The Kuratowski Ryll-Nardzewski Theorem and semismooth Newton methods for Hamilton Jacobi Bellman equations

Transcription

1 The Kuratowski Ryll-Nardzewski Theorem and semismooth Newton methods for Hamilton Jacobi Bellman equations Iain Smears INRIA Paris Linz, November 2016 joint work with Endre Süli, University of Oxford

2 Overview Talk outline 1. Introduction: Howard s algorithm / policy iteration for Hamilton Jacobi Bellman equations. 2. Semismoothness of HJB operators in function spaces. 3. Applications to discontinuous Galerkin FEM approximations of HJB equations with Cordes coefficients.

3 Overview Talk outline 1. Introduction: Howard s algorithm / policy iteration for Hamilton Jacobi Bellman equations. 2. Semismoothness of HJB operators in function spaces. 3. Applications to discontinuous Galerkin FEM approximations of HJB equations with Cordes coefficients.

4 1/28 1. Hamilton Jacobi Bellman Equation F [u] := sup[l α u f α ] = 0 in Ω, α Λ u = 0 on Ω, (HJB) where L α u := a α (x) : D 2 u + b α (x) u c α (x) u. Notation: a α (x) : D 2 u = Assumptions: bounded domain Ω, d aij α (x)u xi x j, b α (x) u = i,j=1 control set Λ is a compact metric space, continuous functions a, b, c and f in x Ω and α Λ. d bi α (x)u xi. Remark: Further assumptions are required for well-posedness of the problem, but not for the semismoothness discussed here. i=1

5 2/28 1. Motivation Howard s algorithm / policy iteration Formal structure 1. Choose an initial guess u For each k 0, choose α k : Ω Λ such that α k (x) argmax α Λ (L α u k f α )(x), x Ω. 3. Then, find u k+1 as a solution of the PDE L α k u k+1 = f α k in Ω, with u k+1 = 0 on Ω, where L α k v := a α k (x) (x) : D 2 v + b α k (x) (x) v c α k (x) v In practice: used in a discrete context after discretization by a numerical method.

6 2/28 1. Motivation Howard s algorithm / policy iteration Formal structure 1. Choose an initial guess u For each k 0, choose α k : Ω Λ such that α k (x) argmax α Λ (L α u k f α )(x), x Ω. 3. Then, find u k+1 as a solution of the PDE L α k u k+1 = f α k in Ω, with u k+1 = 0 on Ω, where L α k v := a α k (x) (x) : D 2 v + b α k (x) (x) v c α k (x) v In practice: used in a discrete context after discretization by a numerical method.

7 3/28 1. Background Classical works: [Bellman,Dynamic Programming, 1957] [Howard, Dynamic Programming and Markov Processes, 1960]. Historical summary from [Puterman & Brumelle, 1979]: Policy iteration is usually attributed to Bellman [...] and Howard [...] Bellman developed the technique, which he called iteration in policy space, to solve several dynamic programming problems. Howard [16] later developed a version of this procedure for Markovian decision problems which he called the policy-iteration method. [Puterman & Brumelle, 1979]: interpretation as Newton Kantorovich method & convergence rates assuming: there is δ (0, 1] such that, for all functions u and v, L αv L αu L(X,Y ) v u δ X where α v and α u are arg-maximisers for v and u. NB: this cannot hold when arg-max operation is non-unique or not continuous.

8 4/28 1. Background On solver algorithms for HJB: [Santos & Rust, 2004] Analysis of policy iteration for finite dimensional MDP problems. [Bokanowski, Maroso, Zidani, 2009]: Superlinear convergence and semismoothness of finite dimensional HJB operators of form min x c α ] = 0 α A N[Bα with matrices B α R N N and vectors x, c R N, (see also discussion of Bellman Isaacs). Variant algorithms and applications: penalty methods [Reisinger & Witte, 2011, 2012], coupled value-policy iteration [Alla, Falcone, Kalise, 2015] Semismooth Newton methods [Ulbrich, 2002], [Hintermüller, Ito, Kunisch, 2002] (primal-dual active set method as as semismooth Newton method)

