Virtual Element Methods for general second order elliptic ... · University of Milano-Bicocca,...
Transcript of Virtual Element Methods for general second order elliptic ... · University of Milano-Bicocca,...
Virtual Element Methodsfor general second order elliptic problems
Alessandro Russo
Department of Mathematics and ApplicationsUniversity of Milano-Bicocca, Milan, Italy
andIMATI-CNR, Pavia, Italy
Workshop onPolytopal Element Methods in Mathematics and Engineering
26–28 October 2015
Georgia Tech, Atlanta
A. Russo (Milan) VEM October 26, 2015 1 / 27
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Outline of the presentation
Outline of the presentation
Virtual Element Spaces
Projectors
VEM approximation of general elliptic equations
Primal formMixed form
Numerical Experiments
Conclusions
Joint work with L. Beirao da Veiga, F. Brezzi, D. Marini
A. Russo (Milan) VEM October 26, 2015 2 / 27
Elliptic problem in primal form
Elliptic problem in primal form
We want to approximate a general second-order elliptic equation in 3Dwith an arbitrary polyhedral mesh with a conforming finite elementmethod of order k .
−div (κ∇u) + β · ∇u + αu = f in Ω ⊂ R3
u = g on ∂Ω
Beirao da Veiga, Brezzi, Marini, R.: Virtual Element Methods forgeneral second order elliptic problems on polygonal meshes, to appear inMathematical Models and Methods in Applied Sciences (alsoarXiv:1412.2646)
We start with the two-dimensional case.
A. Russo (Milan) VEM October 26, 2015 3 / 27
Virtual Element spaces
The local finite element space Vk(E )
Let E be a polygon. We would like to define a finite element space Vk(E )on E such that:
Vk(E ) contains the space Pk(E ) of polynomials of degree less than orequal to k (plus other non-polynomial functions);
a function in Vk(E ) restricted to an edge e is in Pk(e), so that if twopolygons E and E ′ have an edge in common, the two spaces Vk(E )and Vk(E ′) must “glue” in C 0(E ∪ E ′);
given a function vh ∈ Vk(E ), I can compute its L2 projection Π0kvh in
Pk(E ), and this is all the information of vh that I need in order toapproximate my equation.
A. Russo (Milan) VEM October 26, 2015 4 / 27
Virtual Element spaces
Meaning of “I can compute”
In what follows the precise meaning of the statement
I can compute Π0kvh
is:
given the array dofi (vh), I can directly compute Π0kvh
The same applies for all other quantities which are computable from vh.
A. Russo (Milan) VEM October 26, 2015 5 / 27
Virtual Element spaces
The space Vk,k(E )
Given a polygon E , consider the following space (already mentioned inFranco’s talk):
Vk,k(E ) := vh ∈ C 0(E ) such that: vh|e ∈ Pk(e), ∆vh ∈ Pk(E )
If the polygon E has NE edges (and vertices), it is clear that
dimVk,k(E ) = k NE + dimPk(E ) = k NE +(k + 1)(k + 2)
2
Is the space Vk,k(E ) OK for us?
A. Russo (Milan) VEM October 26, 2015 6 / 27
Virtual Element spaces
Degrees of freedom in Vk ,k(E )
Let (xE , yE ) be the centroid of E and hE its diameter. If α = (α1, α2) is a
multiindex we define the scaled monomials of degree |α| = α1 + α2:
mα(x , y) :=
(x − xEhE
)α1(y − yEhE
)α2
.
The set mα, with |α| ≤ k is a basis for Pk(E ).
As degrees of freedom in Vk,k(E ), we choose:
the value of vh at the vertices and at k − 1 equally spaced points oneach edge;
the (scaled) moments1
|E |
∫Evhmα for |α| ≤ k.
A. Russo (Milan) VEM October 26, 2015 7 / 27
Virtual Element spaces
Degrees of freedom in Vk ,k(E )
It can be easily shown that:
the degrees of freedom above are unisolvent in Vk,k(E ).
Short proof: suppose that vh ∈ Vk,k(E ) is zero on the boundary and∫Evh mα = 0 for |α| ≤ k
We show that vh ≡ 0. For,∫E|∇vh|2 = −
∫E
∆vh vh = 0 since ∆vh ∈ Pk(E )
Hence vh ≡ 0.
A. Russo (Milan) VEM October 26, 2015 8 / 27
Virtual Element spaces
Degrees of freedom in Vk ,k(E )
The choice of the scaled moments1
|E |
∫Evhmα among the degrees of
freedom implies that, starting from the degrees of freedom of vh, I cantrivially compute
Π0kvh := L2 projection of vh onto Pk(E ).
