Approximation of mean curvature motion with nonlinear Neumann - PowerPoint PPT Presentation

Approximation of mean curvature motion with nonlinear Neumann conditions Yves Achdou joint work with M. Falcone Laboratoire J-L Lions, Universit´ e Paris Diderot Padova 2012 Y. Achdou HYP2012 Padova

The boundary value problem The PDE �� � � ∂u I − Du ⊗ Du D 2 u ∂t − trace = 0 , in Ω × (0 , T ) , | Du | 2 where Ω is a bounded domain of R d with a W 3 , ∞ boundary. Initial condition u ( x, · ) = u 0 , Y. Achdou HYP2012 Padova

The boundary value problem The PDE �� � � ∂u I − Du ⊗ Du D 2 u ∂t − trace = 0 , in Ω × (0 , T ) , | Du | 2 where Ω is a bounded domain of R d with a W 3 , ∞ boundary. Initial condition u ( x, · ) = u 0 , Boundary condition: the normal vectors to the level sets make a given angle with the outward normal vector ∂u ∂n = θ | Du | on ∂ Ω × (0 , T ) , where θ is a Lipschitz continuous function with | θ ( x ) | ≤ θ < 1. Y. Achdou HYP2012 Padova

Other forms of the PDE ∂t − ∆ u + ( D 2 u Du, Du ) ∂u = 0 | Du | 2 or � Du � ∂u ∂t − div | Du | = 0 . | Du | Y. Achdou HYP2012 Padova

Other forms of the PDE ∂t − ∆ u + ( D 2 u Du, Du ) ∂u = 0 | Du | 2 or � Du � ∂u ∂t − div | Du | = 0 . | Du | Goal Propose a semi-Lagrangian scheme for the PDE (extension of Carlini-Falcone-Ferretti, JCP 2005 and Interfaces and Free Boundaries, 2011) Couple it with a finite difference scheme to deal with the boundary conditions Hereafter, d = 2. Y. Achdou HYP2012 Padova

Some references MCM via viscosity solution techniques Evans-Spruck, J. Diff. Geom., 1991 Evans - Soner-Souganidis, Comm. Pure Appl. Math, 1992, ..... Soner-Touzi, J. Eur. Math. Soc, 2002 and Ann. Prob. 2003 The boundary value problem with nonlinear Neumann cond. Barles, J. Diff. eq. 1999 Ishii-Ishii, SIAM J. Math. Anal. 2001 Numerical methods Osher-Sethian, JCP, 1988 Merriman-Bence-Osher, AMS LN, 1993 Crandall-Lions, Numer. Math., 1996 Barles-Georgelin, SIAM J. Num. Anal., 1995 Catt´ e-Dibos-Koepfler, SIAM J. Num. Anal., 1995 Y. Achdou HYP2012 Padova

Viscosity solutions � � ( I d − p ⊗ p F ( p, X ) = − trace | p | 2 ) X is defined for p � = 0 F and F are the LSC and USC envelopes of F � η + F ( p, X ) if x ∈ Ω , G ( x, η, p, X ) = max( η + F ( p, X ) , p · n − θ | p | ) if x ∈ ∂ Ω � η + F ( p, X ) if x ∈ Ω , G ( x, η, p, X ) = min( η + F ( p, X ) , p · n − θ | p | ) if x ∈ ∂ Ω G is used for subsolutions, G is used for supersolutions Strong comparison principle (Barles 1999) Y. Achdou HYP2012 Padova

Semi-Lagrangian schemes for MCM: (CFF 2011) Representation formula in Ω = R 2 : Soner-Touzi For any regular solution u of the PDE s.t. Du � = 0, u ( x, t ) = E { u 0 ( y ( t ; x, t )) } , where √  � � dy ( s ; x, t ) = 2 P Du ( y ( s ; x, t ) , t − s ) dW ( s ) ,  y (0; x, t ) = x,  P ( q ) = I − qq T 1 � u 2 � − u x 1 u x 2 x 2 i.e. P ( Du ) = . u 2 − u x 1 u x 2 | q | 2 | Du | 2 x 1 Y. Achdou HYP2012 Padova

