Virtual Elements for the Stokes problem L. Beiro da Veiga in - - PowerPoint PPT Presentation

Virtual Elements for the Stokes problem

• L. Beirão da Veiga

in collaboration with: P . Antonietti, D. Mora, V. Gyrya,

• K. Lipnikov, G. Manzini, M. Verani

Polygonal Workshop 2012

17-19 September, Milan

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 1 / 24

Outline

A Virtual Element Method for the Stokes problem [BdV, Gyrya, Lipnikov, Manzini – JCP 2009, SISC 2010] Reduced bubbles [BdV, Lipnikov – SISC 2012] A definite positive formulation [Antonietti, BdV, Mora, Verani – in progress]

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 2 / 24

The Stokes problem

We consider the Stokes problem in two dimensions      − div (ν∇su) − ∇p = f in Ω div u = 0 in Ω u = 0

• n ∂Ω,

where Ω ⊂ R2 is a polygonal domain; the scalar field ν > 0 is assumed constant on Ω (only for simplicity!); the loading f is assumed in [L2(Ω)]2 As usual, the vector field u represents the velocities and the scalar field p the pressures.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 3 / 24

Variational formulation

The variational formulation reads      Find u ∈ V := [H1

0(Ω)]2, q ∈ Q := L2 0(Ω) such that

a(u, v) + (div v, p) = (f, v) ∀v ∈ V (div u, q) = 0 ∀q ∈ Q where the bilinear form a(u, v) =

• Ω

ν∇su : ∇sv dx.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 4 / 24

A Virtual Element Method

We will build a discrete problem in following form      Find uh ∈ Vh, qh ∈ Qh such that ah(uh, vh) + (div vh, ph) =< fh, vh >h ∀vh ∈ Vh (div uh, qh) = 0 ∀qh ∈ Qh where Vh ⊂ V and Qh ⊂ Q are discrete spaces; ah(·, ·) : Vh × Vh → R is a discrete bilinear form approximating the continuous form a(·, ·); < fh, vh >h is a right hand side term approximating the load

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 5 / 24

The local spaces Vh|K

Let Th be a polygonal mesh on Ω; we define the space of velocities Vh element by element.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 6 / 24

The local spaces Vh|K

Let Th be a polygonal mesh on Ω; we define the space of velocities Vh element by element. For all K ∈ Th: Vh|K =

• v ∈ [H1(K)]2 : div v constant, v minimizes ||∇v||2

L2(K),

v|e · te ∈ P1(e), v|e · ne ∈ P2(e) ∀e ∈ ∂K

• .
• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 6 / 24

The local spaces Vh|K

Let Th be a polygonal mesh on Ω; we define the space of velocities Vh element by element. For all K ∈ Th: Vh|K =

• v ∈ [H1(K)]2 : div v constant, v minimizes ||∇v||2

L2(K),

v|e · te ∈ P1(e), v|e · ne ∈ P2(e) ∀e ∈ ∂K

• .

Local degrees of freedom: NOTE: div v =

1 |K|

• ∂K v · nK ds

go back

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 6 / 24

The global spaces Vh and Qh

The global space Vh is built by assembling the local spaces Vh|K as usual. Vh = {v ∈ C0(Ω) : v|K ∈ Vh|K ∀K ∈ Th, v = 0 on ∂Ω} The d.o.f.s are two per internal vertex and one per internal edge. The space Qh is simply given by Qh = {q ∈ L2

0(Ω) : q ∈ P0(K) ∀K ∈ Th},

with one d.o.f. per element.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 7 / 24

The bilinear form ah(·, ·)

The bilinear form ah(·, ·) is built element by element ah(uh, vh) =

• K∈Th

aK

h (uh, vh)

∀ uh, vh ∈ Vh, where aK

h (·, ·) : Vh|K × Vh|K −

→ R are symmetric bilinear forms that mimic aK

h (·, ·) ≃ a(·, ·)|K

by satisfying a stability and a consistency condition.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 8 / 24

