Project 4 Inhomogeneous Dirichlet and Robin Boundary ... · Vinh Ma and Markus Wol Project 4...

Linear FEM for BVPs with inhomogeneous Dirichlet bc’sLinear FEM for BVPs with Robin bc’s

Heater example

Project 4 Inhomogeneous Dirichlet and RobinBoundary Conditions

Vinh Ma and Markus Wolff

July 10, 2014

Vinh Ma and Markus Wolff Project 4 Inhomogeneous Dirichlet and Robin Boundary Conditions


Heater example

Contents

1 Linear FEM for BVPs with inhomogeneous Dirichlet bc’sDirichlet lift ansatzLagrange ansatz

2 Linear FEM for BVPs with Robin bc’s

3 Heater exampleDirichlet lift ansatzLagrange ansatz



Heater example

Dirichlet lift ansatzLagrange ansatz

Model problem

Find u ∈ C 2(Ω) ∩ C (Ω), such that

−∇ · a∇u + cu = f in Ω

u = g on ∂Ω

with

constants a > 0, c ≥ 0,

Lipschitz domain Ω ⊂ R2 with boundary ∂Ω,

sufficiently smooth functions f and g .



Heater example


Dirichlet lift ansatz

Idea: Assume u = u0 + ug with ug |∂Ω = g .

Only need to solve for u0 if ug is known.

⇒ BVP with homogeneous Dirichlet bc’s, since u0|∂Ω = 0.



Heater example


Variational formulation

−∇ · a∇(u0 + ug ) + c(u0 + ug ) = f in Ω

u0 = 0 on ∂Ω

Testing with v ∈ H10 (Ω), integrating and applying Green’s formula

leads to:



Heater example



Find u0 ∈ H10 (Ω), such that

∫Ωa∇u0∇v + cu0v dx =

∫Ωfv − a∇ug∇v − cugv dx

for all v ∈ H10 (Ω).



Heater example


Implementation

Discretize Ω by choosing an appropriate mesh M with nodesx1, . . . , xN .

Let ϕ1, . . . , ϕN denote the basis of hat functions for the FEsubspace of H1(Ω).

Let k1, . . . , kL ⊆ 1, . . . ,N denote the indices of theboundary nodes of the discretized domain, i.e.xk1 , . . . , xkL ∈ ∂Ω.



Heater example


Implementation

Approximate solution by

u0 ≈∑

i∈1,...,N\k1,...,kL u0,iϕi

with u0,i ∈ R.

Choose ug ∈ H1(Ω) such that

ug (xi ) = g(xi ) if i ∈ k1, . . . , kLug (xi ) = 0 else,

i.e.

ug =L∑

i=1

g(xki )ϕki .



Heater example


Implementation

Compute the stiffness matrix K , mass matrix M and the loadvector f (for solution and test functions in H1(Ω)) by

Kij =

∫Ω∇ϕi∇ϕj dx

Mij =

∫Ωϕi ϕj dx

fi =

∫Ωf ϕi dx

for i , j = 1, . . . ,N.



Heater example


Implementation

Compute Ku0 ,Mu0 ∈ R(N−L)×(N−L) by deleting the i-th rowand column in K and M, resp., for all i ∈ k1, . . . , kL(i.e. omit all hat functions which are not in H1

0 (Ω)).

Compute f ∈ RN−L by deleting the i-th entry of the vector

f − (aK + cM)ug

for all i ∈ k1, . . . , kL, whereby ug ∈ RN is given by

(ug )j =

g(xj) for j ∈ k1, . . . , kL0 for j ∈ 1, . . . ,N\k1, . . . , kL.



Heater example


Implementation

Solve the linear system

(aKu0 + cMu0)u0 = f

to get u0 ∈ RN−L, which approximates the solution of theafore-mentioned BVP with homogeneous Dirichlet bc’s at allinner nodes.



Heater example


Lagrange ansatz

Idea: Test with v ∈ H1(Ω) instead of v ∈ H10 (Ω).

Substitute ∇u · n =: λ ∈ H−1/2(∂Ω) in the arising boundaryintegral.

Test the bc’s with µ ∈ H−1/2(∂Ω).

H−1/2(∂Ω) ⊃ L2(δΩ) is the space of the Neumann trace.



Heater example



Find u ∈ H1(Ω), such that

∫Ωa∇u∇v + cuv dx −

∫∂Ω

a∇u · n v ds =

∫Ωfv dx∫

∂Ωuµ ds =

∫∂Ω

gµ ds

for all v ∈ H1(Ω), for all µ ∈ H−1/2(∂Ω).



