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W I S S E N T E C H N I K L E I D E N S C H A F T A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements Matthias Gsell and Olaf Steinbach Institute of Computational Mathematics DD23,

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W I S S E N T E C H N I K L E I D E N S C H A F T www.numerik.math.tugraz.at

A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements

Matthias Gsell and Olaf Steinbach

Institute of Computational Mathematics

DD23, International Conference on Domain Decomposition Methods

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Outline

  • 1. Motivation
  • 2. Approach

– Continuous Formulation – Discretization Strategies

  • 3. Numerical Example
  • 4. Outlook

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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Outline

  • 1. Motivation
  • 2. Approach

– Continuous Formulation – Discretization Strategies

  • 3. Numerical Example
  • 4. Outlook

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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Motivation

See [Berningner, 2008]. To describe the flow of fluid (water) in porous media, we use the Richards equation for the physical pressure p.

Richards Equation

n(x) ∂θ(p(x, t)) ∂t − ∇ ·

  • K(x)

µ kr

  • θ
  • p(x, t)
  • ∇
  • p(x, t) − ̺g z
  • = f(x, t)

Quantities θ . . . saturation n . . . porosity ̺ . . . density µ . . . viscosity K . . . permeability kr . . . relative permeability g . . . gravitational const. f . . . source term Ω Γ

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Motivation

Consider the heat equation to describe the distribution of heat where the thermal conductivity depends on the heat p. Heat Equation

∂p(x, t) ∂t − ∇ ·

  • k
  • p(x, t)
  • ∇p(x, t)
  • = f(x, t)

Quantities k . . . thermal conductivity f . . . heat source

Ω Γ

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Motivation

Consider the heat equation to describe the distribution of heat where the thermal conductivity depends on the heat p. Heat Equation

∂p(x, t) ∂t − ∇ ·

  • k
  • p(x, t)
  • ∇p(x, t)
  • = f(x, t)

Quantities k . . . thermal conductivity f . . . heat source

Ω Γ

Both equtions have same second order term −∇ ·

  • k
  • p(x, t)
  • ∇p(x, t)
  • .

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Motivation

Let Ω ⊂ Rd. For given data f, gN, gD consider the following quasilinear boundary value problem. Model Problem

Find p, such that −∇ ·

  • k
  • p(x)
  • ∇p(x)
  • = f(x)

in Ω p(x) = gD(x)

  • n ΓD

k

  • p(x)
  • ∇p(x) · nΓN = gN(x)
  • n ΓN

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Motivation

Let Ω ⊂ Rd. For given data f, gN, gD consider the following quasilinear boundary value problem. Model Problem

Find p, such that −∇ ·

  • k
  • p(x)
  • ∇p(x)
  • = f(x)

in Ω p(x) = gD(x)

  • n ΓD

k

  • p(x)
  • ∇p(x) · nΓN = gN(x)
  • n ΓN

Assumptions on k : R → R.

  • )

k ∈ L∞(R), i.e. |k(s)| ≤ C < ∞ for almost all s ∈ R.

  • )

there exists a constant c > 0 such that 0 < c ≤ k(s) for almost all s ∈ R.

  • )

k is Lipschitz continuous, i.e. |k(s) − k(t)| ≤ L |s − t|.

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Motivation

Consider the weak formulation of the BVP . Weak Formulation

Find p ∈ H1

gD,ΓD (Ω), such that

  • Ω

k(p) ∇p · ∇v dx =

  • Ω

f v dx +

  • ΓN

gN v dsx is satisfied for all v ∈ H1

0,ΓD (Ω).

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Motivation

Consider the weak formulation of the BVP . Weak Formulation

Find p ∈ H1

gD,ΓD (Ω), such that

  • Ω

k(p) ∇p · ∇v dx =

  • Ω

f v dx +

  • ΓN

gN v dsx is satisfied for all v ∈ H1

0,ΓD (Ω).

We can apply the Kirchhoff transformation to the physical quantity p and introduce the generalized quantity u as u(x) := κ

  • p(x)
  • =

p(x)

  • k(s) ds.

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Motivation

Consider the weak formulation of the BVP . Weak Formulation

Find p ∈ H1

gD,ΓD (Ω), such that

  • Ω

k(p) ∇p · ∇v dx =

  • Ω

f v dx +

  • ΓN

gN v dsx is satisfied for all v ∈ H1

0,ΓD (Ω).

