Virtual Element Methods for general second order elliptic problems

Transcription

1 Virtual Element Methods for general second order elliptic problems Alessandro Russo Department of Mathematics and Applications University of Milano-Bicocca, Milan, Italy and IMATI-CNR, Pavia, Italy Workshop on Polytopal Element Methods in Mathematics and Engineering October 25 Georgia Tech, Atlanta A. Russo (Milan) VEM October 26, 25 / 27

2 Outline of the presentation Outline of the presentation Virtual Element Spaces Projectors VEM approximation of general elliptic equations Primal form Mixed form Numerical Experiments Conclusions Joint work with L. Beirão da Veiga, F. Brezzi, D. Marini A. Russo (Milan) VEM October 26, 25 2 / 27

3 Elliptic problem in primal form Elliptic problem in primal form We want to approximate a general second-order elliptic equation in 3D with an arbitrary polyhedral mesh with a conforming finite element method of order k. { div (κ u) + β u + αu = f in Ω R 3 u = g on Ω Beirão da Veiga, Brezzi, Marini, R.: Virtual Element Methods for general second order elliptic problems on polygonal meshes, to appear in Mathematical Models and Methods in Applied Sciences (also arxiv: ) We start with the two-dimensional case. A. Russo (Milan) VEM October 26, 25 3 / 27

4 Virtual Element spaces The local finite element space V k (E) Let E be a polygon. We would like to define a finite element space V k (E) on E such that: V k (E) contains the space P k (E) of polynomials of degree less than or equal to k (plus other non-polynomial functions); a function in V k (E) restricted to an edge e is in P k (e), so that if two polygons E and E have an edge in common, the two spaces V k (E) and V k (E ) must glue in C (E E ); given a function v h V k (E), I can compute its L 2 projection Π k v h in P k (E), and this is all the information of v h that I need in order to approximate my equation. A. Russo (Milan) VEM October 26, 25 4 / 27

5 Virtual Element spaces Meaning of I can compute In what follows the precise meaning of the statement is: I can compute Π k v h given the array dof i (v h ), I can directly compute Π k v h The same applies for all other quantities which are computable from v h. A. Russo (Milan) VEM October 26, 25 5 / 27

6 Virtual Element spaces The space V k,k (E) Given a polygon E, consider the following space (already mentioned in Franco s talk): V k,k (E) := {v h C (E) such that: v h e P k (e), v h P k (E)} If the polygon E has N E edges (and vertices), it is clear that dim V k,k (E) = k N E + dim P k (E) = k N E + Is the space V k,k (E) OK for us? (k + )(k + 2) 2 A. Russo (Milan) VEM October 26, 25 6 / 27

7 Virtual Element spaces Degrees of freedom in V k,k (E) Let (x E, y E ) be the centroid of E and h E its diameter. If α = (α, α 2 ) is a multiindex we define the scaled monomials of degree α = α + α 2 : ( ) x α ( ) xe y α2 ye m α (x, y) :=. The set {m α, with α k} is a basis for P k (E). h E As degrees of freedom in V k,k (E), we choose: the value of v h at the vertices and at k equally spaced points on each edge; the (scaled) moments v h m α for α k. E E h E A. Russo (Milan) VEM October 26, 25 7 / 27

8 Virtual Element spaces Degrees of freedom in V k,k (E) It can be easily shown that: the degrees of freedom above are unisolvent in V k,k (E). Short proof: suppose that v h V k,k (E) is zero on the boundary and v h m α = for α k We show that v h. For, v h 2 = E Hence v h. E E v h v h = since v h P k (E) A. Russo (Milan) VEM October 26, 25 8 / 27

9 Virtual Element spaces Degrees of freedom in V k,k (E) The choice of the scaled moments v h m α among the degrees of E E freedom implies that, starting from the degrees of freedom of v h, I can trivially compute Π k v h := L 2 projection of v h onto P k (E). It is clear that if I could compute directly the bilinear form on the space V k,k (E), the resulting finite element method would converge with the right optimal rates. For the time being, assume that the L 2 projection onto P k (E) is enough (we have a general way of approximating the bilinear form just using the L 2 projection). A. Russo (Milan) VEM October 26, 25 9 / 27