9 1. Semismooth Newton methods Notation: Let X and Y be sets. We write G : X Y if G is a set-valued map that maps X into the subsets of Y. Definition of semismoothness [Ulbrich, 2002] Let X and Y be Banach spaces. Let F : X Y. Let DF : X L(X, Y ) with non-empty images. We say that F is DF -semismooth on U if, for all x U, lim e X 0 1 e X sup L DF [u+e] F [u + e] F [u] L e Y = 0. Then DF is then called a generalised differential of F on U. Semismoothness + uniform stability of linearizations: sup L 1 L(Y,X ) < L DF [v],v X = local superlinear convergence of semismooth Newton method. 5/28

10 6/28 1. Semismoothness of max(v, 0) and norm-gap Important example from [Ulbrich, 2002], [Hintermüller, Ito, Kunisch, 2002] Let 1 q < r. Let G : L r (Ω) L q (Ω) be defined by G : u max(u, 0). Then G is semismooth from L r (Ω) to L q (Ω) with differential DF [v] the set of all L L (Ω) of the form: 1 if v(x) > 0 L(x) = 0 if v(x) < 0 an arbitrary fixed value if v(x) = 0 Norm gap: the restriction q < r cannot be removed (counter-examples). How to generalise this to HJB operators?

11 7/28 Overview Talk outline 1. Introduction: Howard s algorithm / policy iteration for Hamilton Jacobi Bellman equations. 2. Semismoothness of HJB operators in function spaces. 3. Applications to discontinuous Galerkin FEM approximations of HJB equations with Cordes coefficients.

12 8/28 1. Motivation Howard s algorithm / policy iteration Formal structure 1. Choose an initial guess u For each k 0, choose α k : Ω Λ such that α k (x) argmax α Λ (L α u k f α )(x), x Ω. 3. Then, find u k+1 as a solution of the PDE L α k u k+1 = f α k in Ω, with u k+1 = 0 on Ω, where L α k v := a α k (x) (x) : D 2 v + b α k (x) (x) v c α k (x) v In practice: used in a discrete context after discretization by a numerical method.

13 9/28 2. Semismoothness of HJB operators For FEM applications: let T h be a mesh on Ω. Space X = W 2,r (Ω, T h ), 1 r, with norm: u W 2,r (Ω;T h ) = K T h u r W 2,r (K) Function u W 2,r (Ω, T h ) have element-wise gradient h u and Hessian D 2 hu. For Λ compact and continuous coefficients, F : W 2,r (Ω, T h ) L r (Ω) is well defined and Lipschitz continuous F [u] := sup[l α u f α ]. α Λ 1 r.

14 10/28 2. Semismoothness: argmax set-valued map For each u X, we define u = ( u, h u, D 2 hu ) L r (Ω; R m ) for suitable m. We then view the differential operator F [u] as a composition of x u(x) with the scalar function F : Ω R m R defined by F (x, v) = sup[a α (x) : M + b α (x) p c α (x)z f α (x)], v = (z, p, M) α Λ Define the set-valued map Ω R m (x, v) Λ(x, v) Λ by Λ(x, v) := argmax α Λ [a α (x) : M + b α (x) p c α (x)z f α (x)] Straightforward: Λ(x, v) is non-empty and closed in Λ.