We can apply the Kirchhoff transformation to the physical quantity p and introduce the generalized quantity u as u(x) := κ

  • p(x)
  • =

p(x)

  • k(s) ds.

Therefore we get ∇u(x) = κ′ p(x)

  • ∇p(x) = k
  • p(x)
  • ∇p(x).

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Motivation

Transformed Model Problem

Find u ∈ H1

hD,ΓD (Ω), such that

  • Ω

∇u · ∇v dx =

  • Ω

f v dx +

  • ΓN

gN v dsx is satisfied for all v ∈ H1

0,ΓD (Ω) with hD := κ(gD).

  • )

Unique solvability from Lax–Milgram Lemma.

  • )

Numerical analysis is well known for this problem.

  • )

Apply inverse Kirchhoff transformation to obtain the physical quantity p.

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Motivation

Transformed Model Problem

Find u ∈ H1

hD,ΓD (Ω), such that

  • Ω

∇u · ∇v dx =

  • Ω

f v dx +

  • ΓN

gN v dsx is satisfied for all v ∈ H1

0,ΓD (Ω) with hD := κ(gD).

  • )

Unique solvability from Lax–Milgram Lemma.

  • )

Numerical analysis is well known for this problem.

  • )

Apply inverse Kirchhoff transformation to obtain the physical quantity p. ⇒ We considered nonlinearities of the form k : R → R.

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Motivation

Transformed Model Problem

Find u ∈ H1

hD,ΓD (Ω), such that

  • Ω

∇u · ∇v dx =

  • Ω

f v dx +

  • ΓN

gN v dsx is satisfied for all v ∈ H1

0,ΓD (Ω) with hD := κ(gD).

  • )

Unique solvability from Lax–Milgram Lemma.

  • )

Numerical analysis is well known for this problem.

  • )

Apply inverse Kirchhoff transformation to obtain the physical quantity p. ⇒ We considered nonlinearities of the form k : R → R. ⇒ Try nonlinearities of the form k : R × Ω → R.

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Outline

  • 1. Motivation
  • 2. Approach

– Continuous Formulation – Discretization Strategies

  • 3. Numerical Example
  • 4. Outlook

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Approach

Continuous Formulation

Consider a second order term is of the form −∇ ·

  • k
  • p(x), x
  • ∇p(x)
  • .

Richards equation

Ω1 Ω2 Γ1 Γ2 Γ

Different soil parameter. ⇒ Different permeabilities. Heat equation

Ω1 Ω2 Γ1 Γ

Different material types. ⇒ Different conductivities.

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Approach

Continuous Formulation

Consider a second order term is of the form −∇ ·

  • k
  • p(x), x
  • ∇p(x)
  • .

Richards equation

Ω1 Ω2 Γ1 Γ2 Γ

Different soil parameter. ⇒ Different permeabilities. Heat equation

Ω1 Ω2 Γ1 Γ

Different material types. ⇒ Different conductivities.

  • )

Apply Kirchhoff transformation. ⇒ u(x) := κ

  • p(x)
  • =

p(x)

  • k(s, x) ds

but ∇u(x) = k

  • p(x), x
  • ∇p(x).

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Approach

Continuous Formulation

Consider a second order term is of the form −∇ ·

  • k
  • p(x), x
  • ∇p(x)
  • .

Richards equation

Ω1 Ω2 Γ1 Γ2 Γ

Different soil parameter. ⇒ Different permeabilities. Heat equation

Ω1 Ω2 Γ1 Γ

Different material types. ⇒ Different conductivities.

  • )

Apply Kirchhoff transformation. ⇒ u(x) := κ

  • p(x)
  • =

p(x)

  • k(s, x) ds

but ∇u(x) = k

  • p(x), x
  • ∇p(x).

⇒ new approach to exploit the advantages of the Kirchhoff transformation.

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Approach

Continuous Formulation

Let Ω ⊂ Rd. For given data f, gN, gD consider the following quasilinear boundary value problem. Model Problem

Find p, such that −∇ ·

  • k(p(x), x) ∇p(x)
  • = f(x)

in Ω p(x) = gD(x)

  • n ΓD

k(p(x), x) ∇p(x) · nΓN = gN(x)

  • n ΓN

ΓD ΓN Ω1 Ω2 Ω3 Ω4 Ω5

Assumption:

  • ) k
  • p(x), x
  • =

K

  • i=1

XΩi (x) ki

  • p(x)
  • ) ki ∈ L∞(R) and 0 < ci ≤ ki for ci ∈ R,
  • ) ki is Lipschitz continuous.

