Mixed Hybrid Finite Element Method: an introduction

Mixed Hybrid Finite Element Method: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato in Scienze dell Ingegneria Civile e Ambientale Corso di Metodi Numerici marzo 2008

Outline Some observation on the Finite Element method The Mixed Finite Element method The Mixed Hybrid Finite Element method Some properties of the method Superconvergence results Numerical implementation Numerical results A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 2 / 24

About the Finite Element method It is well known that the classical Finite Element (FE) method minimizes the so called energy functional) in a well chosen space of admissible functions W inf J(w). w W If the functional J( ) is differentiable, the minimum (whenever it exists) will be characterized by a variational equation. The FE method is based on a few simple ideas: the domain Ω R d, d = 2, 3, in which the problem is posed, is partitioned into a set of simple subdomains, called elements. These elements may be triangles, quadrilaterals, tetrahedra. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 3 / 24

A space W of functions defined on Ω is then approximated by simple functions, defined on each element with suitable matching conditions at interfaces. Simple functions are commonly polynomials or functions obtained from polynomials by a change of variables. A FE method can only be considered in relation with a variational principle and a functional space. Changing the variational principle and the space in which it is posed leads to a different FE approximation, even if the solution for the continuous problem can remain the same. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 4 / 24

The Mixed Finite Element method In the Mixed Finite Element (MFE) method two different Finite Element spaces are used. Let us consider the simple elliptic equation: (D c) = f c = 0 in Ω on Ω where D = D(x) is the dispersion tensor, and c represents the concentration. Introducing the dispersive flux G, we can write the Fick law, G = D c. It could be desiderable to approximate G and c simultaneously using a different Finite Element space for each variable. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 5 / 24

With this purpose the elliptic problem is decomposed into a first order system as follows: G + D c = 0 G = f c = 0 in Ω in Ω on Ω The first of the above equations can be written as D 1 G + c = 0 in Ω. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 6 / 24

Multiplying by test functions and integrating by parts we obtain the following weak formulation D 1 G w d c w d = 0 w H(div, Ω) Ω Ω ψ G d = f ψ d ψ L 2 (Ω) Ω Ω where H(div, Ω) = { w L 2 (Ω) d : w L 2 (Ω)} Hilbert space The weak formulation involves the divergence of the solution and test functions and not arbitrary first derivatives. Thus we work with the space H(div, Ω) formed by piecewise polynomial vector functions with continuous normal component. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 7 / 24

In order to define finite element approximations to the solution ( G, c), we need to have finite element subspaces of H(div, Ω) and L 2 (Ω). Let T = {T l } m l=1 be a triangulation of Ω, i.e. Ω = T l T T l with diameter h. The triangulation is admissibile if the intersection of two triangles is either empty, or a vertex, or a complete side. Thus, we have to construct piecewise polynomials spaces W h and Ψ h associated with T l such that W h H(div, Ω) and Ψ h L 2 (Ω). A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 8 / 24

The MFE approximation ( G h, c h ) W h Ψ h is defined by Ω Ω D 1 Gh w d c h w d = 0 w W h Ω ψ Gh d = f ψ d ψ Ψ h. Ω In order to have stability and convergence W h and Ψ h can not be chosen arbitrarily but they have to be related. We assume that W h = Ψ h. Let π 2G be the L 2 - projection of G into Ψ h. The following equality must hold: ( G π 2G)ψ d = 0 G H 1 (Ω) d, ψ Ψ h Ω where H 1 (Ω) is the Sobolev space (H 1 (Ω) = {v L 2 (Ω) : D 1 v L 2 (Ω) D 1 v partial derivative}) A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 9 / 24

We can state the following theorem: Theorem The problem of finding a pair of functions ( G, c) H(div, Ω) L 2 (Ω) such that the weak form of the elliptic problem holds has a unique solution. In addition, c is the solution of the elliptic problem and G = D c. Proof is omitted. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 10 / 24

Numerical implementation of the MFE method A MFE scheme is given by defining the spaces W h and Ψ h approximating W H(div, Ω) and Ψ L 2 (Ω) respectively. We use the Raviart-Thomas spaces defined on a generic element T l Ω as RT k = (P k ) d + xp k where k is an integer 0 and P k is the space of polynomials of degree k. The dimension of RT k is given by { (k + 1)(k + 3) for d = 2 dimrt k = 1(k + 1)(k + 2)(k + 4) for d = 3. 2 A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 11 / 24

Lemma For w h RT k the following relations hold: w h P k w h n Tl R k. where R k is the polynomial space defined on the edges e j of each element T l : R k = {φ : φ L 2 ( T l ), φ ej P k, e j }. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 12 / 24

We consider the Raviart-Thomas space of degree zero, whereby the functions G and c can be approximated by: G G = c c = m g l w l l=1 m c l ψ l where w l and ψ l are vector and scalar basis functions. Since we are considering the RT 0 spaces, w l are first order polynomial of the type: w l = l=1 ( ax + b ay + c while ψ l are P 0 polynomials equal to one on element T l and zero elsewhere. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 13 / 24 ),

Lemma Given a triangle T l with edges e j, j = 1, 2, 3, the following relationships hold for w l W h (T l ): w l P 0 w l n j l R 0 j = 1, 2, 3 where n j l is the outward normal to edge e j of T l and R 0 is the polynomial space defined on edge e j of T l as R 0 = {φ : φ L 2 ( T l ), φ ej P 0, e j }. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 14 / 24

