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 ∇ s u ) − ∇ p = f in Ω div u = 0 in Ω u = 0 on ∂ Ω , where Ω ⊂ R 2 is a polygonal domain; the scalar field ν > 0 is assumed constant on Ω (only for simplicity!); the loading f is assumed in [ L 2 (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 := [ H 1 0 (Ω)] 2 , q ∈ Q := L 2 0 (Ω) such that a ( u , v ) + ( div v , p ) = ( f , v ) ∀ v ∈ V ( div u , q ) = 0 ∀ q ∈ Q where the bilinear form � ν ∇ s u : ∇ s v d x . a ( u , v ) = Ω 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 u h ∈ V h , q h ∈ Q h such that a h ( u h , v h ) + ( div v h , p h ) = < f h , v h > h ∀ v h ∈ V h ( div u h , q h ) = 0 ∀ q h ∈ Q h where V h ⊂ V and Q h ⊂ Q are discrete spaces; a h ( · , · ) : V h × V h → R is a discrete bilinear form approximating the continuous form a ( · , · ) ; < f h , v h > 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 V h | K Let T h be a polygonal mesh on Ω ; we define the space of velocities V h element by element. L. Beirão da Veiga et al. (Univ. of Milan) VEM for Stokes Polygonal Workshop 2012 6 / 24
The local spaces V h | K Let T h be a polygonal mesh on Ω ; we define the space of velocities V h element by element. For all K ∈ T h : � v ∈ [ H 1 ( K )] 2 : div v constant, v minimizes ||∇ v || 2 V h | K = L 2 ( K ) , � v | e · t e ∈ P 1 ( e ) , v | e · n e ∈ P 2 ( e ) ∀ e ∈ ∂ K . L. Beirão da Veiga et al. (Univ. of Milan) VEM for Stokes Polygonal Workshop 2012 6 / 24
The local spaces V h | K Let T h be a polygonal mesh on Ω ; we define the space of velocities V h element by element. For all K ∈ T h : � v ∈ [ H 1 ( K )] 2 : div v constant, v minimizes ||∇ v || 2 V h | K = L 2 ( K ) , � v | e · t e ∈ P 1 ( e ) , v | e · n e ∈ P 2 ( e ) ∀ e ∈ ∂ K . Local degrees of freedom: NOTE: � 1 div v = ∂ K v · n K d s | K | go back L. Beirão da Veiga et al. (Univ. of Milan) VEM for Stokes Polygonal Workshop 2012 6 / 24
The global spaces V h and Q h The global space V h is built by assembling the local spaces V h | K as usual. V h = { v ∈ C 0 (Ω) : v | K ∈ V h | K ∀ K ∈ T h , v = 0 on ∂ Ω } The d.o.f.s are two per internal vertex and one per internal edge. The space Q h is simply given by Q h = { q ∈ L 2 0 (Ω) : q ∈ P 0 ( K ) ∀ K ∈ T h } , 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 a h ( · , · ) The bilinear form a h ( · , · ) is built element by element � a K a h ( u h , v h ) = h ( u h , v h ) ∀ u h , v h ∈ V h , K ∈T h where a K h ( · , · ) : V h | K × V h | K − → R are symmetric bilinear forms that mimic a K 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 a h ( · , · ) Consistency: for all h and for all K ∈ T h it holds a K h ( p , v h ) = a K ( p , v h ) ∀ p ∈ [ P 1 ( K )] 2 , v h ∈ V h | K . Stability: there exist two positive constants α ∗ and α ∗ , independent of h and of K , such that h ( v h , v h ) ≤ α ∗ a K ( v h , v h ) α ∗ a K ( v h , v h ) ≤ a K ∀ v h ∈ V h | K . L. Beirão da Veiga et al. (Univ. of Milan) VEM for Stokes Polygonal Workshop 2012 9 / 24
The bilinear form a h ( · , · ) Consistency: for all h and for all K ∈ T h it holds a K h ( p , v h ) = a K ( p , v h ) ∀ p ∈ [ P 1 ( K )] 2 , v h ∈ V h | K . Stability: there exist two positive constants α ∗ and α ∗ , independent of h and of K , such that h ( v h , v h ) ≤ α ∗ a K ( v h , v h ) α ∗ a K ( v h , v h ) ≤ a K ∀ v h ∈ V h | K . NOTE: the r.h.s. above is computable since integrating by parts � � � a K ( p , v h ) = ν ( ∇ s p ) n K ∀ p ∈ [ P 1 ( K )] 2 , v h ∈ V h | K . · v h d s ∂ 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 u h ∈ V h , q h ∈ Q h such that a h ( u h , v h ) + ( div v h , p h ) = < f h , v h > h ∀ v h ∈ V h ( div u h , q h ) = 0 ∀ q h ∈ Q h where the loading term � n � � � 1 � < f h , v h > h = v h ( V i ) · f d x . n K K ∈T h i = 1 Proposition Assuming standard regularity conditions on the mesh family {T h } h , it holds � � � 1 / 2 ( | u | 2 2 , K + | f | 2 || u − u h || 1 + || p − p h || 0 ≤ C h 1 , K ) . K ∈T h 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: � Ω q h div ( v h ) dx sup ≥ β || q h || L 2 (Ω) ∀ q h ∈ Q h . || v h || H 1 (Ω) v h ∈ V h / { 0 } 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: � Ω q h div ( v h ) dx sup ≥ β || q h || L 2 (Ω) ∀ q h ∈ Q h . || v h || H 1 (Ω) v h ∈ V h / { 0 } 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 Ω h , let the macro-element M = M ( ν ) = { K ∈ T h | 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 e = ∂ K ∩ ∂ K ′ , with K , K ′ ∈ M . that � Single edges e 1 ν e 2 ν e 3 Frontiers 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 ¯ N M indicate the number of edges which connect in ν AND are not a part of a frontier. Let either (a) ¯ N M < 3 or (b) ¯ N M = 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. Single edges e 1 ν e 2 ν e 3 Frontiers L. Beirão da Veiga et al. (Univ. of Milan) VEM for Stokes Polygonal Workshop 2012 14 / 24
Reduced edge bubble space PROPOSITION Let ¯ N M indicate the number of edges which connect in ν AND are not a part of a frontier. Let either (a) ¯ N M < 3 or (b) ¯ N M = 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. Single edges e 1 ν e 2 ν e 3 Frontiers 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
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