Heater example





∫∂Ω

a∇u · n︸︷︷︸=:λ

v ds =

∫Ωfv dx∫

∂Ωuµ ds =

∫∂Ω

gµ ds




Heater example



Find u ∈ H1(Ω), λ ∈ H−1/2(∂Ω) such that


∫∂Ω

aλv ds =

∫Ωfv dx∫

∂Ωuµ ds =

∫∂Ω

gµ ds




Heater example


Implementation

Discretize Ω by choosing an appropriate mesh M with nodesx1, . . . , xN .

Let ϕ1, . . . , ϕN denote the basis of hat functions for the FEspace u ∈ H1(Ω).

Let ψ1, . . . , ψL denote the basis of piecewise constantfunctions for the FE subspace of H−1/2(∂Ω).

Approximate solution by

u ≈N∑i=1

uiϕi and λ ≈L∑

i=1

λiψi

with [u1, . . . , uN ] ∈ RN and [λ1, . . . , λL] ∈ RL.



Heater example


Implementation

Two choices possible for the piecewise constant basis functions ψi :

1 Choice 1: Constant and support on each boundary edge.⇒ Can lead to a singular linear system, if the number ofboundary nodes L is even.

2 Choice 2: Constant around each boundary node with supportbetween the midpoints of the adjacent edges.



Heater example


Implementation


Compute M0 ∈ RL×N by

(M0)ikj =

∫∂Ωϕki ψj ds

for i , j = 1, . . . , L.

Compute g ∈ RL by

gi =

∫∂Ω

g ψi ds

for i = 1, . . . , L.



Heater example


Implementation

Let K and M be the stiffness and mass matrix, resp., and fthe load vector as above.

Solve the linear system[aK + cM −MT

0

M0 0

] [uλ

]=

[fg

].



Heater example


Example

Let the BVP’s solution be u = cos(2πx) cos(2πy).

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1−1

−0.5

0

0.5

1

exact solution: cos(2πx)⋅cos(2πy)



Heater example


Example

Dirichlet lift ansatz: Absolute error.



Heater example


Example

Lagrange ansatz: Absolute error using choice 1 (singular matrix).



Heater example


Example

Lagrange ansatz: Absolute error using choice 2.



Heater example


Comparison

Dirichlet lift ansatz Lagrange ansatzlargest error ≈ 0.03 ≈ 0.03error on boundary no yessize of the LS N-L N+L



Heater example

Linear FEM for BVPs with Robin bc’s – Model problem


−∇ · a∇u + cu = f in Ω

∇u · n + βu = g on ∂Ω

with

constants a > 0, c ≥ 0, β > 0,

Lipschitz domain Ω ⊂ R2 with boundary ∂Ω,

sufficiently smooth functions f and g .



Heater example




∫∂Ω

a∇u · n v ds =

∫Ωfv dx

for all v ∈ H1(Ω).



Heater example




∫∂Ω

a∇u · n︸︷︷︸g−βu

v ds =

∫Ωfv dx .




Heater example



∫Ωa∇u∇v + cuv dx +

∫∂Ω

aβuv ds =

∫Ωfv dx +

∫∂Ω

gv ds




Heater example

Implementation


Compute M∂Ω ∈ RN×N by

(M∂Ω)kikj =

∫∂Ωϕki ϕkj ds

for i , j = 1, . . . , L and set all other entries to be zero.

Compute g ∈ RN by

gki =

∫∂Ω

g ϕki ds

for i = 1, . . . , L and set all other entries to be zero.



Heater example

Implementation

Let K and M be the stiffness and mass matrix, resp., and fthe load vector as usual (i.e. in Ω).

Solve the linear system

(aK + cM + aβM∂Ω)u = f + g .



Heater example

Example

Let the BVP’s solution be u = cos(2πx) cos(2πy).



Heater example

Example

Absolute error of the numerical solution:



Heater example


Heater example


−κ∆u = 0 in Ω (2D cross section of room)

u = uH on ΓH ⊆ ∂Ω (heater)

u = uW on ΓW ⊆ ∂Ω (window)

κ∇u · n + βu = 0 on ΓR ⊆ ∂Ω (ceiling and walls)

∇u · n = 0 on ΓF ⊆ ∂Ω (floor)

with κ, β > 0.



Heater example


Dirichlet lift ansatz

Suppose u = u0 + uH + uW with

u0 ∈ H1ΓH∪ΓW ,0

(Ω),

uH |ΓH= uH and uH |∂Ω\ΓH

= 0 known,

uW |ΓW= uW and uW |∂Ω\ΓW

= 0 known.