Proof. The first relation is very simple to prove: w l = 2a P 0. The second relation says that the restriction of w l to each of the edges of T l coincides with a polynomial of degree zero, i.e. a constant. Let T l be an element with nodes P 1 = (x 1, y 1 ), P 2 = (x 2, y 2 ), P 3 = (x 3, y 3 ). The normal to the edge e 1 = P 1 P 2 is given by: n 1 l = ( y2 y 1 x 1 x 2 ). A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 15 / 24

Continued. Noting that the line passing through the points P 1 and P 2 has equation: x(y 2 y 1 ) + y(x 1 x 2 ) = x 1 (y 2 y 1 ) + y 1 (x 1 x 2 ), the inner product w l n 1 l can be written as w l n 1 l = (ax + b)(y 2 y 1 ) + (ay + c)(x 1 x 2 ) = a[x 1 (y 2 y 1 ) + y 1 (x 1 x 2 )] + b(y 2 y 1 ) + c(x 1 x 2 ) = const. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 16 / 24

About RT0 basis functions On each element T l we choice the RT0 basis functions w l i (i = 1, 2, 3) of the following form: ( ) a w l i i = l x + bl i al iy + ci l It is satisfied the following property (Kronecker property): { w j 1 if j = i l n i l ds = δ ij = 0 otherwise e j where n i l is the outward normal to e i. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 17 / 24

We can now derive the coefficients a i l, bi l and c i l coefficients in the following way (for sake of simplicity, we will omit the sub- super- scripts l, i). Given the triangle T l with edges e 1, e 2, e 3 and nodes P 1 = (x 1, y 1 ), P 2 = (x 2, y 2 ), P 3 = (x 3, y 3 ), the normal to edge e 1 = P 1 P 2 is derived by: x y 1 x 1 y 1 1 x 2 y 2 1 = 0 that is α(x x 1 ) + β(y y 1 ) = 0, where α = y 2 y 1, β = x 1 x 2. The normalized components of the normal to e 1 are then (with no consideration about inner or outward normal) : n x = α α2 + β 2, n y = β α2 + β 2. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 18 / 24

When imposing Kronecker property, we obtain the following result: w j l n i l ds = (ax + b)n x + (ay + c)n y ds = e i a e i x2 x 1 (x 1 n x + y 1 n y ) 1 n y dx + a(x 1 n x + y 1 n y ) x 1 x 2 n y x2 x 1 (bn x + cn y ) 1 n y dx = + (bn x + cn y ) x 1 x 2 n y Since x 1 x 2 = is equal to n y (x2 x 1 ) 2 + (y 2 y 1 ) 2 = α 2 + β 2 we obtain: = δ ij e i w j l n i l ds = a(x 1 α + y 1 β) + (bα + cβ) = δ ij A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 19 / 24

The coefficients a, b, c are chosen by the 2D classical Galerkin functions, N i, i = 1, 2, 3, whose gradients are normal to the edges e i+1 for i = 1, 2 while the gradient of N 3 is normal to e 1. They can be written as: N i = a i + b i x + c i y, 2 T l where T l is the area of the triangle T l that can be written as: T l = 1 1 x 1 y 1 2 1 x 2 y 2 1 x 3 y 3 A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 20 / 24

In particular, the coefficients are: a 1 = x 2 y 2 x 3 y 3 a 2 = x 1 y 1 x 3 y 3 a 3 = x 1 y 1 x 2 y 2 b 1 = 1 y 2 1 y 3 b 2 = 1 y 1 1 y 3 b 3 = 1 y 1 1 y 2 c 1 = 1 x 2 1 x 3 c 2 = 1 x 1 1 x 3 c 3 = 1 x 1 1 x 2 A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 21 / 24

It is simple to see that, for example, for the edge e 1 the following relations hold: αx 1 + βy 2 = a 3, α = b 3, β = c 3 Thus w j n 1 ds = a 3 a + (b 3 b + c 3 c). e 1 The unknowns are now the coefficients a, b, c of w j. Setting j = 1, from Kronecker property and substituting in the previous equation, we get: Note that 3 b i = i=1 a 3 a + b 3 b + c 3 c = 1 a 1 a + b 1 b + c 1 c = 0 a 2 a + b 2 b + c 2 c = 0 3 i=1 c i =0 and that 3 i=1 a i = 2 T l. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 22 / 24

The system gives the following relationship between a and T l : 2 T l a = 1 a = 1 2 T l. Application of divergence theorem to T w 1 dx dy assures that the value of a is the same along the three edges of T l and is positive and equal to 1 2 T l. Indeed w 1 dx dy = w 1 n ds = T T 3 i=1 e i w 1 n i ds = 1 But, T w 1 dx dy = T ( w 1 x + w 1 y ) ds = 2a ds = 2a T l T A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 23 / 24

Again, considering the first equation of the previous system and the measure of T l, we obtain for w 1 : b 1 l = x 3 2 T l, c1 l = y 3 2 T l. Similarly, the coefficients for w 2 and w 3 are: b j l = x j 1 2 T l, cj l = y j 1 2 T l, j = 2, 3 while a j l does not change, a j l = 1. Observe that with this 2 T l choice of basis for the V h space, the components g 1, g 2 and g 3 of G l on the element T l are the edge fluxes that is the velocity (or flux) along each edge of T l. A. Mazzia (DMMMSA) MHFE method a.a. 2007/2008 24 / 24