• It is clear that if I could compute directly the bilinear form on the spaceVk,k(E ), the resulting finite element method would converge with the rightoptimal rates.
• For the time being, assume that the L2 projection onto Pk(E ) is enough(we have a general way of approximating the bilinear form just using theL2 projection).
A. Russo (Milan) VEM October 26, 2015 9 / 27
Virtual Element spaces
Basis functions in Vk ,k(E )
For i = 1, . . . ,Ndof we define ϕi as the function in Vk,k(E ) such that
dofj(ϕi ) = j-th degree of freedom of ϕi =
1 if i = j
0 if i 6= j
We have the usual Lagrange-type expansion
vh =Ndof∑i=1
dofi (vh)ϕi .
A. Russo (Milan) VEM October 26, 2015 10 / 27
Virtual Element spaces
Is the space Vk ,k(E ) OK for the VEM?
. . . yes, but IT HAS TOO MANY DEGREES OF FREEDOM!
We can satisfy the same properties with a (proper) subspace of Vk,k(E ).
Original VEM:
define a projector Π∇k : Vk,k(E )→ Pk(E ) which uses only themoments up to degree k − 2;identify a subspace Vk(E ) ⊂ Vk,k(E ) requiring that∫
Evh mα =
∫E
Π∇k vh mα for |α| = k − 1, k
the dimension of this subspace is k NE + dimPk−2(E ) and the
internal degrees of freedom are the (scaled) moments up to k−2;
in this subspace we can compute Π0kvh.
A. Russo (Milan) VEM October 26, 2015 11 / 27
Virtual Element spaces
Is the space Vk ,k(E ) OK for the VEM?
Serendipity VEM:
define a projector ΠSk : Vk,k(E )→ Pk(E ) which uses only themoments up to degree k − NE ;
identify a subspace Vk(E ) ⊂ Vk,k(E ) requiring that∫Evh mα =
∫E
ΠSk vh mα for |α| = k − NE + 1, . . . , k
the dimension of this subspace is k NE + dimPk−NE(E ) and the
internal degrees of freedom are the (scaled) moments up tok − NE ;
in this subspace we can compute Π0kvh.
A. Russo (Milan) VEM October 26, 2015 12 / 27
Virtual Element spaces
The local finite element space Vk(E )
Let E be a polygon. We have defined a finite element space Vk(E ) on Esuch that:
Vk(E ) contains the space Pk(E ) of polynomials of degree less than orequal to k (plus other non-polynomial functions);
a function in Vk(E ) restricted to an edge e is in Pk(e), so that if twopolygons E and E ′ have an edge in common, the two spaces Vk(E )and Vk(E ′) must “glue” in C 0(E ∪ E ′);
given a function vh ∈ Vk(E ), I can compute its L2 projection Π0kvh in
Pk(E ), and this is all the information of vh that I need in order toapproximate my equation.
A. Russo (Milan) VEM October 26, 2015 13 / 27
Virtual Element spaces
Computing another projector in Vk(E ): Π0k−1∇vh
We show now that we can easily compute Π0k−1∇vh .
To compute Π0k−1∇vh we need to know the moments of ∇vh up to order
k − 1: ∫E
∂vh∂x
mβ = −∫Evh∂mβ
∂x+
∫∂E
vh mβ nx , |β| ≤ k − 1
and both terms are computable directly from the degrees of freedom of vh.
A. Russo (Milan) VEM October 26, 2015 14 / 27
VEM approximation of general elliptic equations
VEM approximation of general elliptic equations
We consider now a general second order elliptic operator with variablecoefficients:
−div (κ∇u) + β · ∇u + αu = f
and we approximate the various local consistency terms as:∫Eκ∇uh · ∇vh
∫Eκ[Π0k−1∇uh
]·[Π0k−1∇vh
]∫E
(β · ∇uh
)vh
∫E
(β ·[Π0k−1∇uh
])Π0kvh∫
Eα uh vh
∫Eα[Π0kuh] [
Π0kvh]
and for the right-hand-side:∫Ef vh
∫Ef Π0
kvh(Π0k−2vh is enough for k ≥ 2
)A. Russo (Milan) VEM October 26, 2015 15 / 27
VEM approximation of general elliptic equations
VEM approximation of general elliptic equations
The approximations above produce, as usual, rank-deficient matrices thatmust be stabilized.
For the stabilization we can take the following term:
S((I − Π0k)uh, (I − Π0
k)vh)
with S(ϕi , ϕj) = δij .