The bilinear form ah(·, ·)

Consistency: for all h and for all K ∈ Th it holds aK

h (p, vh) = aK(p, vh)

∀p ∈ [P1(K)]2, vh ∈ Vh|K. Stability: there exist two positive constants α∗ and α∗, independent of h and of K, such that α∗ aK(vh, vh) ≤ aK

h (vh, vh) ≤ α∗ aK(vh, vh)

∀vh ∈ Vh|K.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 9 / 24

The bilinear form ah(·, ·)

Consistency: for all h and for all K ∈ Th it holds aK

h (p, vh) = aK(p, vh)

∀p ∈ [P1(K)]2, vh ∈ Vh|K. Stability: there exist two positive constants α∗ and α∗, independent of h and of K, such that α∗ aK(vh, vh) ≤ aK

h (vh, vh) ≤ α∗ aK(vh, vh)

∀vh ∈ Vh|K. NOTE: the r.h.s. above is computable since integrating by parts aK(p, vh) = ν

• ∂K
• (∇sp)nK
• · vh ds

∀p ∈ [P1(K)]2, vh ∈ Vh|K.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 9 / 24

The discrete problem

Finally, we can write the Virtual Element problem      Find uh ∈ Vh, qh ∈ Qh such that ah(uh, vh) + (div vh, ph) =< fh, vh >h ∀vh ∈ Vh (div uh, qh) = 0 ∀qh ∈ Qh where the loading term < fh, vh >h=

• K∈Th

1 n

n

• i=1

vh(Vi)

• ·
• K

f dx.

Proposition

Assuming standard regularity conditions on the mesh family {Th}h, it holds ||u − uh||1 + ||p − ph||0 ≤ C h

K∈Th

(|u|2

2,K + |f|2 1,K)

1/2 .

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 10 / 24

Reduced edge bubble space

In particular, an inf-sup condition needs to be shown in order to prove the previous proposition: sup

vh∈Vh/{0}

• Ω qh div (vh) dx

||vh||H1(Ω) ≥ β||qh||L2(Ω) ∀qh ∈ Qh.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 11 / 24

Reduced edge bubble space

In particular, an inf-sup condition needs to be shown in order to prove the previous proposition: sup

vh∈Vh/{0}

• Ω qh div (vh) dx

||vh||H1(Ω) ≥ β||qh||L2(Ω) ∀qh ∈ Qh. The only role of the edge d.o.f.s (and the associated edge-quadratic bubbles) is to guarantee such inf-sup condition. Can those be neglected?

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 11 / 24

Macro elements

We find a sufficient condition for the stability of the discrete system when using only vertex degrees of freedom, i.e. NO edge dofs. We make use of a macro-element technique. For any ν internal vertes of Ωh, let the macro-element M = M(ν) = {K ∈ Th | ν ∈ ∂K}, i.e. the union of all elements sharing vertex ν.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 12 / 24

Reduced edge bubble space

We call a frontier e any collection of at least two adjoint edges such that e = ∂K ∩ ∂K ′, with K, K ′ ∈ M.

ν e1 e2 e3 Single edges Frontiers ν

Note: all the internal boundaries among elements in M can be divided into frontiers and (single) edges.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 13 / 24

Reduced edge bubble space

PROPOSITION Let ¯ NM indicate the number of edges which connect in ν AND are not a part of a frontier. Let either (a) ¯ NM < 3 or (b) ¯ NM = 3 and all the three angles naturally defined by the three respective edges be less or equal than π. Then, if this holds for all ν internal nodes, the inf-sup condition is satisfied.

ν e1 e2 e3 Single edges Frontiers ν

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 14 / 24

Reduced edge bubble space

PROPOSITION Let ¯ NM indicate the number of edges which connect in ν AND are not a part of a frontier. Let either (a) ¯ NM < 3 or (b) ¯ NM = 3 and all the three angles naturally defined by the three respective edges be less or equal than π. Then, if this holds for all ν internal nodes, the inf-sup condition is satisfied.

