1. Introduction. We study the numerical analysis of fully nonlinear secondorder elliptic Hamilton Jacobi Bellman (HJB) equations of the form

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1 DISCONTINUOUS GALERIN FINITE ELEMENT APPROXIMATION OF HAMILTON JACOBI BELLMAN EQUATIONS WITH CORDÈS COEFFICIENTS IAIN SMEARS AND ENDRE SÜLI Abstract. We propose an p-version discontinuous Galerkin finite element metod for fully nonlinear second-order elliptic Hamilton Jacobi Bellman equations wit Cordès coefficients. Te metod is proven to be consistent and stable, wit convergence rates tat are optimal wit respect to mes size, and suboptimal in te polynomial degree by only alf an order. Numerical experiments on problems wit strongly anisotropic diffusion coefficients illustrate te accuracy and computational efficiency of te sceme. An existence and uniqueness result for strong solutions of te fully nonlinear problem, and a semismootness result for te nonlinear operator are also provided. ey words. Hamilton Jacobi Bellman equations, p-version discontinuous Galerkin finite element metods, Cordès condition, fully nonlinear equations, semismoot Newton metods AMS subject classifications. 65N30, 65N12, 65N15, 35J15, 35J66, 35D35, 49M15, 47J25 1. Introduction. We study te numerical analysis of fully nonlinear secondorder elliptic Hamilton Jacobi Bellman (HJB) equations of te form sup [L α u f α ] = 0 in Ω, (1.1) α Λ were Ω is a convex domain in R n, n 2, Λ is a compact metric space, and te L α, α Λ, are elliptic operators of te form L α v = n a α ij v xix j + i,j=1 n b α i v xi c α v. (1.2) HJB equations caracterise te value functions of stocastic control problems, wic arise from applications in engineering, pysics, economics, and finance [10]. Te solution of (1.1) leads to te best coices of controls from te set Λ for steering a stocastic process towards optimising te expected value of a functional. We are interested in consistent, stable, convergent and ig-order metods for multidimensional uniformly elliptic HJB equations wit anisotropic diffusions. Discrete state Markov cain approximations to te underlying stocastic dynamics were amongst te earliest computational approaces to tese problems [18]. Alongside te advent of te notion of a viscosity solution to a fully nonlinear second-order equation [6], it became apparent tat tese Markov cain approximations admit equivalent interpretations as monotone finite difference metods (FDM) [5, 10], i.e. tat satisfy a discrete maximum principle. Tese metods feature a general convergence teory due to Barles and Souganidis [1], and are capable of approximating non-smoot viscosity solutions of certain degenerate problems. Various autors ave commented on te necessarily low-order convergence rates of monotone scemes [22], and on te restrictions tat te coice of stencil imposes on te set of problems amenable to discretization by monotone FDM [7, 16]. For an Matematical Institute, University of Oxford, St. Giles, Oxford OX1 3LB, U, iain.smears@mats.ox.ac.uk Matematical Institute, University of Oxford, St. Giles, Oxford OX1 3LB, U, suli@mats.ox.ac.uk 1 i=1

2 2 I. SMEARS AND E. SÜLI analysis of convergence rates, see [9] and te references terein. Motzkin and Wasow [21] found tat for any coice of stencil, tere exists a uniformly elliptic operator wit no consistent and monotone discretisation; yet, for any set of non-degenerate diffusion coefficients wit uniformly bounded ellipticity constants, tere is a stencil providing a monotone and consistent discretisation. ocan studied te minimum size of suc a stencil as a function of te ellipticity constant in [16]. Conversely, Bonnans and Zidani [5] examined te conditions tat determine te set of problems tat can be discretised wit various stencils: tey found tat te number of conditions on te diffusion coefficient grows bot wit te stencil size and te problem dimension. An algoritm was developed in [4] to compute a monotone discretisation of two dimensional problems, wit a consistency error depending on te stencil widt. Wilst te above considerations concern te notion of consistency of FDM, convergent monotone metods for fully nonlinear problems can also employ notions of consistency from finite element metods (FEM); see [15] for te first convergent monotone FEM for viscosity solutions of parabolic HJB equations. Bömer proposed in [2] a non-monotone H 2 -conforming FEM for fully nonlinear PDE wit linearisations in divergence form; yet linearisations of te HJB operator are usually in non-divergence form wit discontinuous coefficients, and cannot be recast into divergence form. Discontinuous Galerkin finite element metods (DGFEM) allow te approximate solution to be discontinuous between elements of te mes, wit te continuity conditions being enforced only weakly troug te discretised problem [8]. Tis facilitates p-refinement, wic varies bot mes size and polynomial degree, tereby allowing for exponential convergence rates, even for problems wit non-smoot solutions [28]. For problems in non-divergence form, a callenge in te design of DGFEM is to obtain stable inter-element communication. Neverteless, te autors found new tecniques in [26] to obtain stable discretisations of certain linear non-divergence form equations wit discontinuous coefficients, and tese tecniques are taken furter in tis work. We consider ere uniformly elliptic HJB equations tat satisfy te Cordès condition: we provide a concise and accessible proof of existence and uniqueness of a strong solution of equation (1.1) associated to a omogeneous Diriclet boundary condition. Ten, we construct a stable, consistent and convergent p-version DGFEM, for wic we prove convergence rates in a discrete H 2 -type norm tat are optimal wit respect to mes size, and suboptimal in te polynomial degree by only alf an order. As opposed to te monotone metods considered above, our metod is consistent regardless of te coice of mes, tereby permitting p-refinement on very general sape-regular sequences of meses. Our experiments below sow te gains in computational efficiency, flexibility, and accuracy over existing monotone metods. Te Cordès condition, defined in 2 below, encompasses a large range of applications. For example, in two spatial dimensions, te condition amounts to simply requiring uniform ellipticity of te diffusion coefficient and coercivity of te lower order terms, see Examples 1 and 2 of 2. Let us now recount ow te motivation for te Cordès condition stems from genuine PDE-teoretic considerations. Tere is a famous solution algoritm for (1.1), due to Bellman and Howard [3, 23], tat may be understood as follows. Given an approximate solution u k, k N, to (1.1), one finds for eac x Ω an Λ α k (x) = argmax α (L α u k f α )(x). A new approximation u k+1 is sougt as te solution of L α k u k+1 = f α k, were f α k : x f αk(x) (x), and were te coefficients of te linear operator L α k are similarly defined; formally, a solution of (1.1) is a fixed point of tis iteration. It as long been known tat tis metod is in fact a Newton metod for a non-differentiable operator [3, 23], and we

3 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 3 contribute to its analysis by sowing te semismootness in function spaces [27] of te HJB operator. Te question of te well-posedness of te linear PDE to be solved at eac iteration is instructive: tese are non-divergence form elliptic equations wit discontinuous coefficients, and it is known tat well-posedness in te strong sense is not guaranteed by uniform ellipticity alone [11, 19, 25], altoug it is recovered under te Cordès condition [19]. Importantly, we sow ere tat well-posedness of strong solutions extends to HJB equations, under te same condition. Inspired by te analysis of te PDE, te stability of our metod is obtained by relating te residual of te equation to terms measuring te lack of H 2 -conformity of te numerical solution. Te structure of tis article is as follows. After defining te problem in 2, we prove its well-posedness in 3. Te p-version DGFEM framework is prepared in 4 and is followed by te definition and consistency analysis of te metod in 5. We establis te stability of te sceme in 6 and we determine its convergence rates in 7. Section 8 analyses a superlinearly convergent semismoot Newton metod used to solve te discrete problem, and 9 presents te results of numerical experiments tat demonstrate te ig accuracy and computational efficiency of te metod. 