Heater example



−κ∆(u0 + uH + uW ) = f in Ω

u0 = 0 on ΓH

u0 = 0 on ΓW

κ∇u · n + βu0 = 0 on ΓR

∇u · n = 0 on ΓF

Testing the first equation with v ∈ H1ΓH∪ΓW ,0

(Ω), integrating andapplying Green’s formula leads to:



Heater example



∫Ωκ∇u0∇v dx −

∫ΓR∪ΓF

κ∇u · n v ds = −∫

Ωκ∇(uH + uW )∇v dx

whereby

∫ΓR∪ΓF

κ∇u · n v ds =

∫ΓR

κ∇u · n v ds +

∫ΓF

κ∇u · n v ds.



Heater example




∫ΓR∪ΓF

κ∇u · n v ds = −∫


whereby

∫ΓR∪ΓF

κ∇u · n v ds =

∫ΓR

κ∇u · n︸︷︷︸=−βu

v ds +

∫ΓF

κ∇u · n︸︷︷︸=0

v ds

.



Heater example




∫ΓR∪ΓF

κ∇u · n v ds = −∫


whereby

∫ΓR∪ΓF

κ∇u · n v ds = −∫

ΓR

βuv ds.



Heater example



Find u0 ∈ H1ΓH∪ΓW ,0

(Ω), such that

∫Ωκ∇u0∇v dx +

∫ΓR

βuv ds = −∫


for all v ∈ H1ΓH∪ΓW ,0

(Ω).



Heater example


Implementation

Analogously to the cases above.

Need stiffness and mass matrix K and M.

Restrict the matrix M∂Ω to the boundary nodes in ΓR , i.e. setall rows and columns not corresponding to nodes in ΓR to 0.



Heater example


Lagrange ansatz

Test with v ∈ H1(Ω) instead of v ∈ H1ΓH∪ΓR ,0

(Ω).

Use two multipliers λH ∈ H−1/2(ΓH), λW ∈ H−1/2(ΓW ) nowfor substitution of ∇u · n.Test the two inhomogeneous Dirichlet bc’s withµH ∈ H−1/2(ΓH), µW ∈ H−1/2(ΓW ).



Heater example



∫Ωκ∇(u)∇v dx =

∫∂Ωκ∇u · n v ds

=

∫ΓH

κ∇u · n v ds +

∫ΓW

κ∇u · n v ds

+

∫ΓR

κ∇u · n v ds +

∫ΓF

κ∇u · n v ds



Heater example



∫Ωκ∇u∇v dx =

∫∂Ωκ∇u · n v ds

=

∫ΓH

κ∇u · n︸︷︷︸λH

v ds +

∫ΓW

κ∇u · n︸︷︷︸λW

v ds

+

∫ΓR

κ∇u · n︸︷︷︸−βu

v ds +

∫ΓF

κ∇u · n︸︷︷︸=0

v ds



Heater example



Find u ∈ H1(Ω), λH ∈ H−1/2(ΓH), λW ∈ H−1/2(ΓW ), such that

∫Ωκ∇u∇v dx +

∫ΓR

βuv ds =

∫ΓH

κλHv ds +

∫ΓW

κλW v ds∫ΓH

uµH ds =

∫ΓH

uHµH ds∫ΓW

uµW ds =

∫ΓW

uWµW ds

for all v ∈ H1(Ω), µH ∈ H−1/2(ΓH), µW ∈ H−1/2(ΓW ).



Heater example


Implementation

Analogously to the cases above.

Need stiffness and mass matrix K and M.

Need restriction of M∂Ω to boundary nodes in ΓR from before.

Restrict the matrix M0 to the boundary nodes in ΓH , ΓW ,i.e.delete all rows of M0, which correspond to nodes not lying inΓH , ΓW .



Heater example


Numerical results

Dirichlet lift ansatz with β/κ = 0.1:



Heater example


Numerical results

Lagrange ansatz with β/κ = 0.1:



Heater example


Numerical results

Absolute difference between the two solutions:



Heater example


Numerical results




Heater example


Numerical results

Dirichlet lift ansatz with β/κ = 1:



Heater example


Numerical results

Dirichlet lift ansatz with β/κ = 10:



Heater example


Numerical results

Dirichlet lift ansatz with β/κ = 0.1, heater at position x = 0:



Heater example


Numerical results

Heater at position x = 0.1:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1 heater position = 0.1, ||u||2 = 15.5176

−10

−5

0

5

10

15

20

25

30



Heater example


Numerical results


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8


−10

−5

0

5

10

15

20

25

30



Heater example


Numerical results

Heater at position x = 1:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1 heater position = 1, ||u||2 = 19.4654

−10

−5

0

5

10

15

20

25

30



Heater example


Numerical results


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8


−10

−5

0

5

10

15

20

25

30



Heater example


Numerical results




Heater example


Thank you for yourattention!


Project 4 Inhomogeneous Dirichlet and Robin Boundary ... · Vinh Ma and Markus Wol Project 4...

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