Virtual Element Methods F. Brezzi IMATI-C.N.R., Pavia, Italy - PowerPoint PPT Presentation

Virtual Element Methods F. Brezzi IMATI-C.N.R., Pavia, Italy IUSS-Istituto Universitario di Studi Superiori, Pavia, Italy KAU-King Abdulaziz University, Jeddah, Saudi Arabia From joint works with: L. Beira˜ o da Veiga, A. Cangiani G. Manzini, L.D. Marini and A. Russo Milano, 17-19 September, 2012 Workshop on Polygonal and Polyhedral Meshes Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 1 / 71

Outline 1 Traditional FEM 2 Basic Virtual Elements 3 A more precise version 4 Robustness 5 Higher order VEM 6 Linear elasticity 7 Nearly incompressible elasticity 8 Plate bending K-L 9 Conclusions Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 2 / 71

Generalities - Reminders on Classic FEM In FEM the degrees of freedom are used to reconstruct polynomials (or isoparametric images of polynomials) in each element. Ingredients: the geometry of the element (e.g.: triangles) the degrees of freedom; say, N d.o.f. per element in each element, a space of polynomials of dim. N. The ingredients must match Unisolvence N numbers ↔ one and only one polynomial Continuity Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 3 / 71

Virtual Elements on pentagons Assume now that we want to decompose the domain Ω, for instance, in pentagons, obviously not necessarily regular. How to take a polynomial space of dimension 5 (e.g., to be associated to the nodal values)? Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 4 / 71

Virtual Elements for Laplace Equation VEM: We take, as unknowns, the values at the vertices. Then, for every pentagon E and for every fixed set of 5 vertex-values we define the corresponding function (say, ϕ h ) first on ∂ E , by: ϕ h is linear on each edge. Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 5 / 71

Virtual Elements for Laplace Equation VEM: We take, as unknowns, the values at the vertices. Then, for every pentagon E and for every fixed set of 5 vertex-values we define the corresponding function (say, ϕ h ) first on ∂ E , by: ϕ h is linear on each edge. Hence, from the vertex values we have ϕ h on the whole ∂ E . At this point we decide that all our functions, inside E , should be harmonic . Hence, from its 5 nodal values ϕ h is first defined on ∂ E (by linear interpolation), and from its value on ∂ E it is defined inside (by harmonic extension). In summary, the value of ϕ h on E is uniquely determined by its nodal values. Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 5 / 71

Virtual Elements for Laplace Equation In E we have therefore a local space V E , of dimension 5, that contains as a subspace the space P 1 of polynomials of degree ≤ 1, plus two additional functions that are not polynomials (and that, unfortunately, we cannot compute, or, at least, not in a cheap-enough way). E V 1 P 1 x y Classical option: use some numerical integration formula . In VEM, however, we proceed differently . Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 6 / 71

Reminder: implementation of Classic FEM Find u ∈ V ≡ H 1 0 (Ω) s. t. − ∆ u = f . That is: � � ∇ u · ∇ v d Ω = f v d Ω ∀ v ∈ V . Ω Ω Setting V h = continuous piecewise linear functions vanishing at the boundary, we look for u h in V h such that � � ∇ u h · ∇ v h d Ω = f v h d Ω ∀ v h ∈ V h . Ω Ω The final matrix is then computed as the sum of the contributions of the single elements: � � ∇ v j · ∇ v i d Ω = ∇ v j · ∇ v i dE . � A i , j ≡ Ω E E Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 7 / 71

How to use VEM E V 1 P 1 E x y V h := { v ∈ V : v linear on each edge , − ∆ v = 0 in E ∀ E } � � ∂ p 1 a E ( p 1 , v ) = ∂ n v d ℓ =: a E ∇ p 1 · ∇ v dE = h ( p 1 , v ) E ∂ E If u is in P 1 ( E ), then a E ( u , v ) can be computed exactly . We have then to decide how to chose a hE ( u , v ) when both u and v are not in P 1 ( E ). Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 8 / 71

Continuous and discretized problem We consider the continuous problem Find u ∈ V ≡ H 1 0 (Ω) such that � � a ( u , v ) ≡ ∇ u · ∇ v d Ω = f v d Ω ∀ v ∈ V , Ω Ω and its discretized version: Find u h ∈ V h such that a h ( u h , v h ) = ( f h , v h ) ∀ v h ∈ V h , and we look for sufficient conditions on a h that ensure all the good properties that you would have with standard Finite Elements Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements Milano, September 17-19, 2012 9 / 71

The two basic properties H1 (Consistency) a E h ( p 1 , v ) = a E ( p 1 , v ) ∀ E , ∀ v ∈ V E , ∀ p 1 ∈ P 1 ( E ) . ∃ α ∗ , α ∗ > 0 such that: H2 (Stability) h ( v , v ) ≤ α ∗ a E ( v , v ) α ∗ a E ( v , v ) ≤ a E ∀ E , ∀ v ∈ V E . Under Assumptions H1 and H2 the discrete problem has a unique solution. Moreover the Patch Test of order 1 is satisfied : on any patch of elements, if the exact solution is a global polynomial of degree 1 , then the exact solution and the approximate solution coincide . Incidentally: � u − u h � 1 = O ( h ). Milano, September 17-19, 2012 10 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Convergence Theorem Theorem Under the above assumptions H1 and H2 , for every approximation u I of u in V h and for every approximation u p of u that is piecewise in P 1 , we have � � � u − u h � V ≤ C � u − u I � V + � u − u p � h , V + � f − f h � ( V h ) ′ where ( f , v h ) − ( f h , v h ) � f − f h � ( V h ) ′ := sup � v h � V v h ∈ V h Milano, September 17-19, 2012 11 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Proof of convergence Set δ h := u h − u I α ∗ α � δ h � 2 V ≤ α ∗ a ( δ h , δ h ) ≤ a h ( δ h , δ h ) = a h ( u h , δ h ) − a h ( u I , δ h ) � a E = ( f h , δ h ) − h ( u I , δ h ) E � � � a E h ( u I − u p , δ h ) + a E = ( f h , δ h ) − h ( u p , δ h ) E � � � a E h ( u I − u p , δ h ) + a E ( u p , δ h ) = ( f h , δ h ) − E � � � a E h ( u I − u p , δ h ) + a E ( u p − u , δ h ) = ( f h , δ h ) − − a ( u , δ h ) E � � � a E h ( u I − u p , δ h ) + a E ( u p − u , δ h ) = ( f h , δ h ) − − ( f , δ h ) E � � � a E h ( u I − u p , δ h ) + a E ( u p − u , δ h ) = ( f h − f , δ h ) − . E Milano, September 17-19, 2012 12 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

How to satisfy H1 and H2 We construct first on each E an operator 1 from V E into P 1 ( E ) defined by Π ∇ � ( v − Π ∇ a E ( v − Π ∇ 1 v )( V i ) = 0 1 v , p 1 ) = 0 ∀ p 1 V i = vertex of E Note that Π ∇ 1 p 1 = p 1 for all p 1 in P 1 ( E ). Then we set, for all u and v in V E a E h ( u , v ) := a E (Π ∇ 1 u , Π ∇ 1 v ) + S ( u − Π ∇ 1 u , v − Π ∇ 1 v ) where the stabilizing bilinear form S is (for instance) the Euclidean inner product in R 5 . Milano, September 17-19, 2012 13 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Proof of H1 and H2 Consistency: a E h ( p k , v h ) = a E ( p k , Π k v h ) = a E (Π k v h , p k ) = a E ( p k , v h ) Stability (upper bound): a E h ( v h , v h ) ≤ a E (Π k v h , Π k v h )+ c 1 a E ( v h − Π k v h , v h − Π k v h ) = a E ( v h , Π k v h ) + c 1 a E ( v h − Π k v h , v h ) ≤ α ∗ a E ( v h , v h ) Stability (lower bound): a E h ( v h , v h ) ≥ a E (Π k v h , Π k v h )+ c 0 a E ( v h − Π k v h , v h − Π k v h ) ≥ α ∗ ( a E ( v h , Π k v h ) + a E ( v h − Π k v h , v h )) = α ∗ a E ( v h , v h ) Milano, September 17-19, 2012 14 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Robustness of the method (by A. Russo) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Milano, September 17-19, 2012 15 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Robustness of the method (by A. Russo) 0.07 0.06 0.05 0.04 0.03 0.02 0.01 1 0 0 0.5 0.2 0.4 0.6 0.8 0 1 Milano, September 17-19, 2012 16 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Robustness of the method (by A. Russo) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Milano, September 17-19, 2012 17 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Robustness of the method (by A. Russo) 0.07 0.06 0.05 0.04 0.03 0.02 0.01 1 0 0 0.5 0.2 0.4 0.6 0.8 0 1 Milano, September 17-19, 2012 18 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Robustness of the method (by A. Russo) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Milano, September 17-19, 2012 19 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71

Robustness of the method (by A. Russo) 0.07 0.06 0.05 0.04 0.03 0.02 0.01 1 0 0 0.5 0.2 0.4 0.6 0.8 0 1 Milano, September 17-19, 2012 20 / Franco Brezzi (IMATI-CNR & IUSS, Pavia) Virtual Elements 71