Semi-Lagrangian schemes for MCM: (CFF 2011) Representation formula in Ω = R 2 : Soner-Touzi For any regular solution u of the PDE s.t. Du � = 0, u ( x, t ) = E { u 0 ( y ( t ; x, t )) } , where √  � � dy ( s ; x, t ) = 2 P Du ( y ( s ; x, t ) , t − s ) dW ( s ) ,  y (0; x, t ) = x,  P ( q ) = I − qq T 1 � u 2 � − u x 1 u x 2 x 2 i.e. P ( Du ) = . u 2 − u x 1 u x 2 | q | 2 | Du | 2 x 1 The representation formula implies that u ( x, t + ∆ t ) = E { u ( y (∆ t ; x, t + ∆ t ) , t ) } . Y. Achdou HYP2012 Padova

A one dimensional Brownian The projector P ( q ) is of the form � − q 2 � σ ( q ) = 1 P ( q ) = σ ( q ) σ T ( q ) , with q 1 | q | W ( s ) ≡ σ T dW ( s ), For the real valued Brownian � W def. by d � � � √ d � dy ( s ; x, t ) = 2 σ Du ( y ( s ; x, t ) , t − s ) W ( s ) . Y. Achdou HYP2012 Padova

A one dimensional Brownian The projector P ( q ) is of the form � − q 2 � σ ( q ) = 1 P ( q ) = σ ( q ) σ T ( q ) , with q 1 | q | W ( s ) ≡ σ T dW ( s ), For the real valued Brownian � W def. by d � � � √ d � dy ( s ; x, t ) = 2 σ Du ( y ( s ; x, t ) , t − s ) W ( s ) . Remarks σ ( Du ) is tangent to the level sets of u . If d > 2, σ is a d × ( d − 1)-matrix and � W is a ( d − 1)-dimensional Brownian motion. Y. Achdou HYP2012 Padova

Semi-discrete scheme when Du � = 0 (1/2) Euler scheme for the stochastic process: y k ≈ y ( t k ; x, t ) with � � √ ∆ � y k +1 = y k + 2 σ Du ( y ( k ∆ t ; x, t ) , t − k ∆ t ) W k , where ∆ � W k ≈ Gaussian variable with mean value 0 and variance ∆ t . Y. Achdou HYP2012 Padova

Semi-discrete scheme when Du � = 0 (1/2) Euler scheme for the stochastic process: y k ≈ y ( t k ; x, t ) with � � √ ∆ � y k +1 = y k + 2 σ Du ( y ( k ∆ t ; x, t ) , t − k ∆ t ) W k , where ∆ � W k ≈ Gaussian variable with mean value 0 and variance ∆ t . For first order accuracy, it is enough that √ ∆ t } = 1 P { ∆ � W k = ± 2 . For t = t n = n ∆ t and k = 0: � � √ = 1 P y 1 = x ± 2∆ t σ ( Du ( x, t n ) , t n ) 2 . Y. Achdou HYP2012 Padova

Semi-discrete scheme when Du � = 0 (2/2) This leads to the semi-discrete scheme for u : � �� � � � 1 Z + Z − u ( x, t n +1 ) = u n ( x ) , t n + u n ( x ) , t n , 2 where √ Z ± n ( x ) ≡ x ± 2∆ t σ ( Du ( x, t n )) , √ Z + n ( x ) = x + 2∆ t σ ( Du ( x, t n )) Du ( x, t n ) x √ Z − n ( x ) = x − 2∆ t σ ( Du ( x, t n )) contour of u ( · , t n ) passing by x Y. Achdou HYP2012 Padova

Fully-discrete scheme for MCM when Du � = 0 Consider a mesh T h of Ω and call ξ a node of T h . The values u ( ξ, t n ) are approximated by u n h ( ξ ) with � � � � √ √ ( ξ ) = 1 + 1 u n +1 2 I h [ u n ∆ tσ n ( ξ ) 2 I h [ u n ∆ tσ n ( ξ ) h ] ξ + h ] ξ − , h where I h is an interpolation operator, √ σ n ( ξ ) = 2 σ ( D h u n h ( ξ )) , and D h is a discrete version of D . Y. Achdou HYP2012 Padova