This means that we expand Π0kϕi in terms of the ϕj ’s and then we replace
S(ϕi , ϕj) with the identity matrix.
A. Russo (Milan) VEM October 26, 2015 16 / 27
US mesh
49 polygons, 5686 vertices
A. Russo (Milan) VEM October 26, 2015 17 / 27
US mesh
0.68 0.7 0.72 0.74 0.76
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GEORGIA
235 vertices
A. Russo (Milan) VEM October 26, 2015 18 / 27
US mesh
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VEM solution on skeleton
exact solution = 4th degree polynomial
L2 error
k = 2: 1.16× 10−3
k = 3: 1.43× 10−4
k = 4: 2.92× 10−13
(patch test)
A. Russo (Milan) VEM October 26, 2015 19 / 27
Mixed VEM
The continuous problem in mixed form
Ω ⊂ R2 (polygonal) computational domain, κ, γ,b smooth.Assume that the problem
L p := div(−κ(x)∇p + b(x)p) + γ(x) p = f (x) in Ω
p = 0 on Γ
is solvable for any f ∈ H−1(Ω), and
‖p‖1,Ω ≤ C‖f ‖−1,Ω, ‖p‖2,Ω ≤ C‖f ‖0,Ω
Existence and uniqueness as well for the adjoint operator
L∗p := div(−κ(x)∇p)− b(x) · ∇p + γ(x) p
In particular, ∀f ∈ L2(Ω) ∃ϕ ∈ H2(Ω) ∩ H10 (Ω):
L∗ϕ = f , ‖ϕ‖2,Ω ≤ C ∗‖f ‖0,Ω
A. Russo (Milan) VEM October 26, 2015 20 / 27
Mixed VEM
The continuous problem in mixed form
Setting: ν = κ−1, β = κ−1b and νu := −∇p + βp, we have
div u + γ p = f in Ω, p = 0 on Γ
V = H(div,Ω), Q = L2(Ω)
Find (u, p) ∈ V × Q such that
(νu, v)− (p, div v)− (β · v, p) = 0 ∀v ∈ V
(div u, q) + (γp, q) = (f , q) ∀q ∈ Q
A. Russo (Milan) VEM October 26, 2015 21 / 27
Mixed VEM
VEM approximation: the discrete spaces
The spaces: We define, for k integer ≥ 0,
V kh (E ) := v ∈ H(div;E ) ∩ H(rot;E ) : v · n|e ∈ Pk(e) ∀e ∈ ∂E ,
div v ∈ Pk(E ), and rot v ∈ Pk−1(E ).V kh := v ∈ H(div; Ω) such that v|E ∈ V k
h (E ) ∀E ∈ ThQk
h := q ∈ L2(Ω) such that: q|E ∈ Pk(E ) ∀ element E inTh• degrees of freedom in V k
h :∫ev · n pk ds ∀e,∀ pk ∈ Pk(e)∫
Ev · gk−1dx ∀E , ∀gk−1 ∈ Gk−1(E ),∫Ev · g⊥k dx ∀E , ∀g⊥k ∈ G⊥k (E )
(Gk−1 = ∇Pk , G⊥k (E ) = L2(E ) orthogonal of Gk(E ) in (Pk(E ))2)A. Russo (Milan) VEM October 26, 2015 22 / 27
Mixed VEM
The bilinear forms
aEh (v,w) := (νΠ0kv,Π
0kw)0,E + SE (v − Π0
kv,w − Π0kw)
α∗aE (v, v) ≤ SE (v, v) ≤ α∗aE (v, v) ∀v ∈ V k
h
Setah(v,w) :=
∑E
aEh (v,w).
Lemma
The bilinear form ah(·, ·) is continuous and elliptic in (L2)2:
∃M > 0 such that |ah(v,w)| ≤ M‖v‖0‖w‖0 ∀v,w ∈ V kh ,
∃α > 0 such that ah(v, v) ≥ α‖v‖20 ∀v ∈ V k
h ,
with M and α depending on ν but independent of h.
A. Russo (Milan) VEM October 26, 2015 23 / 27
Mixed VEM
The discrete problem
(∗)
Find (uh, ph) ∈ V k
h × Qkh such that
ah(uh, vh)− (ph, div vh)− (β · Π0kvh, ph) = 0 ∀vh ∈ V k
h
(div uh, qh) + (γph, qh) = (f , qh) ∀qh ∈ Qh.