15 10/28 2. Semismoothness: argmax set-valued map Important lemma: The mapping Λ(, ): Ω R m Λ is upper semicontinuous: For every (x, v) Ω R m, and any open neighbourhood U of Λ(x, v), there is an open neighbourhood V of (x, v) such that Λ(y, w) U for all (y, w) V. (x, v) (x n, v n) (x n, v n)! (x, v)

16 11/28 2. Kuratowski Ryll-Nardzewski Theorem Kuratowski Ryll-Nardzewski Let Ω R d be a bounded open set, let Λ be a compact metric space, let Λ(, ): Ω R m Λ be an upper semicontinuous set-valued function, such that Λ(x, v) is non-empty and closed for every (x, v) Ω R m. Then, for any Lebesgue measurable function u: Ω R m, there exists a Lebesgue measurable selection α: Ω Λ such that α(x) Λ ( x, u(x) ) for a.e. x Ω. (Presented here in the form needed for our purposes - original result is rather more general) Kuratowski & Ryll-Nardzewski, Bull. Acad. Polon. Sci., 1965: A general theorem on selectors. A (specialised) proof in Aubin & Cellina, Differential Inclusions, 1984.

17 12/28 2. The generalized differential of HJB operators Recall u(x) = (u(x), h u(x), Dhu(x)) 2 for u W 2,r (Ω; T h ). Define the set of measurable selections Λ[u]: Λ[u] = {α: Ω Λ; α Lebesgue measurable, α(x) Λ(x, u(x)) a.e. in Ω}. Kuratowski Ryll-Nardzewski Thm = Λ[u] is non-empty for all u W 2,r (Ω; T h ). Define the differential DF [u] := {L α = a α : D 2 h + b α h c α, α Λ[u]} The measurability of α Λ[u] implies that L α is well defined in L(W 2,r (Ω; T h ), L r (Ω)).

18 12/28 2. The generalized differential of HJB operators Recall u(x) = (u(x), h u(x), Dhu(x)) 2 for u W 2,r (Ω; T h ). Define the set of measurable selections Λ[u]: Λ[u] = {α: Ω Λ; α Lebesgue measurable, α(x) Λ(x, u(x)) a.e. in Ω}. Kuratowski Ryll-Nardzewski Thm = Λ[u] is non-empty for all u W 2,r (Ω; T h ). Define the differential DF [u] := {L α = a α : D 2 h + b α h c α, α Λ[u]} The measurability of α Λ[u] implies that L α is well defined in L(W 2,r (Ω; T h ), L r (Ω)).

19 13/28 2. Semismoothness of HJB operators DF [u] := {L α = a α : D 2 h + b α h c α, α Λ[u]} Theorem [S. & Süli, SINUM, 2014] Let 1 q < r. The operator F : W 2,r (Ω; T h ) L q (Ω) is DF -semismooth on W 2,r (Ω; T h ): 1 lim sup F [u + e] F [u] L α e L e W 2,r (Ω;Th ) 0 e q (Ω) = 0. W 2,r (Ω;T h ) L α DF [u+e] Remark: The sup implies any choice of measurable selection is sufficient. I. S. & E. Süli, SIAM J. Numer. Anal. 2014: Discontinuous Galerkin finite element approximation of Hamilton Jacobi Bellman equations with Cordes coefficients.

20 14/28 2. Semismoothness of HJB operators: Proof Suppose the claim is false: there exists {e j } j=0, e j W 2,r (Ω;T h ) 0, α j Λ[u + e j ], and ρ > 0 such that, j 0: 1 F [u + e j ] F [u] L α j e j L e j q (Ω) > ρ W 2,r (Ω;T h ) ( ) We will find a subsequence of {e j } j=0 such that ( ) is contradicted.

21 15/28 2. Semismoothness of HJB operators: Proof 1. Passing to a subsequence, we have (e j, h e j, D 2 he j ) 0 pointwise a.e. in Ω. 2. Key inequality: using the definition of F, we can show (pointwise): F [u +e j ](x) F [u](x) L α j e j (x) L α e j (x) L α j e j (x) α Λ(x, u(x)), a.e. x. This implies F [u + e j ] F [u] L α j e j G j ( e j + e j + D 2 he j ), ( ) where the function G j is defined by: G j := inf { a α a α j + b α b α j + c α c α j }. α Λ(,u( )) Remark: G j is measurable (can be written as a composition of a lower semicontinous function with measurable functions).

