Texte

Some applications of Nitsche's method

Roland Becker (University of Pau, France)CIMPA Summer school IIT Mumbai July 2015

I. Nitsche’s method1.Motivation2.Definition of Nitsche’s method3.A priori error analysis4.A posteriori error analysis5.Relation with Mortaring/Lagrange multipliers6.Variants of the method7.Convection-Diffusion8.Optimal control9.Porous media

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1.Motivation

✤ Complicated boundary conditions

✤ Optimal boundary control

✤ B.C. for singularly perturbed problems

✤ Interface problems

2

‘Complicated’ boundary conditionsNavier-Stokes flow in symmetrical domain

symmetry axis ⇢

✓@v

@t+ v ·rv

◆� µ�v +rp = f

div v = 0

v · n = 0 on �

µ@v

@n⇥ n = 0 on �

Symmetry boundary condition:

not very easy to implement in general

Nonlinear boundary conditions: (avalanche of water, blood flow in a vein):vn = 0, ��t 2 ↵@(|vt|)

(vn := v · n, vt := v � vnn)

(� = 2µ"(v)� pI, �n := �n · n, �t = �n� �nn)

(@ (a) :=�b| (h)� (a) � (b� a) · h 8h 2 Rd

)

3

Optimal controlElliptic (right-hand-side) control

Variational framework for optimization:

��u = f + q in ⌦

u = 0 on @⌦

inf1

2ku� uk2X + ↵

1

2kqk2Y

Functional spaces: u, u 2 X = H10 (⌦), q 2 Y = L2(⌦).

L(u, z, q) := J(q, u) + hf, ziL2(⌦) � b(q, z)� a(u, z)

J(q, u) :=1

2ku� uk2X +

↵

2kqk2Y

a(u, z) :=

Z

⌦ru ·rz

b(q, z) := hq, ziL2(⌦)

4

Variational framework of optimal controlVariational framework for

optimization:

Optimality system

Gradient of reduced functional

a(u, v) = hf, vi � b(q, v) 8v 2 X

a(v, z) = J 0u(q, u)(v) 8v 2 X

J 0q(q, u)(r) = b(r, z) 8r 2 Y

S : Y ! X q ! u(q) =: S(q)

j(q) := J(q, S(q))

j0(q)(r) = J 0q(q, u)(r)� b(r, z)

L(u, z, q) := J(q, u) + hf, ziL2(⌦) � b(q, z)� a(u, z)

J(q, u) :=1

2ku� uk2X +

↵

2kqk2Y

a(u, z) :=

Z

⌦ru ·rz

b(q, z) := hq, ziL2(⌦)

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Advantages of variational framework Xh ⇢ X, Yh ⇢ Y

same as on continuous level:

•key for error estimates•discretization and optimization commute (!)

•i.e. •(1) discretizing the optimality system (first optimize)•(2) deriving the optimality system in the discrete setting (first discretize)

•give the same result•important, since algorithms developed on the continuous level can be applied for the discrete system, mesh-independency principles...

Sh : Yh ! Xh q ! uh(q) =: Sh(q)

jh(q) := J(q, Sh(q))

j0h(q)(r) = J 0q(q, uh)(r)� b(r, zh)

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Optimal boundary control

Elliptic boundary control

May not be straightforward for X ⇢ H1(⌦) and classical P kh approximations

since uh = Ihqh on @⌦ is not variational

��u = f in ⌦

u = q on @⌦

inf1

2ku� uk2X +

↵

2kqk2Y

Functional spaces X , Y ?

Variational formulation ?

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Singularly perturbed problems

�"�u+ v ·ru = f in ⌦

u = 0 on @⌦

Convection-Diffusion (for example in stationary heat flow)

" : real positive number, possibly very small

v : given velocity

If " > 0 we can look for a solution in H10 (⌦), but general estimates

depend on " !

For robustness, we have to investigate the limit problem " = 0:

v ·ru = f.

This problem has a completely different theory, and at least the boundary

conditions can only be imposed on the inflow boundary

@⌦in := {x 2 @⌦| v(x) · n(x) < 0}

8

Singularly perturbed problems

What happens when diffusion gets small ?

The solution develops a boundary layer at the outflow

v

The boundary layer can only be resolved on a sufficiently fine mesh, what to do if the mesh is not fine enough ?

9

Interface problems

� div(kru) = f in ⌦

u = 0 on @⌦

k > 0 and piecewise constant

Describes heterogenous media

Solution cannot be smooth if k discontinuous !