Theorem
Assume that the continuous problem has a unique solution p. Then, for hsufficiently small, (∗) has a unique solution (uh, ph) ∈ V k
h × Qkh , and
‖p − ph‖0 ≤ Chk+1(‖u‖k+1 + ‖p‖k+1
),
‖u− uh‖0 ≤ Chk+1(‖u‖k+1 + ‖p‖k+1
),
‖ div(u− uh)‖0 ≤ Chk+1(|f |k+1 + ‖p‖k+1
)with C a constant depending on ν,β, and γ but independent of h.
A. Russo (Milan) VEM October 26, 2015 24 / 27
Mixed VEM
Superconvergence results
Theorem
Let ph be the discrete solution, and let pI ∈ Qkh be the interpolant of p.
Then, for h sufficiently small,
‖pI − ph‖0 ≤ C hk+2(‖u‖k+1 + ‖p‖k+1 + |f |k+1
),
where C is a constant depending on ν, β, and γ but independent of h.
A. Russo (Milan) VEM October 26, 2015 25 / 27
Numerical Experiments
Joining meshes 1
joined832 polygons, Ndof=6849, h
max = 1.77e−01, h
mean = 4.08e−02
00.2
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1−2
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VEM solution
A. Russo (Milan) VEM July 14, 2014 33 / 43
Numerical Experiments
Joining meshes 1
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1error
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A. Russo (Milan) VEM July 14, 2014 34 / 43
Numerical Experiments
Joining meshes 2
joined852 polygons, Ndof=7033, h
max = 1.77e−01, h
mean = 4.03e−02
0
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1−2
−1
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VEM solution
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Numerical Experiments
Joining meshes 2
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1VEM solution
−10
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−6
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−4
−3
A. Russo (Milan) VEM July 14, 2014 36 / 43
Numerical Experiments
Joining meshes 3
joined1573 polygons, Ndof=10389, h
max = 3.73e−01, h
mean = 3.46e−02
A. Russo (Milan) VEM July 14, 2014 37 / 43
Numerical Experiments
Joining meshes 3
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−2
A. Russo (Milan) VEM July 14, 2014 38 / 43
Numerical results
div(−κ∇p + bp) + γp = f in [0, 1]× [0, 1]
κ(x , y) =
(x2 + y2 −xy−xy x2 + y2
)b(x , y) = [x , y ] γ(x , y) = x2 + y2
Exact solution:
p(x , y) = x2 + y2 + cos(7y) sin(3x) + 2, u = −κ∇p + bp
Donatella Marini (Pavia) VEM variable Ferrara 2015 11 / 23
Exact solution pe
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x
x2+y2+cos(y 7.0) sin(x 3.0)+2.0
y
exact solution
Errors: uniform triangulation, k = 1
10−2
10−1
100
10−4
10−3
10−2
10−1
1.94
1.98
2.00
k = 1
1
2
|| u − uh ||
0,Ω
10−2
10−1
100
10−6
10−5
10−4
10−3
10−2
1.97
1.99
2.00
2.93
2.98
3.00
k=1
1
2
|| p − pI ||
0,Ω|| p
I − p
h ||
0,Ω
Donatella Marini (Pavia) VEM variable Ferrara 2015 13 / 23
Errors: uniform triangulation, k = 3
10−2
10−1
100
10−8
10−7
10−6
10−5
10−4
10−3
3.96
3.99
4.00
k = 3
1
4
|| u − uh ||
0,Ω
10−2
10−1
100
10−9
10−8
10−7
10−6
10−5
10−4
10−3
3.97
3.99
4.00
4.95
4.99
5.00
k=3
1
4
|| p − pI ||
0,Ω|| p
I − p
h ||
0,Ω
Donatella Marini (Pavia) VEM variable Ferrara 2015 14 / 23
Voronoi mesh, k = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
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1
lloyd−25−0
10−2
10−1
100
10−4
10−3
10−2
10−1
1.98
1.95
1.89
k = 1
1
2
|| u − uh ||
0,Ω
10−2
10−1
100
10−4
10−3
10−2
10−1
2.54
1.95
1.97
2.89
2.54
3.09
k=1
1
2
|| p − pI ||
0,Ω|| p
I − p
h ||
0,Ω
Donatella Marini (Pavia) VEM variable Ferrara 2015 15 / 23
Voronoi mesh, k = 1
10−2
10−1
100
10−4
10−3
10−2
10−1
1.98
1.95
1.89
k = 1
1
2
|| u − uh ||
0,Ω
10−2
10−1
100
10−4
10−3
10−2
10−1
2.54
1.95
1.97
2.89
2.54
3.09
k=1
1
2
|| p − pI ||
0,Ω|| p
I − p
h ||
0,Ω
Donatella Marini (Pavia) VEM variable Ferrara 2015 15 / 23
Voronoi mesh, k = 3
10−2
10−1
100
10−7
10−6
10−5
10−4
10−3
10−2
4.25
3.63
3.66
k = 3
1
4
|| u − uh ||
0,Ω
10−2
10−1
100
10−7
10−6
10−5
10−4
10−3
10−2
4.96
3.95
3.50
5.43
4.08
4.83
k=3
1
4
|| p − pI ||
0,Ω|| p
I − p
h ||
0,Ω
Donatella Marini (Pavia) VEM variable Ferrara 2015 16 / 23