2. Statement of te problem. Let Ω be a bounded convex polytopal open set in R n, n 2, and let Λ be a compact metric space. It will always be assumed tat Ω and Λ are nonempty. Convexity of Ω implies tat te boundary Ω is Lipscitz; see [12]. Let te real-valued functions a ij = a ji, b i, c and f belong to C(Ω Λ) for all i, j = 1,..., n. For eac α Λ, we consider te function a α ij : x a ij(x, α), x Ω. Te functions b α i, cα and f α are defined in a similar way. Define te matrix-valued functions a α := (a α ij ) and define te vector-valued functions bα := (b α 1,..., b α n), were α Λ. Te bounded linear operators L α : H 2 (Ω) L 2 (Ω) are defined by L α v := n a α ij v xix j + i,j=1 n b α i v xi c α v, v H 2 (Ω), α Λ. (2.1) i=1 Compactness of Λ and continuity of te coefficients a, b, c and f imply tat te fully nonlinear operator F, defined by F : v F [v] := sup [L α v f α ], (2.2) α Λ is well-defined as a mapping from H 2 (Ω) to L 2 (Ω). Te problem considered is to find u H 2 (Ω) H 1 0 (Ω) tat is a strong solution of te HJB equation subject to a omogeneous Diriclet boundary condition F [u] = 0 in Ω, u = 0 on Ω. (2.3) Well-posedness of problem (2.3) is establised in 3 under te following ypoteses. It is assumed tat tere exist positive constants ν ν suc tat n ν ξ 2 a α ij(x) ξ i ξ j ν ξ 2 ξ R n, x Ω, α Λ. (2.4) i,j=1 Te function c α is supposed to be non-negative on Ω, for eac α Λ. We assume te Cordès condition: tere exist λ > 0 and ε (0, 1) suc tat, for eac α Λ, a α 2 + b α 2 /2λ + (c α /λ) 2 (Tr a α + c α /λ) 2 1 n + ε in Ω, (2.5)

4 4 I. SMEARS AND E. SÜLI were represents te Euclidian norm for vectors and te Frobenius norm for matrices. In te special case b α 0 and c α 0 for eac α Λ, we set λ = 0 and te Cordès condition (2.5) is replaced by: tere exists ε (0, 1) suc tat, for eac α Λ, a α 2 (Tr a α ) 2 1 n 1 + ε in Ω. (2.6) Conditions (2.5) and (2.6) are related troug te observation tat te term c α /λ may be viewed as te (n + 1, n + 1) entry of an (n + 1) (n + 1) matrix wit principal n n sub-matrix a α, wic explains te difference in te rigt and sides of te inequalities in (2.5) and (2.6). Te parameter λ serves to make te Cordès condition invariant under rescaling te coordinates. It will be seen below tat it is often easy to coose an appropriate value for λ. Example 1. We sow ow te Cordès condition (2.5) arises in practice in an example from stocastic control problems [10]. We consider a problem were te controls permit te coice of orientation and angle between two Wiener diffusions. Let Ω be a domain in R 2 and let Λ = [0, π/3] SO(2), were SO(2) is te set of 2 2 rotation matrices. Te diffusions act along te directions σ1 α and σ2 α, were ( ) σ α := (σ1 α σ2 α ) := R T 1 sin θ, α = (θ, R) Λ. (2.7) 0 cos θ In stocastic control problems, we ave a α := σ α (σ α ) T /2 and usually c α c 0 > 0 is a fixed constant [10]. Ten, Tr a α = 1 and a α 2 = (1 + sin 2 θ)/2 7/8; so condition (2.6) olds wit ε = 1/7. Momentarily assuming tat b α 0, by coosing te value λ = 8 7 c 0 tat minimises te left-and side in (2.5), we find tat condition (2.5) also olds wit ε = 1/7. For non-zero b α, te Cordès condition olds for ε < 1/7 wenever b α 2 /c 0 is sufficiently small; tis amounts to a standard coercivity assumption. Example 1 is considered furter in te numerical experiments of 9.1. Observe tat for any coice of Cartesian coordinates on R 2, for θ = π/3 tere is an R SO(2) suc tat a α is not diagonally dominant. Terefore, te classical monotone usner Dupuis FDM is not applicable ere [5]. Example 2. For problems in two dimensions, i.e. n = 2, te uniform ellipticity condition (2.4) is sufficient for te Cordès condition (2.6). Indeed, for eac α Λ, we ave ν 2 det a α, and a α 11 + a α 22 2 ν, so, for ε = ν 2 /(2 ν 2 ν 2 ), we get (a α 11) 2 + 2(a α 12) 2 + (a α 22) 2 (a α 11 + aα 22 )2 1 2ν 2 ν2 (a α aα 22 )2 2 ν 2 = ε. (2.8) Te above examples demonstrate tat te results of tis paper are relevant to a very broad class of problems, including some tat require large stencils for monotone FDM; significant furter evidence for tis observation is found in 9. Define te strictly positive function γ : Ω Λ R >0 by γ(x, α) := Tr a α (x) + c α (x)/λ a α (x) 2 + b α (x) 2 /2λ + (c α (x)/λ) 2. (2.9) In te special case b α 0 and c α 0 for all α Λ, we take λ = 0 and define γ(x, α) := Tr aα (x) a α (x) 2. (2.10)