Fully-discrete scheme for MCM when Du � = 0 Consider a mesh T h of Ω and call ξ a node of T h . The values u ( ξ, t n ) are approximated by u n h ( ξ ) with � � � � √ √ ( ξ ) = 1 + 1 u n +1 2 I h [ u n ∆ tσ n ( ξ ) 2 I h [ u n ∆ tσ n ( ξ ) h ] ξ + h ] ξ − , h where I h is an interpolation operator, √ σ n ( ξ ) = 2 σ ( D h u n h ( ξ )) , and D h is a discrete version of D . Questions Which scheme in the regions where | D h u n h | is small? Which scheme near the boundary? Y. Achdou HYP2012 Padova

A subdivision of Ω depending on a small parameter δ thickness: δ : 1 ≫ δ ≫ h quasiuniform Ω unstructured mesh: h n n ξ + δσ n ( ξ ) ξ ω 1 ξ − δσ n ( ξ ) ω 2 boundary nodes: finite difference scheme strongly internal nodes: semi-lagrangian scheme the triangles have acute angles in this region In the layers ω k , one can define a system of orthogonal coordinates by projecting the points orthogonally onto ∂ Ω. Similarly, one can lift the outward unit vector n into ω k . Y. Achdou HYP2012 Padova

The scheme in the layer ω ℓ The nonlinear Neumann condition is not only imposed at the nodes on ∂ Ω, but also at all the boundary nodes in ω ℓ . We use the lifting of n and a monotone scheme: B ℓ ( ξ i , u n +1 , [ u n +1 ] ℓ , [[ u n ]]) = 0 , for all i s.t. ξ i ∈ ω ℓ i where [ u ] ℓ = { u j , 1 ≤ j ≤ N h , j � = i, ξ j ∈ ω ℓ } , [[ u ]] = { u j , 1 ≤ j ≤ N h , ξ j is strongly internal } . For example, a first order Godunov like scheme can be used. Y. Achdou HYP2012 Padova

The scheme in the layer ω ℓ The nonlinear Neumann condition is not only imposed at the nodes on ∂ Ω, but also at all the boundary nodes in ω ℓ . We use the lifting of n and a monotone scheme: B ℓ ( ξ i , u n +1 , [ u n +1 ] ℓ , [[ u n ]]) = 0 , for all i s.t. ξ i ∈ ω ℓ i where [ u ] ℓ = { u j , 1 ≤ j ≤ N h , j � = i, ξ j ∈ ω ℓ } , [[ u ]] = { u j , 1 ≤ j ≤ N h , ξ j is strongly internal } . For example, a first order Godunov like scheme can be used. Given the values at the strongly internal nodes, this is a system of nonlinear equations which can be solved by combining Gauss-Seidel sweeps with different orderings. Y. Achdou HYP2012 Padova

The scheme at the strongly internal nodes (1/2) Ingredients I h : Lagrange interpolation operator associated with P 1 finite elements Y. Achdou HYP2012 Padova

The scheme at the strongly internal nodes (1/2) Ingredients I h : Lagrange interpolation operator associated with P 1 finite elements D h : discrete gradient reconstructed at the mesh nodes. For example, � | τ | [ D h v ]( ξ i ) = | ω ξ i | D ( I h [ v ] | τ ) . τ ∈T h,i Set D n i = [ D h u n ]( ξ i ). Y. Achdou HYP2012 Padova

The scheme at the strongly internal nodes (1/2) Ingredients I h : Lagrange interpolation operator associated with P 1 finite elements D h : discrete gradient reconstructed at the mesh nodes. For example, � | τ | [ D h v ]( ξ i ) = | ω ξ i | D ( I h [ v ] | τ ) . τ ∈T h,i Set D n i = [ D h u n ]( ξ i ). Two internal regions: given two positive numbers C and s , the two sets of indices J n 1 and J n 2 are defined as follows: J n 1 = { i : ξ i is strongly internal and | D n i | ≥ Ch s } , J n 2 = { i : ξ i is strongly internal and | D n i | < Ch s } . Y. Achdou HYP2012 Padova