22 16/28 2. Semismoothness of HJB operators: Proof Recall G j := inf α Λ(,u( )) { a α a α j + b α b α j + c α c α j }. Recall also α j (x) Λ(x, u(x) + e j (x)) for a.e. x Ω. (x, u) (x, u(x)+e j(x)) (x, u(x)+e j(x))! (x, u(x)) Upper semi-continuity of Λ(, ) leads to G j 0 pointwise a.e. in Ω.

23 17/28 2. Semismoothness of HJB operators: Proof Also, we have a uniform bound sup j 0 G j L (Ω) C because all coefficients are uniformly bounded on Ω Λ. Lebesgue s Dominated Convergence Theorem: lim G j L j s (Ω) = 0 for any 1 s <. Recall ( ): F [u + e j ] F [u] L α j e j G j ( e j + e j + Dhe 2 j ), ( ) If q < r, Hölder s inequality implies that we can find s [1, ) s.t. 1 0 < ρ F [u + e j ] F [u] L α j e j L e j q (Ω) G j L s (Ω) 0. W 2,r (Ω;T h ) Remarks: the norm gap appears because there are cases where G j L (Ω) 0.

24 17/28 2. Semismoothness of HJB operators: Proof Also, we have a uniform bound sup j 0 G j L (Ω) C because all coefficients are uniformly bounded on Ω Λ. Lebesgue s Dominated Convergence Theorem: lim G j L j s (Ω) = 0 for any 1 s <. Recall ( ): F [u + e j ] F [u] L α j e j G j ( e j + e j + Dhe 2 j ), ( ) If q < r, Hölder s inequality implies that we can find s [1, ) s.t. 1 0 < ρ F [u + e j ] F [u] L α j e j L e j q (Ω) G j L s (Ω) 0. W 2,r (Ω;T h ) Remarks: the norm gap appears because there are cases where G j L (Ω) 0.

25 18/28 2. Semismoothness of HJB operators DF [u] := {L α = a α : D 2 h + b α h c α, α Λ[u]} Theorem [S. & Süli, SINUM, 2014] Let 1 q < r. The operator F : W 2,r (Ω; T h ) L q (Ω) is DF -semismooth on W 2,r (Ω; T h ): 1 lim sup F [u + e] F [u] L α e L e W 2,r (Ω;Th ) 0 e q (Ω) = 0. W 2,r (Ω;T h ) L α DF [u+e] Remark: The sup implies any choice of measurable selection is sufficient. I. S. & E. Süli, SIAM J. Numer. Anal. 2014: Discontinuous Galerkin finite element approximation of Hamilton Jacobi Bellman equations with Cordes coefficients.

26 19/28 Overview Talk outline 1. Introduction: Howard s algorithm / policy iteration for Hamilton Jacobi Bellman equations. 2. Semismoothness of HJB operators in function spaces. 3. Applications to discontinuous Galerkin FEM approximations of HJB equations with Cordes coefficients.

27 20/28 3. Cordes condition From now on, we suppose Ω is bounded Lipschitz domain a α uniformly elliptic, c α 0, uniformly over Ω. Policy iteration: L α k u k+1 = f α k, α k Λ[u k ] Is this linear PDE well-posed for general uniformly elliptic coefficients a α? In general: no! Due to discontinuities in a α : Non-uniqueness of strong solutions [Gilbarg & Trudinger, 2001] Non-uniquess of viscosity solutions [Nadirashvili, 1997], [Safonov, 1999]. However: yes under a further assumption: Cordes condition (next slide) : well-posedness in H 2 (Ω) H0 1 (Ω).

28 20/28 3. Cordes condition From now on, we suppose Ω is bounded Lipschitz domain a α uniformly elliptic, c α 0, uniformly over Ω. Policy iteration: L α k u k+1 = f α k, α k Λ[u k ] Is this linear PDE well-posed for general uniformly elliptic coefficients a α? In general: no! Due to discontinuities in a α : Non-uniqueness of strong solutions [Gilbarg & Trudinger, 2001] Non-uniquess of viscosity solutions [Nadirashvili, 1997], [Safonov, 1999]. However: yes under a further assumption: Cordes condition (next slide) : well-posedness in H 2 (Ω) H0 1 (Ω).