ν e1 e2 e3 Single edges Frontiers ν

NOTE: the original result is more general (see paper).

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 14 / 24

Reduced edge bubble space

For example the above result shows that following meshes are uniformly stable without the need of any bubble degree of freedom. For instance, (hexagonal) Voronoi meshes always satisfy the inf-sup.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 15 / 24

Sample numerical tests

We compute the discrete inf-sup constant β = β(h) for different sizes h for meshes of the following kind ADDITIONAL OPTION: selected bubbles can be added (right figure).

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 16 / 24

Computed value of β

1/h Poly-mesh Quads Quads + bubbles 8 4.25e-2 1.78e-2 1.29e-1 16 3.72e-2 4.65e-3 1.30e-1 32 3.48e-2 1.18e-3 1.29e-1 64 3.38e-2 2.96e-4 1.32e-1 128 3.35e-2 7.50e-5 1.30e-1 NOTE: orthogonal space to null mode taken in Quads.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 17 / 24

A convergence test

We consider the poly-mesh and solve a problem with solution u(x, y) =

• r(x) sin(a y)

r ′(x) cos(a y)/a

• ,

p(x, y) = xy2, where r(x) = (1 − x) sin(a x), r ′ is its first derivative, and a = 2.2 π. RELATIVE ERRORS 1/h ||u − uh||0,h ||uI − uh||1,h ||pI − ph||0 8 1.24e-1 1.71e-1 1.97e-0 16 3.21e-2 6.83e-2 5.42e-1 32 7.74e-3 2.73e-2 1.79e-1 64 1.92e-3 1.26e-2 6.79e-2 128 4.77e-4 6.13e-3 2.91e-2

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 18 / 24

A characterization of the div-free subspace

Let the subspace Zh ⊂ Vh Zh = {vh ∈ Vh : (div vh, qh) = 0 ∀qh ∈ Qh}

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 19 / 24

A characterization of the div-free subspace

Let the subspace Zh ⊂ Vh Zh = {vh ∈ Vh : (div vh, qh) = 0 ∀qh ∈ Qh} = {vh ∈ Vh : div vh = 0}, since div vh is piecewise constant.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 19 / 24

A characterization of the div-free subspace

Let the subspace Zh ⊂ Vh Zh = {vh ∈ Vh : (div vh, qh) = 0 ∀qh ∈ Qh} = {vh ∈ Vh : div vh = 0}, since div vh is piecewise constant. Then, with respect to the velocity variable, the original VEM stokes problem      Find uh ∈ Vh, qh ∈ Qh such that ah(uh, vh) + (div vh, ph) =< fh, vh >h ∀vh ∈ Vh (div uh, qh) = 0 ∀qh ∈ Qh is equivalent to Find uh ∈ Zh such that ah(uh, vh) =< fh, vh >h ∀vh ∈ Zh

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 19 / 24

A characterization of the div-free subspace

Let the subspace Zh ⊂ Vh Zh = {vh ∈ Vh : (div vh, qh) = 0 ∀qh ∈ Qh} = {vh ∈ Vh : div vh = 0}, since div vh is piecewise constant. Then, with respect to the velocity variable, the original VEM stokes problem      Find uh ∈ Vh, qh ∈ Qh such that ah(uh, vh) + (div vh, ph) =< fh, vh >h ∀vh ∈ Vh (div uh, qh) = 0 ∀qh ∈ Qh is equivalent to Find uh ∈ Zh such that ah(uh, vh) =< fh, vh >h ∀vh ∈ Zh We will characterize Zh using a discrete stream space Wh Zh = {curlwh : wh ∈ Wh}.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 19 / 24

The local spaces Wh

We define the scalar space Wh ⊂ H2(Ω) element by element. (Ω simply connected)

