A C 1 virtual element method for the Cahn-Hilliard equation with - PowerPoint PPT Presentation

A C 1 virtual element method for the Cahn-Hilliard equation with polygonal meshes Marco Verani MOX, Department of Mathematics, Politecnico di Milano Joint work with: P. F. Antonietti (MOX Politecnico di Milano, Italy) L. Beir˜ ao da Veiga (Universit` a di Milano, Italy) S. Scacchi (Universit` a di Milano, Italy) Polytopal Element Methods, Georgia Tech, October 26th, 2015 M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 1 / 28

Outline 1 Cahn-Hilliard equation 2 C 1 -VEM for Cahn-Hilliard 3 Numerical results 4 Conclusions M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 2 / 28

Cahn-Hilliard (CH) problem Let Ω ⊂ R 2 be an open bounded domain. Let ψ ( x ) = (1 − x 2 ) 2 / 4 and φ ( x ) = ψ ′ ( x ).  � � φ ( u ) − γ 2 ∆ u ( t ) ∂ t u − ∆ = 0 in Ω × [0 , T ] ,   u ( · , 0) = u 0 ( · ) in Ω ,  � �  φ ( u ) − γ 2 ∆ u ( t ) ∂ n u = ∂ n = 0 on ∂ Ω × [0 , T ] , where ∂ n denotes the (outward) normal derivative and γ ∈ R + , 0 < γ ≪ 1, represents the interface parameter. M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 3 / 28

Cahn-Hilliard (CH) problem Let Ω ⊂ R 2 be an open bounded domain. Let ψ ( x ) = (1 − x 2 ) 2 / 4 and φ ( x ) = ψ ′ ( x ).  � � φ ( u ) − γ 2 ∆ u ( t ) ∂ t u − ∆ = 0 in Ω × [0 , T ] ,   u ( · , 0) = u 0 ( · ) in Ω ,  � �  φ ( u ) − γ 2 ∆ u ( t ) ∂ n u = ∂ n = 0 on ∂ Ω × [0 , T ] , where ∂ n denotes the (outward) normal derivative and γ ∈ R + , 0 < γ ≪ 1, represents the interface parameter. CH models the evolution of interfaces specified as the level set of a smooth continuos function u exhibiting large gradients across the interface [Korteweg 1901, Cahn and Hilliard 1958, Ginzburg and Landau 1965, van der Waals 1979] Many applications: phase separation in binary alloys, tumor growth, origin of Saturn’s rings, separation of di-block copolymers, population dynamics, image processing, clustering of mussels ... M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 3 / 28

Huge literature on numerical methods for CH... M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 4 / 28

Huge literature on numerical methods for CH... Here we limit ourvselves to FEM as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 4 / 28

Huge literature on numerical methods for CH... Here we limit ourvselves to FEM as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. CH is a fourth order nonlinear problem ❀ C 1 FEM (see [Elliott Zheng, 1986], [Elliott French, 1987]). M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 4 / 28

Huge literature on numerical methods for CH... Here we limit ourvselves to FEM as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. CH is a fourth order nonlinear problem ❀ C 1 FEM (see [Elliott Zheng, 1986], [Elliott French, 1987]). To avoid the well known difficulty met in the implementation of C 1 FEM, another possibility is the use of non-conforming (see [Elliott French, 1989] or discontinuous (see [Wells Kuhl Garikipati, 2006]) methods. M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 4 / 28

Huge literature on numerical methods for CH... Here we limit ourvselves to FEM as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. CH is a fourth order nonlinear problem ❀ C 1 FEM (see [Elliott Zheng, 1986], [Elliott French, 1987]). To avoid the well known difficulty met in the implementation of C 1 FEM, another possibility is the use of non-conforming (see [Elliott French, 1989] or discontinuous (see [Wells Kuhl Garikipati, 2006]) methods. Alternatively, use mixed methods (see e.g. [Elliot French Milner, 1989], [Elliott Larsson, 1992] and [Kay Styles Suli, 2009] for the continuous and discontinuous setting, respectively). M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 4 / 28

Huge literature on numerical methods for CH... Here we limit ourvselves to FEM as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. CH is a fourth order nonlinear problem ❀ C 1 FEM (see [Elliott Zheng, 1986], [Elliott French, 1987]). To avoid the well known difficulty met in the implementation of C 1 FEM, another possibility is the use of non-conforming (see [Elliott French, 1989] or discontinuous (see [Wells Kuhl Garikipati, 2006]) methods. Alternatively, use mixed methods (see e.g. [Elliot French Milner, 1989], [Elliott Larsson, 1992] and [Kay Styles Suli, 2009] for the continuous and discontinuous setting, respectively). Recently, isogeometric analysis [G´ omez Calo Bazilevs Hughes, 2008]. M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 4 / 28