29 20/28 3. Cordes condition From now on, we suppose Ω is bounded Lipschitz domain a α uniformly elliptic, c α 0, uniformly over Ω. Policy iteration: L α k u k+1 = f α k, α k Λ[u k ] Is this linear PDE well-posed for general uniformly elliptic coefficients a α? In general: no! Due to discontinuities in a α : Non-uniqueness of strong solutions [Gilbarg & Trudinger, 2001] Non-uniquess of viscosity solutions [Nadirashvili, 1997], [Safonov, 1999]. However: yes under a further assumption: Cordes condition (next slide) : well-posedness in H 2 (Ω) H0 1 (Ω).

30 21/28 3. Cordes condition From now on, we assume that Ω is convex: Cordes condition: Case 1: without advection and reaction Assume that there exists ε (0, 1] s. t. a α (x) 2 (Tr a α (x)) 2 1 d 1 + ε x Ω, α Λ. (Cordes 0) Cordes condition: Case 2: extension to b α 0 and c α 0 Assume that there exist λ > 0 and ε (0, 1] s. t. a α 2 + b α 2 /2λ + (c α /λ) 2 (Tr a α + c α /λ) 2 1 d + ε in Ω, α Λ. (Cordes 1) Thm: [Cordes 1956], [Maugeri, Palagachev, Softova 2000] There is C = C(ε) s.t. for any measurable α Λ: (L α ) 1 L 2 H 2 H 1 0 C.

31 22/28 3. Cordes condition Well-posedness theorem [S. & Süli, SINUM, 2014] Under these assumptions, there exists a unique u H 2 (Ω) H 1 0 (Ω) that solves (HJB) pointwise a.e. in Ω. Remarks If dimension d = 2, (Cordes 0) uniform ellipticity. Proof is based on Cordes condition and Miranda Talenti inequality for convex domains: u H 2 (Ω) := D 2 u L 2 (Ω) u L 2 (Ω) u H 2 (Ω) H 1 0 (Ω).

32 23/28 3. Applications to numerical scheme Construction an hp-version discontinuous Galerkin finite element scheme in [S. & Süli, SINUM SINUM Num. Math. 2016]: Discrete Stability in H 2 : u h is uniquely defined, and stable in discrete norm H 2 (Ω;T h ) + Jh. Consistency for true solution : if u H s (Ω, T h ) with s > 5/2. Near-best approximation w.r.t H 2 -conforming subspaces. Convergence rates: if u H s (Ω, T h ) with s > 5/2: u u h H 2 (Ω;T h ) hmin(p+1,s) 2 u p s 5/2 H s (Ω). Main idea: based on approximating a strongly monotone operator formulation of the PDE: A(u; v) = sup[γ α (L α u f α )]( v λv) dx = 0 v H 2 (Ω) H0 1 (Ω); Ω α Λ

33 24/28 3. Applications to numerical scheme Discrete linearised problems: A k h(u k+1 h, v h ) = K T h γ α k f α k, L λ v h K v h V h,p, where the bilinear form A k h : V h,p V h,p R is defined by A k h(w h, v h ) := K Th (γ α k L α k w h, L λ v h K + linear stabilization terms from A h ( ; ). Superlinear convergence [S. & Süli, SINUM, 2014] There exists R > 0, possibly depending on h and p, such that if u h u 0 h H 2 (Ω;T h ) < R, then superlinear convergence: u h u k+1 h H lim 2 (Ω;T h ) k u h uh k = 0 H 2 (Ω;T h )

