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

0.2 0.4 0.6 0.8 1 1.2

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


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.


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.



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.



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?



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.




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.




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).



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 .



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.



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.



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.



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.




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



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.


US mesh

49 polygons, 5686 vertices


US mesh

0.68 0.7 0.72 0.74 0.76

0.16

0.18

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0.24

0.26

GEORGIA

235 vertices


US mesh

0

0.5

1

0

0.5

1−0.3

−0.2

−0.1

0

0.1

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)


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


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


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.


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.


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.



Joining meshes 1

joined832 polygons, Ndof=6849, h

max = 1.77e−01, h

mean = 4.08e−02

00.2

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0

0.5

1−2

−1

0

1

2

3

VEM solution

A. Russo (Milan) VEM July 14, 2014 33 / 43


Joining meshes 1

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1error

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Joining meshes 2


max = 1.77e−01, h

mean = 4.03e−02

0

0.5

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−1

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3

VEM solution



Joining meshes 2

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1VEM solution

−10

−9

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−3



Joining meshes 3


max = 3.73e−01, h

mean = 3.46e−02



Joining meshes 3

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1VEM solution

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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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0.81

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1.5

2

2.5

3

3.5

4

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


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


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

0.1

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0.8

0.9

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


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


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


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


Poisson problem; load= Dirac mass at center

Virtual elements of degree 4: mesh and solution

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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



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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.


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]


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