FEniCS Course - FEniCS Project

FEniCS Course
Lecture 10: Discontinuous Galerkin
methods for elliptic equations
Contributors
André Massing
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The discontinuous Galerkin (DG) method uses
discontinuous basis functions
u
u h
t
V h = P k (T h ) = {v h ∈ L 2 (Ω) : v h | T ∈ P k (T ) ∀ T ∈ T h }
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The discontinuous Galerkin (DG) method uses
discontinuous basis functions
u
u h
t
V h = P k (T h ) = {v h ∈ L 2 (Ω) : v h | T ∈ P k (T ) ∀ T ∈ T h }
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The discontinuous Galerkin (DG) method uses
discontinuous basis functions
u
u h
t
V h = P k (T h ) = {v h ∈ L 2 (Ω) : v h | T ∈ P k (T ) ∀ T ∈ T h }
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The DG method eases mesh adaptivity
u
u h
t
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The DG method eases mesh adaptivity
u
u h
t
3 / 10

The DG method eases mesh adaptivity
u
u h
t
3 / 10

The DG method eases space adaptivity
u
u h
t
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The DG method eases space adaptivity
u
u h
t
4 / 10

The DG method eases space adaptivity
u
u h
t
4 / 10

DG-FEM Notation
Interface facets
Boundary facet
Average 〈v〉 = 1 2 (v+ + v − )
〈v〉 = [v] = v
Jump [v] = (v + − v − )
Jump identity
[(∇ h v)w h ] = [∇ h v]〈w h 〉 + 〈∇ h v〉[w h ]
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The symmetric interior penalty method (SIP)
a h (u h , v h ) = ∑ ∫
∇u h · ∇v h dx − ∑ ∫
〈∇u h 〉 · n[v h ] dS
T ∈T
T
F ∈F
F
} {{ }
Consistency
− ∑ ∫
〈∇v h 〉 · n[u h ] dS + ∑ γ
[u h ][v h ] dS
F ∈F
F
h F
F ∈F
∫F
} {{ } } {{ }
Symmetry
Penalty
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The symmetric interior penalty method (SIP)
a h (u h , v h ) = ∑ ∫
∇u h · ∇v h dx − ∑ ∫
〈∇u h 〉 · n[v h ] dS
T ∈T
T
F ∈F
F
} {{ }
Consistency
− ∑ ∫
〈∇v h 〉 · n[u h ] dS + ∑ γ
[u h ][v h ] dS
F ∈F
F
h F
F ∈F
∫F
} {{ } } {{ }
Symmetry
Penalty
6 / 10

The symmetric interior penalty method (SIP)
a h (u h , v h ) = ∑ ∫
∇u h · ∇v h dx − ∑ ∫
〈∇u h 〉 · n[v h ] dS
T ∈T
T
F ∈F
F
} {{ }
Consistency
− ∑ ∫
〈∇v h 〉 · n[u h ] dS + ∑ γ
[u h ][v h ] dS
F ∈F
F
h F
F ∈F
∫F
} {{ } } {{ }
Symmetry
Penalty
6 / 10

The symmetric interior penalty method (SIP)
a h (u h , v h ) = ∑ ∫
∇u h · ∇v h dx − ∑ ∫
〈∇u h 〉 · n[v h ] dS
T ∈T
T
F ∈F
F
} {{ }
Consistency
− ∑ ∫
〈∇v h 〉 · n[u h ] dS + ∑ γ
[u h ][v h ] dS
F ∈F
F
h F
F ∈F
∫F
} {{ } } {{ }
Symmetry
Penalty
∫
l h (v h ) = fv h dx − ∑
〈∇v h 〉 · ng dS + ∑ γ
gv h dS
Ω
F ∈F b ∫F
h F
∫F
F ∈F b
6 / 10

The symmetric interior penalty method (SIP)
a h (u h , v h ) = ∑ ∫
∇u h · ∇v h dx − ∑ ∫
〈∇u h 〉 · n[v h ] dS
T ∈T
T
F ∈F
F
} {{ }
Consistency
− ∑ ∫
〈∇v h 〉 · n[u h ] dS + ∑ γ
[u h ][v h ] dS
F ∈F
F
h F
F ∈F
∫F
} {{ } } {{ }
Symmetry
Penalty
∫
l h (v h ) = fv h dx − ∑
〈∇v h 〉 · ng dS + ∑ γ
gv h dS
Ω
F ∈F b ∫F
h F
∫F
F ∈F b
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Split of SIP form into interior and boundary
contribution
a h (u h , v h ) = ∑ T ∈T
∫
− ∑
T
F ∈F i ∫F
∇u h · ∇v h dx − ∑
F ∈F i ∫F
〈∇u h 〉 · n[v h ] dS
} {{ }
Consistency
〈∇v h 〉 · n[u h ] dS + ∑ γ
[u h ][v h ] dS
} {{ }
Symmetry
− ∑
∇u h · nv h ds
F ∈F b ∫F
} {{ }
Consistency
+ ∑ γ
u h v h ds
h F
∫F
F ∈F b
− ∑
h F
∫F
F ∈F i
} {{ }
Penalty
F ∈F b ∫F
∇v h · nu h ds
} {{ }
Symmetry
} {{ }
Penalty 7 / 10

Useful FEniCS tools (I)
Access facet normals and local mesh size:
n = FacetNormal ( mesh )
h = CellSize ( mesh )
Restriction:
f = Function (V)
f(’+’)
grad (f)(’+’)
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Useful FEniCS tools (II)
Average and jump:
# define it yourself
h_avg = (h(’+’) + h(’-’))/2
# or use built -in expression
avg (h)
jump (v)
jump (v, n)
Integration on interior facets:
... *dS
alpha / h_avg * dot ( jump (v, n), jump (u, n))*dS
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Exercise
Solve our favorite Poisson problem given
• Domain:
Ω = [0, 1] × [0, 1],
∂Ω D = ∂Ω
• Source and boundary values:
f(x, y) = 200 cos(10πx) cos(10πy)
g D (x, y) = cos(10πx) cos(10πy)
Mission: Solve this PDE numerically by using the SIP method.
Print the errornorm for both the L 2 and the H 1 norm for various
mesh sizes. For a UnitSquareMesh(128,128) the error should be
0.0009166 and 0.1962, respectively.
Extra mission: Implement the NIP variant, solve the same
problem and compare the H 1 and L 2 error for a range of
meshes UnitSquareMesh(N,N), N = 2 j , j = 2, · · · , 7. Can you
determine the order of convergence?
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