10 Virtual Element spaces Basis functions in V k,k (E) For i =,..., N dof we define ϕ i as the function in V k,k (E) such that { if i = j dof j (ϕ i ) = j-th degree of freedom of ϕ i = if i j We have the usual Lagrange-type expansion v h = dof i (v h ) ϕ i. N dof i= A. Russo (Milan) VEM October 26, 25 / 27

11 Virtual Element spaces Is the space V k,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 V k,k (E). Original VEM: define a projector Π k : V k,k(e) P k (E) which uses only the moments up to degree k 2; identify a subspace V k (E) V k,k (E) requiring that v h m α = Π k v h m α for α = k, k E E the dimension of this subspace is k N E + dim P k 2 (E) and the internal degrees of freedom are the (scaled) moments up to k 2; in this subspace we can compute Π k v h. A. Russo (Milan) VEM October 26, 25 / 27

12 Virtual Element spaces Is the space V k,k (E) OK for the VEM? Serendipity VEM: define a projector Π S k : V k,k(e) P k (E) which uses only the moments up to degree k N E ; identify a subspace V k (E) V k,k (E) requiring that v h m α = Π S k v h m α for α = k N E +,..., k E E the dimension of this subspace is k N E + dim P k NE (E) and the internal degrees of freedom are the (scaled) moments up to k N E ; in this subspace we can compute Π k v h. A. Russo (Milan) VEM October 26, 25 2 / 27

13 Virtual Element spaces The local finite element space V k (E) Let E be a polygon. We have defined a finite element space V k (E) on E such that: V k (E) contains the space P k (E) of polynomials of degree less than or equal to k (plus other non-polynomial functions); a function in V k (E) restricted to an edge e is in P k (e), so that if two polygons E and E have an edge in common, the two spaces V k (E) and V k (E ) must glue in C (E E ); given a function v h V k (E), I can compute its L 2 projection Π k v h in P k (E), and this is all the information of v h that I need in order to approximate my equation. A. Russo (Milan) VEM October 26, 25 3 / 27

14 Virtual Element spaces Computing another projector in V k (E): Π k v h We show now that we can easily compute Π k v h. To compute Π k v h we need to know the moments of v h up to order k : v h x m m β β = v h E x + v h m β nx, β k E E and both terms are computable directly from the degrees of freedom of v h. A. Russo (Milan) VEM October 26, 25 4 / 27

15 VEM approximation of general elliptic equations VEM approximation of general elliptic equations We consider now a general second order elliptic operator with variable coefficients: div (κ u) + β u + αu = f and we approximate the various local consistency terms as: κ u h v h κ [ Π k u ] [ ] h Π k v h E E ( ) ( [ ]) β uh vh β Π k u h Π k v h E E α u h v h and for the right-hand-side: f v h E E E E α [ Π k u ] [ ] h Π k v h f Π k v h ( Π k 2 v h is enough for k 2 ) A. Russo (Milan) VEM October 26, 25 5 / 27

16 VEM approximation of general elliptic equations VEM approximation of general elliptic equations The approximations above produce, as usual, rank-deficient matrices that must be stabilized. For the stabilization we can take the following term: with S(ϕ i, ϕ j ) = δ ij. S((I Π k )u h, (I Π k )v h) This means that we expand Π k ϕ i in terms of the ϕ j s and then we replace S(ϕ i, ϕ j ) with the identity matrix. A. Russo (Milan) VEM October 26, 25 6 / 27

17 US mesh 49 polygons, 5686 vertices A. Russo (Milan) VEM October 26, 25 7 / 27

18 US mesh vertices GEORGIA A. Russo (Milan) VEM October 26, 25 8 / 27

19 US mesh VEM solution on skeleton exact solution = 4 th degree polynomial...2 L 2 error k = 2:.6 3 k = 3:.43 4 k = 4: (patch test) A. Russo (Milan) VEM October 26, 25 9 / 27

20 Mixed VEM The continuous problem in mixed form Ω R 2 (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 = on Γ is solvable for any f H (Ω), and p,ω C f,ω, p 2,Ω C f,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 L 2 (Ω) ϕ H 2 (Ω) H (Ω): L ϕ = f, ϕ 2,Ω C f,ω A. Russo (Milan) VEM October 26, 25 2 / 27

