Download - Fenics Discontinuous Galerkin
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FEniCS CourseLecture 10: Discontinuous Galerkinmethods for elliptic equations
ContributorsAndre Massing
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The discontinuous Galerkin (DG) method uses
discontinuous basis functions
u
uh
t
Vh = Pk(Th) = {vh L2() : vh|T P k(T ) T Th}
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The discontinuous Galerkin (DG) method uses
discontinuous basis functions
u
uh
t
Vh = Pk(Th) = {vh L2() : vh|T P k(T ) T Th}
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The discontinuous Galerkin (DG) method uses
discontinuous basis functions
u
uh
t
Vh = Pk(Th) = {vh L2() : vh|T P k(T ) T Th}
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The DG method eases mesh adaptivity
u
uh
t
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The DG method eases mesh adaptivity
u
uh
t
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The DG method eases mesh adaptivity
u
uh
t
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The DG method eases space adaptivity
u
uh
t
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The DG method eases space adaptivity
u
uh
t
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The DG method eases space adaptivity
u
uh
t
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DG-FEM Notation
Interface facets
Average v = 12(v+ + v)Jump [v] = (v+ v)
Boundary facet
v = [v] = v
Jump identity
[(hv)wh] = [hv]wh+ hv[wh]5 / 10
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The symmetric interior penalty method (SIP)
ah(uh, vh) =TT
Tuh vh dx
FF
Fuh n[vh] dS Consistency
FF
Fvh n[uh] dS Symmetry
+FF
hF
F
[uh][vh] dS Penalty
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The symmetric interior penalty method (SIP)
ah(uh, vh) =TT
Tuh vh dx
FF
Fuh n[vh] dS Consistency
FF
Fvh n[uh] dS Symmetry
+FF
hF
F
[uh][vh] dS Penalty
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The symmetric interior penalty method (SIP)
ah(uh, vh) =TT
Tuh vh dx
FF
Fuh n[vh] dS Consistency
FF
Fvh n[uh] dS Symmetry
+FF
hF
F
[uh][vh] dS Penalty
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The symmetric interior penalty method (SIP)
ah(uh, vh) =TT
Tuh vh dx
FF
Fuh n[vh] dS Consistency
FF
Fvh n[uh] dS Symmetry
+FF
hF
F
[uh][vh] dS Penalty
lh(vh) =
fvh dx
FFb
Fvh ng dS +
FFb
hF
Fgvh dS
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The symmetric interior penalty method (SIP)
ah(uh, vh) =TT
Tuh vh dx
FF
Fuh n[vh] dS Consistency
FF
Fvh n[uh] dS Symmetry
+FF
hF
F
[uh][vh] dS Penalty
lh(vh) =
fvh dx
FFb
Fvh ng dS +
FFb
hF
Fgvh dS
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Split of SIP form into interior and boundary
contribution
ah(uh, vh) =TT
Tuh vh dx
FFi
Fuh n[vh] dS Consistency
FFi
Fvh n[uh] dS Symmetry
+FFi
hF
F
[uh][vh] dS Penalty
FFb
Fuh nvh ds
Consistency
FFb
Fvh nuh ds
Symmetry
+FFb
hF
Fuhvh ds
Penalty7 / 10
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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: = [0, 1] [0, 1], D =
Source and boundary values:
f(x, y) = 200 cos(10pix) cos(10piy)
gD(x, y) = cos(10pix) cos(10piy)
Mission: Solve this PDE numerically by using the SIP method.Print the errornorm for both the L2 and the H1 norm for variousmesh sizes. For a UnitSquareMesh(128,128) the error should be0.0009166 and 0.1962, respectively.
Extra mission: Implement the NIP variant, solve the sameproblem and compare the H1 and L2 error for a range ofmeshes UnitSquareMesh(N,N), N = 2j , j = 2, , 7. Can youdetermine the order of convergence?
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