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Approach

Continuous Formulation

Let Ω ⊂ Rd. For given data f, gN, gD consider the following quasilinear boundary value problem. Model Problem

Find p, such that −∇ ·

  • k(p(x), x) ∇p(x)
  • = f(x)

in Ω p(x) = gD(x)

  • n ΓD

k(p(x), x) ∇p(x) · nΓN = gN(x)

  • n ΓN

Nonlinear Variational Formulation

Find p ∈ H1

gD,ΓD (Ω), such that

  • a(p, v) =
  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ H1

0,ΓD (Ω)

The linear form a(·, ·) is given by

  • a(p, v) :=
  • Ω

k(p)∇p · ∇v dx.

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Approach

Continuous Formulation

Use Primal Hybrid Formulation to exploit the structure of the nonlinearity.

See [Raviart, 1977]. Introduce X :=

  • v ∈ L2(Ω) | vi ∈ H1(Ωi), i = 1, . . . , N
  • ,

and M0,ΓN :=

  • µ ∈

N

  • i=1

H

1 2 (∂Ωi)

′

| ∃ q ∈ H0,ΓN (div, Ω) : q · ni = µ on ∂Ωi, i = 1, . . . , N

  • .

Ω1 Ω2 Γ1 Γ2 Γ Ω1 Ω2 Γ1 Γ

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Approach

Continuous Formulation

The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation

Find (p, λ) ∈ X × M0,ΓN , such that

  • a(p, v) + b(v, λ) =
  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ X b(p, µ) = − gD, µ∂Ω ∀ µ ∈ M0,ΓN

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Approach

Continuous Formulation

The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation

Find (p, λ) ∈ X × M0,ΓN , such that

  • a(p, v) + b(v, λ) =
  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ X b(p, µ) = − gD, µ∂Ω ∀ µ ∈ M0,ΓN

The linear form a(·, ·) and the bilinear form b(·, ·) are given by

  • a(p, v) :=

N

  • i=1

ai(pi, vi) =

N

  • i=1
  • Ωi

ki(pi) ∇pi · ∇vi dx b(p, µ) :=

N

  • i=1

bi(p, µ) = −

N

  • i=1

γ0

∂Ωi pi, µ∂Ωi

The bilinear form b(·, ·) “enforces” the test function v and the solution u to be continuous (in a weak sense) on the interface and therefore to be in H1(Ω).

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Approach

Continuous Formulation

The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation

Find (p, λ) ∈ X × M0,ΓN , such that

  • a(p, v) + b(v, λ) =
  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ X b(p, µ) = − gD, µ∂Ω ∀ µ ∈ M0,ΓN

Apply the Kirchhoff transformation in each subdomain separately. ⇒ ui(x) := κi

  • pi(x)
  • =

pi (x)

  • ki(s) ds

and ∇ui(x) = ki

  • pi(x)
  • ∇pi(x)

⇒ pi(x) := κ−1

i

  • ui(x)
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Approach

Continuous Formulation

The variational problem can be written in the equivalent formulation. Primal Hybrid Variational Formulation

Find (p, λ) ∈ X × M0,ΓN , such that

  • a(p, v) + b(v, λ) =
  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ X b(p, µ) = − gD, µ∂Ω ∀ µ ∈ M0,ΓN

Rewrite the linear form a(·, ·) and the bilinear form b(·, ·) in terms of u.