⌦ = ⌦1 [ ⌦2

� = ⌦1 \ ⌦2

k|⌦i = ki

[u] = 0, [k@u

@n] = k1

@u1

@n1+ k2

@u2

@n2= 0

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Interface problems

�x

Solutions are not smooth at the interface: 1D

Example. Let ⌦ =] � 1, 1[, k = k1 if x 0 and K = k2 else. If f = 0,

u(�1) = �1, and u(1) = 1, the solution is piecewise linear, i.e.

u(x) = �1 + p1(x+ 1) for x 0 and else u(x) = 1 + p2(x� 1)

with

k1p1 = k2p2 and � 1 + p1 = 1� p2, i.e. p1 + p2 = 2,

such that

(1 +k1

k2)p1 = 2 i.e. p1 =

2k2k1 + k2

, p2 =2k1

k1 + k2, � =

k1k2

(k1 + k2)/2.

([k@u

@n] = 0) ([u] = 0)

� div(kru) = 0, u(�1) = �1, u(1) = 1

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Interface problems

⌦1

⌦2

Example. Suppose that ⌦1 ⇢ ⌦2.

The interface system reads

��u1 = f/k1 in ⌦1, ��u2 = f/k2 in ⌦2

u1 = u2,k1k2

@nu1 = @nu2 on �.

Let u2|@⌦2\� = 0, k1 = 1 and let k2 ! 1. Then u2 solves

��u2 = 0 in ⌦2, @nu2 = 0 on �, u2 = 0 on @⌦2 \ �.

thus u2 = 0 and we end up with a Dirichlet problem for u1.

Example. Suppose that ⌦1 ⇢ ⌦2.

The interface system reads

�k1�u1 = f1 in ⌦1, � k2�u2 = f2 in ⌦2

u1 = u2, @nu1 =k2k1

@nu2 on �.

Let f2 = 0, u2|@⌦2\� = 0, k1 = 1 and let k2 ! 0. Then u1 solves

��u1 = f1 in ⌦1, @nu1 = 0 on �, u1 = u2 on �.

thus we end up with a Neumann problem for u1. u2|� is the Steklov-Poincare

multiplier.

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Interface problems

✤ If the mesh coincides with the interface: All methods are OK ?!

✤ If not ?✤ bad approximation

✤ pollution effect

✤ We want methods robust w.r.t✤ jump in coefficient

✤ cut geometry

✤ least modification of FE spaces

✤ Interesting for domain decomposition/ fictitious domains

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ReferencesThere is a saying "The boundary is the invention of the devil" attributed to Werner Karl Heisenberg in connection with his research on turbulence modelling. We paraphrase Heisenberg's words and formulate the question: "Is the no-slip boundary in incompressible fluid dynamics an invention of the devil?"

Jens Frehse and Joseph Malek. Problems due to the no-slip boundary in incompressible fluid dynamics. In Geometric analysis and nonlinear partial differential equations, pages 559–571. Springer, Berlin, 2003.

Alan Turing is reported as saying that PDE’s are made by God, the boundary conditions by the Devil ! The situation has changed, Devil has changed places...We can say that the main challenges are in the interfaces, with Devil not far away from them...”Jacques-Louis Lions

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References

J. Nitsche. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Univ. Hamburg, 36:9–15, 1971.D. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19:742–760, 1982.R. Stenberg. On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math., 63(1-3):139–148, 1995. R. B., P. Hansbo, and R. Stenberg. A finite element method for domain decomposition with non-matching grids. M2AN, 37(2):209–225, 2003.A. Hansbo and P. Hansbo. An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng., 191(47-48):5537–5552, 2002.

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2.Definition of Nitsche’s method

✤ Notation

✤ General approach

✤ Abstract error analysis

✤ The penalty method

✤ Nitsche’s method

✤ 1D

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Notation

H = family of admissible simplicial finite element meshes

h 2 H : covering ⌦ exactly (no curved boundaries)

Kh : set of cells, dK(⇢K) = diameter (inscribed circle radius) of K (K 2 Kh)

Sh = S@h [ S int

h set of sides (edges/faces)

shape-regularity: max

h2Hmax

K2Kh

dK⇢K

< 1

P kh = space of continuous FE (k � 1)

Dkh = space of (completely) discontinuous FE (k � 0)

h also denotes the piecewise constant function h|K = dK

(and h|S = dK if S ⇢ K)

a . b , 9constant(?) C such that a C b

a & b , 9constant(?) C such that a � C b

a ⇡ b , a . b and a & b

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General approach

��u = f in ⌦, u = uD on @⌦,

f 2 L2(⌦), uD 2 H1/2(@⌦) given data.

