[PPT] - Serendipity Virtual Element Methods L. Beir ao da Veiga, F. Brezzi, PowerPoint Presentation

SLIDE 1

Serendipity Virtual Element Methods

L. Beir˜

ao da Veiga, F. Brezzi, L.D. Marini, A. Russo

IMATI-C.N.R., Pavia, Italy Polytopal Element Methods in Mathematics and Engineering Atlanta, October 26-th, 2015

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 1 / 31

SLIDE 2

Outline

1

Origins

2

Basic idea of VEMs

3

General VEMs in 2 and 3 dimensions

4

VEMs and FEMs

5

Serendipity VEM The reduced degrees of freedom The operator DS The operator RS The Serendipity VEM spaces S-VEM, FEM, and S-FEM

6

Numerical results

7

Conclusions

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 2 / 31

SLIDE 3

In the beginning...MFD Assume that: E = pentagon and we want P2-accuracy.

~

E

~

point value average

We consider 11 degrees of freedom (values at vertexes and midpoints, plus the average on E). For every v ∈ C 0( ¯ E) we define Dv ∈ R11 to be the nodal values and the average of v: Dv =

(Dv)1, ..., (Dv)10, (Dv)11
Beir˜

ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 3 / 31

SLIDE 4

MFD-2

~

Our (local) discrete space will be R11 For P and Q in R11 we want [P, Q]E . to mimic, say

E

∇p · ∇q dE If p is (any) function p.w. P2 on ∂E, with Dp = P and q2 a polynomial of degree ≤ 2 such that Dq2 = Q then

E

∇p·∇q2=−

E

p ∆q2+

∂E

p∂q2 ∂n ≃−P11

E

∆q2+Simpson can be computed without knowing p (but only P = Dp).

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 4 / 31

SLIDE 5

MFD-3 According to the previous slide, we can compute [Dp, Dq]E ≃

E

∇p · ∇qdE using only the values of Dp and Dq in R11 whenever either p or q is a polynomial of degree ≤ 2. This obviously defines an 6 × 11 matrix R, from R11 to R6 that gives the values of [P, D(q2)]E for P ∈ R11 and q2 ∈ P2 ∼ R6. Then we can define a sort of scalar product in R11, for all P and all Q in R11, as [P, Q]E :=(RP)T·(RQ)+ Stabilization (on the kernel of R)

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 5 / 31

SLIDE 6

On the other side...FdV-based methods “Darcy problem”: u = −∇p, divu = f + B.C. From now on: p = 0 on the boundary. Following B. M. Fraeijs de Veubeke (1965) we observe that: given an approximation λh of p at the inter-element boundaries, and another approximation ph of p inside each element we can deduce an approximation uh of u by requiring, in each element E

E

uh · v =

E

ph divv +

∂E

λh (v · n) for all v. This defines an approximate gradient Gh : (λh, ph) → uh = Gh(λh, ph)

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 6 / 31

SLIDE 7

Use of FdV Darcy problem: u = −∇p, divu = f , p = 0 at ∂Ω

E

uh · v =

E

ph divv +

∂E

λh (v · n) for all v. You can then add a discretized conservation equation

E

divuh q =

E

f q for all q. Then you must close the system, with as many equations as there are λh’s, requiring “continuity” (for uh · n or for ph, or for a combination of the two) at the interelement boundaries. You can get zillions of methods.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 7 / 31

SLIDE 8

Approximate Gradient methods Remember that given a λh at the inter-element edges (or faces), and a ph inside each element, we can define an approximate gradient Gh : (λh, ph) → uh = Gh(λh, ph) by requiring, in each element E

E

Gh · v =

E

ph divv +

∂E

λh (v · n) for all v where v ranges over the space where you look for Gh. Then you look for a pair (λh, ph) such that

E
E

Gh(λh, ph) · Gh(µh, qh) =

Ω

f qh for all (λh, qh).

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 8 / 31

SLIDE 9

Approximate Gradient methods-2 Actually, you also need to add a stabilizing term like

E
E

Gh(λh, ph) · Gh(µh, qh) + C h−1

e

e

(λh − ph)(µh − qh) =

Ω

f qh, ∀µh ∀qh. Various methods distinguish themselves for the space where you look for the approximate gradient and for the type of stabilization used. You can get zillions of methods

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 9 / 31

SLIDE 10

Basic idea of VEMs

~

E

~

point value average

We take again 11 degrees of freedom (values at vertexes and midpoints, + the average on E). We define the space VE := {v| such that v|e ∈ P2(e) ∀ edge e, and ∆v ∈ P0(E)} It is easy to see that our 11 d.o.f.s are VE-unisolvent.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 10 / 31

SLIDE 11

Basic idea of VEMs-2 To every v ∈ VE we associate ΠE

2 v ∈ P2(E) defined by

E

∇(ΠE

2 v − v) · ∇q2 = 0 for all q2 ∈ P2(E)

E

(ΠE

2 v − v)dE = 0.