  • a(p, v) =

N

  • i=1
  • Ωi

ki(pi) ∇pi · ∇vi dx =

N

  • i=1
  • Ωi

∇ui · ∇vi dx =: a(u, v) b(p, µ) = −

N

  • i=1

γ0

∂Ωi pi, µ∂Ωi = −

N

  • i=1

γ0

∂Ωi κ−1

i

(ui), µ∂Ωi =: c(u, µ)

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Approach

Continuous Formulation

We end up with the following variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation

Find (u, λ) ∈ X × M0,ΓN , such that a(u, v) + b(v, λ) =

  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ X c(u, µ) = − gD, µ∂Ω ∀ µ ∈ M0,ΓN

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Approach

Continuous Formulation

We end up with the following variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation

Find (u, λ) ∈ X × M0,ΓN , such that a(u, v) + b(v, λ) =

  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ X c(u, µ) = − gD, µ∂Ω ∀ µ ∈ M0,ΓN

The above variational problem is equivalent to the variational problem Nonlinear Variational Formulation

Find p ∈ H1

gD,ΓD (Ω), such that

  • a(p, v) =
  • f, v
  • 0,Ω +
  • gN, v
  • 0,ΓN

∀ v ∈ H1

0,ΓD (Ω)

which is uniquely solvable.

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Approach

Discretization Strategies

How to discretize the corresponding spaces? See [Wohlmuth, 2001].

refine subdomains

− − − − − − − − − − → For each triangulation Th,i of Ωi define Xh,i := S1

h,i(Th,i) =

  • u ∈ C(Ωi) | u|T ∈ P1(T)

∀ T ∈ Th,i

  • and set

Xh :=

N

  • i=1
  • Xh,i ∩ H1

0,∂Ωi ∩ΓD (Ωi)

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Approach

Discretization Strategies

How to discretize the corresponding spaces? See [Wohlmuth, 2001].

Ω1 Ω2 ΓD ΓN Γ21 ΓN ΓC Γ21 ΓN ΓC Γ12 ˚ ϕ2,1 ˚ ϕ2,2 ˚ ϕ2,3 ˚ ϕ2,4 ψ1 ψ2 ψ3 ψ4 ˚ ϕ1,1 ˚ ϕ1,2 Y h

0 (I21 h ) = W h 0,(2,1)

Y h

0 (I12 h )

M h

(2,1)

Define M :=

  • m = (k, l) | Γkl = ∂Ωk ∩ ∂Ωl = ∅ with ′′Th,k finer than T ′′

h,l

  • .

For each interface Γm define Mh,m := S0

h,m(I′ h,m) =

  • λ ∈ L2(Γm) | λ|E ∈ P0(E)

∀ E ∈ I′

h,m

  • and set

Mh :=

  • m∈M

Mh,m

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Approach

Discretization Strategies

Assume ˜ uh := uh + uh,D with uh ∈ Xh. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation

Find (uh, λh) ∈ Xh × Mh, such that a(uh, vh) + b(vh, λh) =

  • f, vh
  • 0,Ω +
  • gN, vh
  • 0,ΓN

− a(uh,D, vh) ∀ vh ∈ Xh c(uh, µh) = 0 ∀ µ ∈ Mh

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Approach

Discretization Strategies

Assume ˜ uh := uh + uh,D with uh ∈ Xh. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation

Find (uh, λh) ∈ Xh × Mh, such that a(uh, vh) + b(vh, λh) =

  • f, vh
  • 0,Ω +
  • gN, vh
  • 0,ΓN

− a(uh,D, vh) ∀ vh ∈ Xh c(uh, µh) = 0 ∀ µ ∈ Mh

In the discrete setting the bilinear form b(·, ·) can be written as b(uh, µh) := −

N

  • i=1
  • uh,i, µh
  • 0,∂Ωi =
  • m∈M
  • uh,k − uh,l, µh
  • 0,Γm

=

  • m∈M
  • uhΓm, µh
  • 0,Γm .

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Approach

Discretization Strategies

Assume ˜ uh := uh + uh,D with uh ∈ Xh. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation

Find (uh, λh) ∈ Xh × Mh, such that a(uh, vh) + b(vh, λh) =

  • f, vh
  • 0,Ω +
  • gN, vh
  • 0,ΓN

− a(uh,D, vh) ∀ vh ∈ Xh c(uh, µh) = 0 ∀ µ ∈ Mh

The same can be done for the linear form c(·, ·). c(uh, µh) := −

N

  • i=1
  • κ−1

i

(uh,i + uh,D,i), µh

  • 0,∂Ωi

=

  • m∈M
  • κ−1

k (uh,k + uh,D,k) − κ−1 l

(uh,l + uh,D,l), µh

  • 0,Γm

=

  • m∈M
  • κ−1(uh + uh,D)Γm, µh
  • 0,Γm .

