω1 ∪ Ω2 , where ΓN = {0} × (0, 1) and ΓD = ∂Ω \ ΓN . The local diffusion coefficients are chosen as in [1, Chapter 1]. The remaining data f, gN,i , gD,i are given appropriately.
Fig. 3: (discontinuous) solution u of problem (4) and its (continuous) inverse Kirchhoff transformations pi = κ−1 i (ui ), i = 1, 2.
Acknowledgements This work was supported by the Austrian Science Fund (FWF) within the IGDK 1754.
References [1] H. B ERNINGER, Domain Decomposition Methods for Elliptic Problems with Jumping Nonlinearities and Application to the Richards Equation, PhD thesis, Freie Universität Berlin, 2008. [2] D. B OFFI, F. B REZZI, and M. F ORTIN, Mixed finite element methods and applications, Springer, Heidelberg, 2013. [3] P. A. R AVIART and J. M. T HOMAS, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp. 31(138), 391–413 (1977). [4] B. I. W OHLMUTH, Discretization methods and iterative solvers based on domain decomposition, Springer, 2001. c 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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