Note that the quantity (= right-hand side)

E

∇v · ∇q2 = −∆q2

E

v +

∂E

v ∂q2 ∂n is computable (from the dofs) ∀v ∈ VE and ∀q2 ∈ P2.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 11 / 31

SLIDE 12

Basic idea of VEMs-3 Discretized problem: find ph ∈ V such that

E
E

∇ΠE

2 ph·∇ΠE 2 qh+ S(ph, qh) =

E
E

f ΠE

2 qh ∀qh ∈ V

where the stabilizing term S(ph, qh) can be taken as S(ph, qh) :=

E
D(qh − ΠE

2 qh)

T ·

D(ph − ΠE

2 ph)

where D is the degrees of freedom vector defined before.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 12 / 31

SLIDE 13

Panorama In summary: AD-HOC functions (e.g. Rational, Baricentric,...); one field and numerical integration DG: One field, discontinuous MFD: Only dofs, no functions HDG WG HHO: Two/three polynomial fields (λ, p, (u)) VEM: One field, solution of a PDE.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 13 / 31

SLIDE 14

Original VEM-General order For k integer ≥ 1 we define Vk(E)={ϕ∈C 0( ¯ E): ϕ|e ∈Pk(e)∀ edge e, ∆ϕ ∈ Pk−2(E)}. The degrees of freedom in Vk(E) are taken as

the values of ϕ at the vertices,
e

ϕ qds for all q ∈ Pk−2(e) ∀ edge e,

E

ϕ qdE for all q ∈ Pk−2(E). It is immediate to verify that the degrees of freedom are unisolvent

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 14 / 31

SLIDE 15

Classical VEM-General case For k integer ≥ 1 and k∆ integer ≥ −1 we define Vk,k∆(E)={ϕ∈C 0( ¯ E): ϕ|e ∈Pk(e)∀ edge e, ∆ϕ ∈ Pk∆(E)}. The degrees of freedom in Vk(E) are taken as

the values of ϕ at the vertices,
e

ϕ qds for all q ∈ Pk−2(e) ∀ edge e,

E

ϕ qdE for all q ∈ Pk∆(E). It is immediate to verify that the degrees of freedom are

unisolvent. In general it is better to take k∆ ≥ k − 2.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 15 / 31

SLIDE 16

Classical VEM-General case -3D For k integer ≥ 1,and k∆, kf integers ≥ −1 we define Vk,kf ,k∆(E)={ϕ∈C 0( ¯ E): ϕ|f ∈Vk,kf ∀ face f , ∆ϕ ∈ Pk∆(E)} The following degrees of freedom are unisolvent

the values of ϕ at the vertices,
e

ϕ qds for all q ∈ Pk−2(e) ∀ edge e,

f

ϕ qds for all q ∈ Pkf (f ) ∀ face f ,

E

ϕ qdE for all q ∈ Pk∆(E).

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 16 / 31

SLIDE 17

VEM and FEM on triangles: degrees of freedom

VEM k=3 FEM k=2 FEM k=1 FEM k=3 VEM k=1 VEM k=2

Figure: Triangles: Classical FEM and Original VEM

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 17 / 31

SLIDE 18

VEM and FEM on quads: degrees of freedom

VEM k=3 FEM k=2 FEM k=1 FEM k=3 VEM k=1 VEM k=2

Figure: Quads: Classical FEM and Original VEM

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 18 / 31

SLIDE 19

The reduced dofs We choose an S with (k + 1)(k + 2)/2 ≤ S ≤ NE (NE=dimensiion ov Vk,k), and assume that the degrees of freedom of Vk,k are ordered in such a way that the first S δ1, δ2, ... δS have the following properties:

(B)

They include all the boundary dofs

(S )

∀pk ∈ Pk(E) we have: {δ1(pk) = δ2(pk) = ... = δS(pk) = 0} ⇒ {pk ≡ 0} Property (B) is there to ensure conformity, and is easy.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 19 / 31

SLIDE 20

The reduced dofs - Examples Once (B) is satisfied, you know that a polynomial pk that satisfies {δ1(pk) = δ2(pk) = ... = δS(pk) = 0} must be identically zero on the boundary. Let’s deal with (S ). To get Property (S ), on top of the bounday dofs:

n a triangle, you must include as many internal dofs

as the dimension of Pk−3,

n a square, you must include as many internal dofs

as the dimension of Pk−4,

n a regular n-gon you must include as many internal

dofs as the dimension of Pk−n. In general (even on very distorted polygons): you must have as many internal dofs as there are Pk-bubbles.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 20 / 31

SLIDE 21

VEMS on quads As we are allowing dramatic distortions, the number of Pk-bubbles depends also on the geometry of the element. Here are some examples on quadrilaterals for k = 4.