Poisson problem; load= Dirac mass at center
Finite elements of degree 4: mesh and solution
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Generated by conn2mesh on 12−Mar−2015 18:45:11
Donatella Marini (Pavia) VEM variable Ferrara 2015 17 / 23
Poisson problem; load= Dirac mass at center
Virtual elements of degree 4: mesh and solution
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joined
Donatella Marini (Pavia) VEM variable Ferrara 2015 18 / 23
Advection dominated case
−ε∆p + β · ∇p = 0 in [0, 1]× [0, 1]p = (1− cos(2πy)/2 on x = 0, p = 0 on x = 1∂p
∂n= 0 on y = 0, y = 1
Test case: ε = 10−2, β = (1,−1/3).Virtual elements of degree 1: mesh and solution
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VEM solution − k=1
Donatella Marini (Pavia) VEM variable Ferrara 2015 19 / 23
Advection dominated case
−ε∆p + β · ∇p = 0 in [0, 1]× [0, 1]p = (1− cos(2πy)/2 on x = 0, p = 0 on x = 1∂p
∂n= 0 on y = 0, y = 1
Test case: ε = 10−2, β = (1,−1/3).Virtual elements of degree 1: mesh and solution
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VEM solution − k=1
Donatella Marini (Pavia) VEM variable Ferrara 2015 19 / 23
Advection dominated case
−ε∆p + β · ∇p = 0 in [0, 1]× [0, 1]p = (1− cos(2πy)/2 on x = 0, p = 0 on x = 1∂p
∂n= 0 on y = 0, y = 1
Test case: ε = 10−2, β = (1,−1/3).Virtual elements of degree 1: mesh and solution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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VEM solution − k=1
Donatella Marini (Pavia) VEM variable Ferrara 2015 19 / 23
Advection dominated case
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Donatella Marini (Pavia) VEM variable Ferrara 2015 20 / 23
Advection dominated case
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VEM solution − k=1
Donatella Marini (Pavia) VEM variable Ferrara 2015 20 / 23
Advection dominated case
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1
joined
Donatella Marini (Pavia) VEM variable Ferrara 2015 21 / 23
Advection dominated case
0
0.2
0.4
0.6
0.8
1 00.2
0.40.6
0.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
VEM solution − k=1
Donatella Marini (Pavia) VEM variable Ferrara 2015 21 / 23
Advection dominated case
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
joined
Donatella Marini (Pavia) VEM variable Ferrara 2015 22 / 23
Advection dominated case
0
0.2
0.4
0.6
0.8
1 00.2
0.40.6
0.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
VEM solution − k=1
Donatella Marini (Pavia) VEM variable Ferrara 2015 22 / 23
The VEM paradigm
The VEM paradigm
We believe that the Virtual Element Method has a wide range ofapplicability. The keypoints of VEM are:
The definition of the local finite element spaces and of the associateddegrees of freedom. These spaces contain polynomials plus otherfunctions which are not computable.
The costruction of variuos projectors onto polynomial spaces.
The definition of a consistent and stable bilinear form using theseprojectors which is the VEM approximation of the exact bilinear form.
A general theorem that guarantees convergence for a consistent andstable approximate bilinear form.
A. Russo (Milan) VEM October 26, 2015 26 / 27
Basic references
Basic References
Virtual Element Methods for general second order elliptic problemson polygonal meshes, Beirao da Veiga, Brezzi, Marini, R.: to appear inMath. Models and Methods in Applied Sciences (also arXiv:1506.07328[math.NA])
Mixed Virtual Element Methods for general second order ellipticproblems on polygonal meshes, Beirao da Veiga, Brezzi, Marini, R.: toappear in ESAIM: Math. Modelling and Numerical Analysis (alsoarXiv:1412.2646 [math.NA])
The Hitchhiker’s Guide to the Virtual Element Method Beirao daVeiga, Brezzi, Marini, R., Math. Models and Methods in Applied SciencesVol. 24, No. 8 (2014), pp. 1541–1573.
Thanks for your [email protected]
A. Russo (Milan) VEM October 26, 2015 27 / 27