34 25/28 3. Numerics: experiment 1: h-refinement Experiment 1 : Test of high order convergence rates under h-refinement, fixed p. ( ) a α = sin 2 θ sin θ cos θ 2 R sin θ cos θ cos 2 R θ with α = (θ, R) Λ = [0, π 3 ] SO(2). Remark: a α becomes increasingly anisotropic as θ π/3; rotation matrices R SO(2) prevent monotone schemes from aligning the grid with the anisotropy u uh H 2 (Ω;Th) /64 1/32 1/16 1/8 Mesh size p = 2 p = 3 p = 4 p = 5 1/4 1/2 uh u k h H 2 (Ω;Th) h = 1/4 h = 1/ h = 1/16 h = 1/32 h = 1/64 Converged Iteration number k

35 26/28 3. Numerics: experiment 2: hp-refinement Experiment 2: test of exponential convergence rates under hp-refinement Let Ω = (0, 1) 2, b α := (0, 1), c α := 10 and define ( ) a α := α 20 1 α, α Λ := SO(2), λ = (Cordes 1) holds with ε (nearly degenerate, strongly anisotropic). Solution: ( ) ( u(x, y) = (2x 1) e 1 2x 1 1 y + 1 ) ey/δ, δ := = O(ε) e 1/δ 1 Near-degenerate and anisotropic diffusion. Sharp boundary layer. Non-smooth solution.

36 26/28 3. Numerics: experiment 2: hp-refinement Experiment 2: test of exponential convergence rates under hp-refinement Let Ω = (0, 1) 2, b α := (0, 1), c α := 10 and define ( ) a α := α 20 1 α, α Λ := SO(2), λ = (Cordes 1) holds with ε (nearly degenerate, strongly anisotropic). Solution: ( ) ( u(x, y) = (2x 1) e 1 2x 1 1 y + 1 ) ey/δ, δ := = O(ε) e 1/δ 1 Boundary layer adapted meshes with p-refinement: 2 p K 10, from 100 to 1320 DoFs.

37 27/28 3. Numerics: experiment 2: hp-refinement Relative Error Broken H 2 norm Broken H 1 norm DoF

38 27/28 3. Numerics: experiment 2: hp-refinement 10 2 uh u k h H 2 (Ω;T h ) uh H 2 (Ω;Th ) p = 3 p = 4 p = 5 p = 6 p = Iteration number k

39 Conclusions Conclusions: General semismoothness result for HJB operators with Λ a general compact metric space and continuous coefficients. Usage of the measurable selection theorem of Kuratowski Ryll-Nardzewski. Application to DGFEM for HJB equations with Cordes coefficients: superlinear convergence of the semismooth Newton method Numerical experiments showing fast convergence and weak dependence of the iteration counts on h and p. Linear nondivergence form PDE: Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordes coefficients, I. S. & E. Süli, SIAM J. Numer. Anal Elliptic HJB: Discontinuous Galerkin finite element approximation of Hamilton Jacobi Bellman equations with Cordes coefficients, I. S. & E. Süli, SIAM J. Numer. Anal Parabolic HJB: Discontinuous Galerkin finite element methods for time-dependent Hamilton Jacobi Bellman equations with Cordes coefficients, I. S. & E. Süli, Numerische Mathematik Thank you! 28/28

40 28/28 3. Numerical scheme Numerical scheme: solve A h (u h ; v h ) = 0 for all v h V h,p A h (u h ; v h ) := F γ[u h ], L λ v h K + J h (u h, v h ) K Th + 1 B h (u h, v h ) L λ u h, L λ v h K. 2 K Th

41 28/28 3. Numerical scheme Numerical scheme: solve A h (u h ; v h ) = 0 for all v h V h,p A h (u h ; v h ) := F γ[u h ], L λ v h K + J h (u h, v h ) K Th + 1 B h (u h, v h ) L λ u h, L λ v h K. 2 K Th F γ[u h ], L λ v h K := sup [γ α (L α u h f α )] ( v h λv h ) dx. K α Λ Remark: the term γ α rescales the operators L α without changing the true solution.