3 bubbles Number of P −bubbles on quadrilaterals

4

* * * * *

1 bubble 1 bubble

Figure: Allowed distortions for quadrilaterals

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 21 / 31

SLIDE 22

The lazy choice and the stingy choice As we have seen, the minimum number of internal degrees

f freedom that have to be kept depends on k, and on the

geometry of the element. Typically: is the dimension of Pk−η where η is the minimum number of straight lines necessary to cover all the boundary of the element. In practice, in the code, you can either check every element to compute its η (stingy choice), or treat every element as if it was a triangle (lazy choice). One or the other choice could be preferable, depending on the circumstances.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 22 / 31

SLIDE 23

The operator DS We assume now that, for a given k, we are given a set δ1, δ2, ... δS of degrees of freedom having the properties B and S , and we define the operator DS DS : Vk,k(E) → RS defined by DSϕ := (δ1(ϕ), ..., δS(ϕ)). Needless to say, the operator DS has the properties

DS can be computed using only the d.o.f δ1, ..., δS ,
DS q = 0 ⇒ q = 0

for all q ∈ Pk . S ≡ {DS is injective when applied to Pk}

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 23 / 31

SLIDE 24

The operator R We are now going to use DS to construct another

perator, RS, as follows: ∀ ϕ ∈ Vk,k(E) we define

RS ϕ ∈ Pk by (∗) (DSRSϕ − DSϕ, DSq)RS = 0 ∀q ∈ Pk, where (· , ·)RS is the “Euclidean scalar product” in RS. Property S ensures that the matrix (DSp, DSq)RS p, q ∈ Pk is nonsingular, so that for every ϕ (and hence for every r.h.s. (DS(ϕ), DSq)RS) (∗) has a unique solution RSϕ. RS is a projector Vk,k → Pk

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 24 / 31

SLIDE 25

The Serendipity VEM spaces The operator Rs that we constructed has the properties

RS is computable using only the d.o.f. δ1, ..., δS ,
RSqk = qk for all qk ∈ Pk ,

that allow us to construct the Serendipity VEM space V S

k (E)={ϕ∈Vk,k(E) s.t.δr(ϕ)=δr(RSϕ) ∀r =S+1,.., NE}

From the first S dofs we can compute RS and then using RS we compute all the others. And Pk ⊆ V S

k !

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 25 / 31

SLIDE 26

FEM and S-VEM: triangles

VEMS k=3 FEM k=2 FEM k=1 FEM k=3 VEMS k=1 VEMS k=2 Figure: Triangles: Classical FEM and Serendipity VEM

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 26 / 31

SLIDE 27

S-FEMS and S-VEM: quads

VEMS k=4 FEMS k=1 FEMS k=2 FEMS k=3 FEMS k=4 VEMS k=1 VEMS k=2 VEMS k=3

Figure: Quads: S-FEM (Arnold-Awanou) and S-VEM

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 27 / 31

SLIDE 28

Numerical results: Meshes

Figure: Trapezoidal mesh Figure: Voronoi-Lloyd mesh

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 28 / 31

SLIDE 29

Numerical results:Qk-FEM, S-FEM, and S-VEM on quads

10

−2

10

−1

10 10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

1 4

mean diameter relative L2 error

stingy serendipity iso−Qk

Figure: Trapezoidal mesh k = 3

10

−2

10

−1

10 10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

1 5

mean diameter relative L2 error

stingy serendipity iso−Qk

Figure: Trapezoidal mesh k = 4

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 29 / 31

SLIDE 30

Classical VEM and S-VEM (stingy, lazy) on V-Lloyd

10

−2

10

−1

10 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

1 4

mean diameter relative L2 error

stingy lazy classical VEM

Figure: Voronoi-Lloyd mesh k = 3

10

−2

10

−1

10 10

−9

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

1 5

mean diameter relative L2 error

stingy lazy classical VEM

Figure: Voronoi-Lloyd k = 4

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 30 / 31

SLIDE 31

Final comments and conclusions Serendipity Virtual Elements allow a drastic reduction

f the number of internal degrees of freedom.

In 3 dimensions, the degrees of freedom internal to faces are also reduced, making a big difference. As the older enhanced VEMs, the Serendipity VEMs allow an immediate computation of the L2-projection (of trial and test functions). On triangles we (finally!) reproduce classical FEMs. On quads, S-VEM imitate Serendipity FEM, without paying for distortions. And they preserve all the “geometric freedom” of the

riginal VEMs.

Beir˜ ao da Veiga, Brezzi, Marini, Russo (vv) VEM October 26-th, 2015 31 / 31