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Approach

Discretization Strategies

Assume ˜ uh := uh + uh,D with uh ∈ Xh. We obtain the following modified discrete variational problem. Transformed Nonlinear Primal Hybrid Variational Formulation

Find (uh, λh) ∈ Xh × Mh, such that a(uh, vh) + b(vh, λh) =

  • f, vh
  • 0,Ω +
  • gN, vh
  • 0,ΓN

− a(uh,D, vh) ∀ vh ∈ Xh c(uh, µh) = 0 ∀ µ ∈ Mh

To solve the nonlinear problem, we apply Newton’s method and solve a sequence of linear problems.

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Approach

Discretization Strategies

Linearized Formulation

For a given wh ∈ Xh find (uh, λh) ∈ Xh × Mh, such that a(uh, vh) + b(vh, λh) =

  • f, vh
  • 0,Ω +
  • gN, vh
  • 0,ΓN

− a(uh,D, vh) c′[wh](uh, µh) = c′[wh](wh, µh) − c(wh, µh) for all (vh, µh) ∈ Xh × Mh.

The linearized bilinear form c′[·](·, ·) is given by c′[wh](uh, µh) :=

  • m∈M
  • κ−1

k ′

wh,k + uh,D,k

  • uh,k − κ−1

l ′

wh,l + uh,D,l

  • uh,l, µh
  • 0,Γm

=

  • m∈M
  • κ−1′

wh + uh,D

  • uhΓm, µh
  • 0,Γm .

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Approach

Discretization Strategies

To obtain solvability of the variational problem we have to show that sup

vh∈Xh

b(vh, µh) vhX ≥ γb µhMh for all µh ∈ Mh, sup

vh∈Xh

c′[wh](vh, µh) vhX ≥ γc µhMh for all µh ∈ Mh, and sup

vh∈KerB

a(uh, vh) vhX ≥ γa uhX for all uh ∈ KerC′[wh], sup

vh∈KerC′[wh]

a(uh, vh) > 0 for all uh ∈ KerB, with positive constants γb, γc and γa. See [Nicolaides, 1982].

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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Outline

  • 1. Motivation
  • 2. Approach

– Continuous Formulation – Discretization Strategies

  • 3. Numerical Example
  • 4. Outlook

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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Numerical Example

R R R Ω1 Ω2 Ω3 Ω4 ΓD ΓN

  • Consider the instationary Richards equation.
  • Domain:

Ω = [0, 1] × [0, 2] ⊂ R2

  • Soil types:

Ω1 . . . sand, Ω2 . . . sandy loam, Ω3 . . . loam, Ω4 . . . sand

  • Boundary conditions:

hD(x, t) =      κ−1

1 (−0.5 (1 − t))

x ∈ ΓD,1, t < 1 κ−1

2 (−0.5 (1 − t))

x ∈ ΓD,2, t < 1 0.0 x ∈ ΓD, t ≥ 1 hN(x, t) = 0.0 x ∈ ∂Ω \ ΓD

  • Time discretization parameter:

Timestep : τ = 0.02 Timesteps : T = 1500

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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Numerical Example

Triangulation of the four domains (zoomed in cutout). VIDEO : Solution

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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SLIDE 39

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Outline

  • 1. Motivation
  • 2. Approach

– Continuous Formulation – Discretization Strategies

  • 3. Numerical Example
  • 4. Outlook

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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Outlook

Summary

Transform quasiliner PDEs to linear PDEs Apply discretization and linearization methods Numerical example

Outlook

Numerical analysis of the discrete linearized SPP Convergence analysis Preconditioners

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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SLIDE 41

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Outlook

Summary

Transform quasiliner PDEs to linear PDEs Apply discretization and linearization methods Numerical example

Outlook

Numerical analysis of the discrete linearized SPP Convergence analysis Preconditioners

Thank you for your attention!

Matthias Gsell, Institute of Computational Mathematics 2015-07-06

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[1] H. Berninger. Domain Decomposition Methods for Elliptic Problems with Jumping Nonlinearities and Application to the Richards Equation. PhD thesis, Freie Universität Berlin, 2008. [2] R. A. Nicolaides. Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal., 19(2):349–357, 1982. [3] P .-A. Raviart and J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic equations.

  • Math. Comp., 31(138):391–413, 1977.

[4] Barbara I. Wohlmuth. Discretization methods and iterative solvers based on domain decomposition, volume 17 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2001.

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