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 20 / 24

The local spaces Wh

We define the scalar space Wh ⊂ H2(Ω) element by element. (Ω simply connected) For all K ∈ Th: Wh|K =

• w ∈ H2(K) : minimizes ||∇curlw||2

L2(K),

w|e ∈ P3(e), ∂nw|e ∈ P1(e) ∀e ∈ ∂K

• .
• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 20 / 24

The local spaces Wh

We define the scalar space Wh ⊂ H2(Ω) element by element. (Ω simply connected) For all K ∈ Th: Wh|K =

• w ∈ H2(K) : minimizes ||∇curlw||2

L2(K),

w|e ∈ P3(e), ∂nw|e ∈ P1(e) ∀e ∈ ∂K

• .

Local degrees of freedom: NOTE: curlwh · n ∈ P2(e) ∀e ∈ ∂K curlwh · t ∈ P1(e) ∀e ∈ ∂K

• riginal
• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 20 / 24

A stream formulation

We can write the original problem as

• Find uh ∈ Zh such that

ah(uh, vh) =< fh, vh >h ∀vh ∈ Zh

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 21 / 24

A stream formulation

Due to the previous construction, it is equivalent to

• Find ψh ∈ Wh such that

ah(curlψh, curlψhwh) =< fh, curlwh >h ∀wh ∈ Wh

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 21 / 24

A stream formulation

Due to the previous construction, it is equivalent to

• Find ψh ∈ Wh such that

ah(curlψh, curlψhwh) =< fh, curlwh >h ∀wh ∈ Wh Let AK represent the local stiffness matrix associated to aK

h (·, ·).

We simply build a matrix CK associated to the curlh discrete operator: W(K)

✲

curl V(K) Wh|K

✲

curlh Vh|K

❄ ❄

Interp. Interp. Then the new local matrixes become (CK)TAKCK.

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 21 / 24

Sample numerical tests

We consider the Stokes problem with known solution u1(x, y) = − cos(2πx) sin(2πy) + sin(2πy), u2(x, y) = sin(2πx) cos(2πy) − sin(2πx), p(x, y) = xy2 − 1 6. and test for two families of meshes:

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 22 / 24

Sample numerical tests

We have the following results:

Mesh 1/h d.o.f. (new) d.o.f. (orig.) |u − uh|1 8 273 696 2.3822 16 1212 2951 1.2100 T 1

h

32 5040 12011 0.6274 64 20739 48898 0.3160 8 96 145 3.7972 16 384 545 2.1249 T 2

h

32 1536 2113 0.9274 64 6144 8321 0.3762 128 24576 33025 0.1665

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 23 / 24

Sample numerical tests

We have the following results:

Mesh 1/h d.o.f. (new) d.o.f. (orig.) |u − uh|1 8 273 696 2.3822 16 1212 2951 1.2100 T 1

h

32 5040 12011 0.6274 64 20739 48898 0.3160 8 96 145 3.7972 16 384 545 2.1249 T 2

h

32 1536 2113 0.9274 64 6144 8321 0.3762 128 24576 33025 0.1665

+ smaller system (note: analogous in terms of non-zero entries) + definite positive system

• condition number is worst (but preconditioning of D.P

. systems..)

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 23 / 24

Conclusions

We have shown a low order VE method for Stokes We have exploited the VEM flexibility to gain advantages. Reduced bubble formulation:

easy ot chek rule; for convex polygons with 5 − 6 or more edges a pure nodal formulation is often stable.

Characterization of the divergence free discrete space:

allows for a definite positive formulation; can be applied also to the higher order VEM below.

Arbitrary order elements for Stokes (incompressible elasticity) also exist and are very efficient, see [BdV, Brezzi, Marini – submitted].

• L. Beirão da Veiga et al. (Univ. of Milan)

VEM for Stokes Polygonal Workshop 2012 24 / 24