21 Mixed VEM The continuous problem in mixed form Setting: ν = κ, β = κ b and νu := p + βp, we have div u + γ p = f in Ω, p = on Γ V = H(div, Ω), Q = L 2 (Ω) Find (u, p) V Q such that (νu, v) (p, div v) (β v, p) = v V (div u, q) + (γp, q) = (f, q) q Q A. Russo (Milan) VEM October 26, 25 2 / 27

22 Mixed VEM VEM approximation: the discrete spaces The spaces: We define, for k integer, Vh k (E) := {v H(div; E) H(rot; E) : v n e P k (e) e E, div v P k (E), and rot v P k (E)}. Vh k := {v H(div; Ω) such that v E Vh k (E) E T h} Qh k := {q L2 (Ω) such that: q E P k (E) element E int h } degrees of freedom in Vh k : v n p k ds e, p k P k (e) e v g k dx E, g k G k (E), E v g k dx E, g k G k (E) E (G k = P k, G k (E) = L2 (E) orthogonal of G k (E) in (P k (E)) 2 ) A. Russo (Milan) VEM October 26, / 27

23 Mixed VEM The bilinear forms a E h (v, w) := (νπ k v, Π k w),e + S E (v Π k v, w Π k w) Set α a E (v, v) S E (v, v) α a E (v, v) a h (v, w) := E ah E (v, w). v V k h Lemma The bilinear form a h (, ) is continuous and elliptic in (L 2 ) 2 : M > such that a h (v, w) M v w v, w Vh k, α > such that a h (v, v) α v 2 v Vh k, with M and α depending on ν but independent of h. A. Russo (Milan) VEM October 26, / 27

24 Mixed VEM The discrete problem Find (u h, p h ) Vh k Qk h such that ( ) a h (u h, v h ) (p h, div v h ) (β Π k v h, p h ) = (div u h, q h ) + (γp h, q h ) = (f, q h ) q h Q h. v h V k h Theorem Assume that the continuous problem has a unique solution p. Then, for h sufficiently small, ( ) has a unique solution (u h, p h ) V k h Qk h, and p p h Ch k+( u k+ + p k+ ), u u h Ch k+( u k+ + p k+ ), div(u u h ) Ch k+( f k+ + p k+ ) with C a constant depending on ν, β, and γ but independent of h. A. Russo (Milan) VEM October 26, / 27

25 Mixed VEM Superconvergence results Theorem Let p h be the discrete solution, and let p I Qh k Then, for h sufficiently small, be the interpolant of p. p I p h C h k+2( u k+ + p k+ + f k+ ), where C is a constant depending on ν, β, and γ but independent of h. A. Russo (Milan) VEM October 26, / 27

26 Numerical Experiments Joining meshes joined 832 polygons, Ndof=6849, h max =.77e, h mean = 4.8e 2 VEM solution A. Russo (Milan) VEM July 4, / 43

27 Numerical Experiments Joining meshes error A. Russo (Milan) VEM July 4, / 43

28 Numerical Experiments Joining meshes 2 joined 852 polygons, Ndof=733, h max =.77e, h mean = 4.3e 2 VEM solution A. Russo (Milan) VEM July 4, / 43

29 Numerical Experiments Joining meshes VEM solution A. Russo (Milan) VEM July 4, / 43

30 Numerical Experiments Joining meshes 3 joined 573 polygons, Ndof=389, h max = 3.73e, h mean = 3.46e 2 A. Russo (Milan) VEM July 4, / 43

31 Numerical Experiments Joining meshes VEM solution A. Russo (Milan) VEM July 4, / 43

32 Numerical results div( κ p + bp) + γp = f in [, ] [, ] κ(x, y) = ( x 2 + y 2 xy ) xy x 2 + y 2 b(x, y) = [x, y] γ(x, y) = x 2 + y 2 Exact solution: p(x, y) = x 2 + y 2 + cos(7y) sin(3x) + 2, u = κ p + bp Donatella Marini (Pavia) VEM variable Ferrara 25 / 23