42 28/28 3. Numerical scheme Numerical scheme: solve A h (u h ; v h ) = 0 for all v h V h,p A h (u h ; v h ) := F γ[u h ], L λ v h K + J h (u h, v h ) K Th + 1 B h (u h, v h ) L λ u h, L λ v h K. 2 K Th Jump penalisation with µ F pk 2 /h K and η F pk 4 /h3 K for F K: J h (u h, v h ) := [ ] µf T u h, T v h F + η F u h, v h F F F i,b h + µ F u h n F, v h n F F. F F i h

43 28/28 3. Numerical scheme Numerical scheme: solve A h (u h ; v h ) = 0 for all v h V h,p A h (u h ; v h ) := F γ[u h ], L λ v h K + J h (u h, v h ) K Th + 1 B h (u h, v h ) L λ u h, L λ v h K. 2 K Th L λ u h, L λ v h K := ( u h λu h ) ( v h λv h ) dx. K

44 3. Numerical scheme Numerical scheme: solve A h (u h ; v h ) = 0 for all v h V h,p A h (u h ; v h ) := F γ[u h ], L λ v h K + J h (u h, v h ) K Th + 1 B h (u h, v h ) L λ u h, L λ v h K. 2 K Th Weak enforcement of Miranda Talenti inequality: B h (u h, v h ) := [ D 2 u h, D 2 v h K + 2λ u h, v h K + λ 2 ] u h, v h K K T h + [ ] divt T {u h }, v h n F F + div T T {v h }, u h n F F F F i h F F i,b h λ [ T { u h n F }, T v h F + T { v h n F }, T u h F ] F F i,b h [ { u h n F }, v h F + { v h n F }, u h F ] λ [ {u h }, v h n F F + {v h }, u h n F F ] F F i h 28/28

45 28/28 3. Numerical scheme Numerical scheme: solve A h (u h ; v h ) = 0 for all v h V h,p A h (u h ; v h ) := F γ[u h ], L λ v h K + J h (u h, v h ) K Th + 1 B h (u h, v h ) L λ u h, L λ v h K. 2 K Th Consistency, Stability, Convergence rates [S. & Süli, SINUM, 2014] Discrete Stability: u h is uniquely defined, and stable in discrete norm H 2 (Ω;T h ) + Jh. Consistency: A h (u; v h ) = 0 for the solution u, if u H s (Ω, T h ) with s > 5/2. Near-best approximation w.r.t H 2 -conforming subspaces. Convergence rates: u u h H 2 (Ω;T h ) hs u p s 1/2 H s (Ω) if u H s (Ω, T h ) with s > 5/2.

46 28/28 3. Discrete Semismooth Newton method Algorithm 1. Choose an initial guess u 0 h V h,p. 2. Given u k h V h,p, k N, choose α k Λ[u k h ]. 3. Solve A k h(u k+1 h, v h ) = K T h γ α k f α k, L λ v h K v h V h,p, where the bilinear form A k h : V h,p V h,p R is defined by A k h(w h, v h ) := K Th (γ α k L α k w h, L λ v h K + linear stabilization terms from A h ( ; ). Well-posedness : the bilinear forms A k h(, ) have a uniform coercivity constant. Superlinear convergence [S. & Süli, SINUM, 2014] There exists R > 0, possibly depending on h and p, such that if u h u 0 h H 2 (Ω;T h ) < R, then u k h u h superlinearly.

47 28/28 3. Discrete Semismooth Newton method Algorithm 1. Choose an initial guess u 0 h V h,p. 2. Given u k h V h,p, k N, choose α k Λ[u k h ]. 3. Solve A k h(u k+1 h, v h ) = K T h γ α k f α k, L λ v h K v h V h,p, where the bilinear form A k h : V h,p V h,p R is defined by A k h(w h, v h ) := K Th (γ α k L α k w h, L λ v h K + linear stabilization terms from A h ( ; ). Well-posedness : the bilinear forms A k h(, ) have a uniform coercivity constant. Superlinear convergence [S. & Süli, SINUM, 2014] There exists R > 0, possibly depending on h and p, such that if u h u 0 h H 2 (Ω;T h ) < R, then u k h u h superlinearly.