Consider the Poisson (model) problem:

The standard weak formulation is:

Find u 2 uD +H10 (⌦) such that for all v 2 H1

0 (⌦)Z

⌦

ru ·rv =

Z

⌦

fv,

uD 2 H1(⌦) : �(uD) = uD, kruDk = kuDkH1/2(@⌦)

infu2uD+H1

0 (⌦)J(u)

J(u) :=1

2

Z

⌦

|ru|2 �Z

⌦

fu

Equivalent to energy minimization:

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General approach

infuh2uDh+Vh\H1

0 (⌦)J(uh)

Vh = P kh , uDh 2 Vh uDh = IhuD?

Standard discrete energy minimization:

vh 2 Vh \H10 (⌦) vh 62 Vh \H1

0 (⌦)

If we want to relax the constraint of boundary condition, we need to modify the energy (on the discrete level) !

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General approach

Remind that for the standard method inf

uh2uDh+Vh\H10 (⌦)

J(uh) ! usth

suppose uDh = uD

J(usth )� J(u) = J 0(u)(u� ust

h| {z }2H1

0 (⌦)

)

| {z }=0!

+1

2J 00(u� ust

h , u� usth ) =

1

2kr(u� ust

h )k2

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General approach

Jh(uh) :=1

2

Z

⌦

|ruh|2 �Z

⌦

fuh +

Z

@⌦

(uh � uD)Lh(uh)

Lh affine operator: Lh(u) = Lh(0) + L0h(u)

Modified energy (discrete):

Discrete minimization (still convex-quadratic):

infuh2Vh

Jh(uh)

The minimizer satisfies:Z

⌦

ruh ·rvh +

Z

@⌦

(uhL0h(vh) + L0

h(uh)vh)| {z }

=:ah(uh,vh)

=

Z

⌦

fvh +

Z

@⌦

(uDL0h(vh)� Lh(0)vh)

| {z }=:lh(vh)

For example (Penalty method):Lh(uh) = h(uh � uD), h ⇡ h�m

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Abstract error analysisSymmetric discrete variational problem:

uh 2 Vh : ah(uh, vh) = lh(vh) 8vh 2 Vh.

Consistency error (depending on interpolation)

Ch(u, Ih) := supvh2Vh\{0}

ah(Ihu, vh)� lh(vh)

k|vh|kh

k|uh|kh norm on Vh

coercivity: 9↵ > 0 : ah(uh, uh) � ↵k|uh|k2h 8uh 2 Vh.

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Abstract error analysisProposition. Let us suppose the coercivity condition. Then we have (for any

Ih : V ! Vh) the error estimate

k|u� uh|kh k|u� Ihu|kh +1

↵Ch(u, Ih). (1)

Therefore, the consistency error should be bounded by the interpolation error

Proof. Let vh = Ihu � uh. Suppose k|vh|kh > 0 (otherwise we have by the

triangle inequality k|u� uh|kh k|u� Ihu|kh). Then

↵k|Ihu� uh|k2h ah(Ihu� uh, vh) = ah(Ihu, vh)� lh(vh)

supwh2Vh\{0}

ah(Ihu, wh)� lh(wh)

k|wh|khk|vh|kh = Ch(u)k|vh|kh

Division by k|vh|kh gives the result.

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The penalty method

Lh(uh) = h

2(uh � uD), L0

h(vh) = h

2vh.

J.-P. Aubin. Approximation of elliptic boundary-value problems. Wiley-Interscience , New York-London-Sydney, 1972. Pure and Applied Mathematics, Vol. XXVI.I. Babuska. The finite element method with penalty. Math. Comp., 27:221–228, 1973.

Coercivity:

ah(uh, uh) =

Z

⌦

ruh ·ruh + 2

Z

@⌦

uhL0h(uh) =

Z

⌦

|ruh|2 +Z

@⌦

hu2h

| {z }=:k|uh|k2h

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Poincaré

k|uh|k2h =

Z

⌦

|ruh|2 +Z

@⌦

hu2h

Remember: we have a nonconforming method..

Equivalent to full norm for any strictly positive weight:

ch k|uh|kh kuhkH1(⌦) Ch k|uh|kh

…gives discrete well-posedness. But when are the constants h-independent ?

Scaling:

h = h�m

⌦ ! ⌦, x = �x

dx = �

ddx, ruh = �

�2ruh, h = �h

Z

⌦

|ruh|2 dx| {z }

⇡�d⇥��2

!z }| {⇡

Z

@⌦

h

�mu

2h

| {z }⇡�d�1⇥��m

) m = 1

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The penalty method

Z

⌦

fvh = �Z

⌦

�uvh =

Z

⌦

ru ·rvh �Z

@⌦

@u

@nvh

Suppose that u satisfies:

Then we have:

ah(Ihu, vh)� lh(vh) =

✓Z

⌦

rIhu ·rvh +

Z

@⌦

(IhuL0h(vh) + vhL

0h(Ihu)