Huge literature on numerical methods for CH... Here we limit ourvselves to FEM as the main properties (and limitations) of these schemes are instrumental to motivate the introduction of our new approach. CH is a fourth order nonlinear problem ❀ C 1 FEM (see [Elliott Zheng, 1986], [Elliott French, 1987]). To avoid the well known difficulty met in the implementation of C 1 FEM, another possibility is the use of non-conforming (see [Elliott French, 1989] or discontinuous (see [Wells Kuhl Garikipati, 2006]) methods. Alternatively, use mixed methods (see e.g. [Elliot French Milner, 1989], [Elliott Larsson, 1992] and [Kay Styles Suli, 2009] for the continuous and discontinuous setting, respectively). Recently, isogeometric analysis [G´ omez Calo Bazilevs Hughes, 2008]. Here we propose a C 1 -VEM scheme (see, e.g., [Beir˜ ao da Veiga Brezzi Cangiani Manzini Marini Russo, 2013] for an intro to VEM for second order pbs and [Brezzi Marini, 2013] for plate bending pb.) M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 4 / 28

Weak formulation of CH � a ∆ ( v , w ) = ( ∇ 2 v ) : ( ∇ 2 w ) d x ∀ v , w ∈ H 2 (Ω) , Ω � a ∇ ( v , w ) = ∀ v , w ∈ H 1 (Ω) , ( ∇ v ) · ( ∇ w ) d x Ω � a 0 ( v , w ) = ∀ v , w ∈ L 2 (Ω) , v w d x Ω M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 5 / 28

Weak formulation of CH � a ∆ ( v , w ) = ( ∇ 2 v ) : ( ∇ 2 w ) d x ∀ v , w ∈ H 2 (Ω) , Ω � a ∇ ( v , w ) = ∀ v , w ∈ H 1 (Ω) , ( ∇ v ) · ( ∇ w ) d x Ω � a 0 ( v , w ) = ∀ v , w ∈ L 2 (Ω) , v w d x Ω and the semi-linear form � φ ′ ( z ) ∇ v · ∇ w d x ∀ z , v , w ∈ H 2 (Ω) . r ( z ; v , w ) = Ω M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 5 / 28

Weak formulation of CH � a ∆ ( v , w ) = ( ∇ 2 v ) : ( ∇ 2 w ) d x ∀ v , w ∈ H 2 (Ω) , Ω � a ∇ ( v , w ) = ∀ v , w ∈ H 1 (Ω) , ( ∇ v ) · ( ∇ w ) d x Ω � a 0 ( v , w ) = ∀ v , w ∈ L 2 (Ω) , v w d x Ω and the semi-linear form � φ ′ ( z ) ∇ v · ∇ w d x ∀ z , v , w ∈ H 2 (Ω) . r ( z ; v , w ) = Ω Let � � v ∈ H 2 (Ω) : ∂ n u = 0 on ∂ Ω V = The weak formulation reads as: find u ( · , t ) ∈ V such that � a 0 ( ∂ t u , v ) + γ 2 a ∆ ( u , v ) + r ( u ; u , v ) = 0 ∀ v ∈ V , u ( · , 0) = u 0 ( · ) . M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 5 / 28

Towards C 1 -VEM discretization of CH Ω h decomposition of Ω into polygons E M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 6 / 28

Towards C 1 -VEM discretization of CH Ω h decomposition of Ω into polygons E Intermediate local space ( see [Brezzi Marini, 2013]): � � v ∈ H 2 ( E ) : ∆ 2 v ∈ P 2 ( E ) , v | ∂ E ∈ C 0 ( ∂ E ) , v | e ∈ P 3 ( e ) ∀ e ∈ ∂ E , V h | E = � ∇ v | ∂ E ∈ [ C 0 ( ∂ E )] 2 , ∂ n v | e ∈ P 1 ( e ) ∀ e ∈ ∂ E , M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 6 / 28

Towards C 1 -VEM discretization of CH Ω h decomposition of Ω into polygons E Intermediate local space ( see [Brezzi Marini, 2013]): � � v ∈ H 2 ( E ) : ∆ 2 v ∈ P 2 ( E ) , v | ∂ E ∈ C 0 ( ∂ E ) , v | e ∈ P 3 ( e ) ∀ e ∈ ∂ E , V h | E = � ∇ v | ∂ E ∈ [ C 0 ( ∂ E )] 2 , ∂ n v | e ∈ P 1 ( e ) ∀ e ∈ ∂ E , Linear Operators: D1 ❀ evaluation of v at the vertexes of E ; D2 ❀ evaluation of ∇ v at the vertexes of E . M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 6 / 28

The operator Π ∆ E E : � The projection operator Π ∆ V h | E → P 2 ( E ) is defined by � a ∆ E (Π ∆ E v h , q ) = a ∆ E ( v h , q ) ∀ q ∈ P 2 ( E ) (Π ∆ ( E v h , q ) ) E = ( ( v h , q ) ) E ∀ q ∈ P 1 ( E ) , for all v h ∈ � V h | E where � ∀ v h , w h ∈ C 0 ( E ) . ( ( v h , w h ) ) E = v h ( ν ) w h ( ν ) ν vertexes of ∂ E The operator Π ∆ E is uniquely determined by D 1 and D 2, i.e. Π ∆ E v h depends only on the values of v h and ∇ v h at the vertices of E . M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 7 / 28

Virtual local space � � � � v ∈ � Π ∆ W h | E = V h | E : E ( v h ) q d x = v h q d x ∀ q ∈ P 2 ( E ) . E E [Ahmad Alsaedi Brezzi Marini Russo, 2013] M. Verani (MOX - PoliMi) VEM Cahn-Hilliard POEMs 2015 8 / 28