33 exact solution x 2 +y 2 +cos(y 7.) sin(x 3.) y x.6.8

34 Errors: uniform triangulation, k = k = 2 k= u u h,ω p p I,Ω p I p h,ω Donatella Marini (Pavia) VEM variable Ferrara 25 3 / 23

35 Errors: uniform triangulation, k = 3 3 k = 3 3 k=3 u u h,ω p p I,Ω p I p h,ω Donatella Marini (Pavia) VEM variable Ferrara 25 4 / 23

36 Voronoi mesh, k = lloyd Donatella Marini (Pavia) VEM variable Ferrara 25 5 / 23

37 Voronoi mesh, k = k = k= u u h,ω p p I,Ω p I p h,ω Donatella Marini (Pavia) VEM variable Ferrara 25 5 / 23

38 Voronoi mesh, k = 3 2 k = 3 2 k=3 u u h,ω p p I,Ω p I p h,ω Donatella Marini (Pavia) VEM variable Ferrara 25 6 / 23

39 Poisson problem; load= Dirac mass at center Finite elements of degree 4: mesh and solution Generated by conn2mesh on 2 Mar 25 8:45: Donatella Marini (Pavia) VEM variable Ferrara 25 7 / 23

40 Poisson problem; load= Dirac mass at center Virtual elements of degree 4: mesh and solution joined Donatella Marini (Pavia) VEM variable Ferrara 25 8 / 23

41 Advection dominated case ε p + β p = in [, ] [, ] p = ( cos(2πy)/2 on x =, p = on x = p = on y =, y = n Donatella Marini (Pavia) VEM variable Ferrara 25 9 / 23

42 Advection dominated case ε p + β p = in [, ] [, ] p = ( cos(2πy)/2 on x =, p = on x = p = on y =, y = n β = (, /3). Test case: ε = 2, Virtual elements of degree : mesh and solution Donatella Marini (Pavia) VEM variable Ferrara 25 9 / 23

43 Advection dominated case ε p + β p = in [, ] [, ] p = ( cos(2πy)/2 on x =, p = on x = p = on y =, y = n β = (, /3). Test case: ε = 2, Virtual elements of degree : mesh and solution joined VEM solution k= Donatella Marini (Pavia) VEM variable Ferrara 25 9 / 23

44 Advection dominated case joined Donatella Marini (Pavia) VEM variable Ferrara 25 2 / 23

45 Advection dominated case VEM solution k= Donatella Marini (Pavia) VEM variable Ferrara 25 2 / 23

46 Advection dominated case joined Donatella Marini (Pavia) VEM variable Ferrara 25 2 / 23

47 Advection dominated case VEM solution k= Donatella Marini (Pavia) VEM variable Ferrara 25 2 / 23

48 Advection dominated case joined Donatella Marini (Pavia) VEM variable Ferrara / 23

49 Advection dominated case VEM solution k= Donatella Marini (Pavia) VEM variable Ferrara / 23

50 The VEM paradigm The VEM paradigm We believe that the Virtual Element Method has a wide range of applicability. The keypoints of VEM are: The definition of the local finite element spaces and of the associated degrees of freedom. These spaces contain polynomials plus other functions which are not computable. The costruction of variuos projectors onto polynomial spaces. The definition of a consistent and stable bilinear form using these projectors which is the VEM approximation of the exact bilinear form. A general theorem that guarantees convergence for a consistent and stable approximate bilinear form. A. Russo (Milan) VEM October 26, / 27

51 Basic references Basic References Virtual Element Methods for general second order elliptic problems on polygonal meshes, Beirão da Veiga, Brezzi, Marini, R.: to appear in Math. Models and Methods in Applied Sciences (also arxiv: [math.na]) Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes, Beirão da Veiga, Brezzi, Marini, R.: to appear in ESAIM: Math. Modelling and Numerical Analysis (also arxiv: [math.na]) The Hitchhiker s Guide to the Virtual Element Method Beirão da Veiga, Brezzi, Marini, R., Math. Models and Methods in Applied Sciences Vol. 24, No. 8 (24), pp Thanks for your attention! alessandro.russo@unimib.it A. Russo (Milan) VEM October 26, / 27