◆

�✓Z

⌦

fvh +

Z

@⌦

(uDL0h(vh)� Lh(0)vh)

◆

=

Z

⌦

r(Ihu� u) ·rvh +

Z

@⌦

vh

✓Lh(Ihu) +

@u

@n

◆+

Z

@⌦

(Ihu� u)L0h(vh)

=

Z

⌦

r(Ihu� u) ·rvh +

Z

@⌦

vh

✓ h

2(Ihu� u) +

@u

@n

◆+

Z

@⌦

(Ihu� u) h

2vh

=

Z

⌦

r(Ihu� u) ·rvh +

Z

@⌦

vh

✓ h(Ihu� u) +

@u

@n

◆

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The penalty method

This can however be improved to: Ch(u, Ih) . hmin{k,m,k+(m�1)/2}

OK for m=k=1 (with higher regularity), but ill conditioned for higher order, suboptimal in L2 (not shown here)

Z

@⌦

vh

✓ h(Ihu� u) +

@u

@n

◆=

Z

@⌦

(Ihu� u) hvh +

Z

@⌦

vh@u

@n

0

BBBB@

✓Z

@⌦

h(Ihu� u)2◆ 1

2

| {z }.h�m

2 +k+12

+

✓Z

@⌦

�1h (

@u

@n)2◆ 1

2

| {z }.h

m2

1

CCCCA

✓Z

@⌦

hv2h

◆ 12

To do better, Lh should be (a better) approximation of �@u@n

To do better, Lh should be (a better) approximation of �@u@n

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Nitsche’s method

•The method is globally conservative

J. Nitsche. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Univ. Hamburg, 36:9–15, 1971.

Take vh = 1 !

ah(uh, 1) = lh(1) , �Z

@⌦

✓@uh

@n� h(uh � uD)

◆=

Z

⌦

f

✓ah(uh, 1) =

Z

⌦

ruh ·r1 +

Z

@⌦

(uhL0h(1) + L0

h(uh)) =

Z

@⌦

huh �@uh

@n

◆

Improved flux: �h :=@uh

@n� h(uh � uD)

(for penalty: �h = � h(uh � uD))

Z

@⌦

�h · n =

Z

⌦

f

Lh(uh) = �@uh

@n+ h

2(uh � uD), L0

h(vh) = �@uh

@n+ h

2uh.

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Nitsche’s method

To make it simple:

consistencysymmetry

stability

h =�

h� > 0, we will see later

Z

⌦

ruh ·rvh +

Z

@⌦

(uhL0h(vh) + L0

h(uh)vh) =

Z

⌦

fvh +

Z

@⌦

(uDL0h(vh)� Lh(0)vh)

()Z

⌦

ruh ·rvh �Z

@⌦

@uh

@nvh �

Z

@⌦

uh@vh@n

+

Z

@⌦

huhvh =

Z

⌦

fvh �Z

@⌦

uD

✓@vh@n

� hvh

◆

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Nitsche’s method in 1D

Z

⌦

ruh ·rvh �Z

@⌦

@uh

@nvh �

Z

@⌦

uh@vh@n

+

Z

@⌦

�

huhvh =

Z

⌦

fvh �Z

@⌦

uD

✓@vh@n

� �

hvh

◆

u0 � u1

h

� u0 � u1

h

� u0

h

+ �

u0

h

=h

2(f(x0) + f(x1))�

uD

h

+ �

uD

h

u1 � u2

h

+u1 � u0

h

+u0

h

=h

2(f(x0) + 2f(x1) + f(x1)) +

uD

h

u2 � u3

h

+u2 � u1

h

=h

2(f(x1) + 2f(x2) + f(x3))

(and trapezoidal rule)

(� � 1)u0

h

=h

2(f(x0) + f(x1)) + (� � 1)

uD

h

2u1 � u2

h

=h

2(f(x0) + 2f(x1) + f(x1)) +

uD

h

�u1 + 2u2 � u3

h

=h

2(f(x1) + 2f(x2) + f(x3))

FD-scheme:

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3.A priori error analysis

✤ Stability

✤ Consistency

✤ L2 estimate

31

Stability

ah(uh, uh) =

Z

⌦

ruh ·ruh + 2

Z

@⌦

uhL0h(uh) =

Z

⌦

ruh ·ruh +

Z

@⌦

hu2h

| {z }k|uh|k2h

�2

Z

@⌦

uh@uh

@n

Z

@⌦

uh@uh

@n

✓Z

@⌦

hu2h

◆1/2 ✓Z

@⌦

�1h (

@uh

@n)2◆1/2

By definition:

Inverse estimate :

h =�

h� > 0,

kh1/2@uh

@nk@⌦ Cinvkruhk

Clearly we should have:(�h =

�

h)

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