5 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 5 As above, for eac α Λ, we define γ α : x γ(x, α), x Ω. It follows from te continuity assumptions on te coefficients and from te uniform ellipticity condition (2.4) tat γ C(Ω Λ). Furtermore, non-negativity of c, continuity of te coefficients and (2.4) imply tat tere is a positive constant γ 0 > 0 suc tat γ γ 0 on Ω Λ. Define te operator F γ : H 2 (Ω) L 2 (Ω) by F γ [v] := sup [γ α (L α v f α )]. (2.11) α Λ It will be seen below tat te HJB equation (2.3) is in fact equivalent to te problem F γ [u] = 0 in Ω, u = 0 on Ω. For λ as above, let te operator L λ be defined by L λ v := v λv, v H 2 (Ω). (2.12) Te following inequality generalises results in [19, 26] tat were used to analyse linear PDE satisfying te Cordès condition. It is key to our analysis of HJB equations. Lemma 1. Let Ω be a bounded open subset of R n and suppose tat (2.4) olds, and suppose tat eiter (2.5) olds wit λ > 0, or tat (2.6) olds wit b α 0, c α 0 for all α, and λ = 0. Ten, for any open set U Ω and u, v H 2 (U), w := u v, te following inequality olds a.e. in U: F γ [u] F γ [v] L λ (u v) 1 ε D 2 w 2 + 2λ w 2 + λ 2 w 2. (2.13) Proof. It will be clear ow to adapt te following arguments to treat te simpler situation were b α 0, c α 0 and λ = 0. So, we consider te case were (2.5) olds wit λ > 0. First, set w := u v. Note tat we ave te identity F γ [u] L λ u = sup α Λ [γ α L α u L λ u γ α f α ]. Also, for bounded sets of real numbers, {x α } α and {y α } α, we ave sup α x α sup α y α sup α x α y α. Terefore, F γ [u] F γ [v] L λ w sup γ α L α w L λ w α Λ sup γ α a α I n D 2 w + γ α b α w + λ c α γ α w, α Λ were I n is te n n identity matrix. Te Caucy Scwarz inequality wit a parameter gives ( ) D2 F γ [u] F γ [v] L λ w C α w 2 + 2λ w 2 + λ 2 w 2, were, for eac α Λ, sup α Λ C α := γ α a α I n 2 + γ α 2 bα 2 2λ + λ cα γ α 2 λ 2. (2.14) Expanding te square terms in (2.14) gives ( ) C α = n + 1 2γ α Tr a α + cα + γ α ( a 2 α 2 + bα 2 λ 2λ + cα 2 ) λ 2. Te definition of γ in (2.9) and te Cordès condition (2.5) imply tat C α 1 ε on U for every α Λ, tus completing te proof of (2.13). In te following analysis, we sall write a b for a, b R to signify tat tere exists a constant C suc tat a C b, were C is independent of te mes size and polynomial degrees used to define te finite element spaces below, but oterwise possibly dependent on oter fixed quantities, suc as te constants in (2.4) and (2.5) or te sape-regularity parameters of te mes, for example.

6 6 I. SMEARS AND E. SÜLI 3. Analysis of te PDE. For λ 0 as above, define te semi-norm H 2 (Ω),λ on H 2 (Ω) by u 2 H 2 (Ω),λ := u 2 H 2 (Ω) + 2λ u 2 H 1 (Ω) + λ2 u 2 L 2 (Ω). (3.1) If λ > 0, ten tis defines a norm on H 2 (Ω). Te following result follows from te Miranda Talenti estimate; see [12, 19, 26]. Recall tat L λ u = u λu. Teorem 2. Let Ω be a bounded convex open subset of R n. Ten, for any λ 0 and any u H 2 (Ω) H 1 0 (Ω), te following inequalities old: u H2 (Ω),λ L λ u L2 (Ω), u H2 (Ω) C L λ u L2 (Ω), (3.2a) (3.2b) were C is a positive constant depending only on n and diam Ω. Proof. In [26, Teorem 2], it is sown tat on bounded convex domains, we ave te Miranda Talenti estimate u H 2 (Ω) u L 2 (Ω) for any u H 2 (Ω) H0 1 (Ω). Te identity Ω u u dx = Ω u 2 dx, based on integration by parts, gives L λ u 2 L 2 (Ω) = ( u λu) 2 dx = u 2 L 2 (Ω) + 2λ u 2 H 1 (Ω) + λ2 u 2 L 2 (Ω). (3.3) Ω Te Miranda Talenti estimate and (3.3) give (3.2a). Te bound (3.2b) follows from (3.3) and te estimate u H 2 (Ω) C(n, diam Ω) u L 2 (Ω) sown in [26, Tm. 2]. Teorem 3. Let Ω be a bounded convex open subset of R n, and let Λ be a compact metric space. Let te data a, b, c, f be continuous on Ω Λ and satisfy (2.4) and eiter (2.5) wit λ > 0 or (2.6) wit c 0, b 0 and λ = 0. Ten, tere exists a unique strong solution u H 2 (Ω) H0 1 (Ω) of te HJB equation (2.3). Moreover, u is also te unique solution of F γ [u] = 0 in Ω, u = 0 on Ω. Proof. First, set H := H 2 (Ω) H0 1 (Ω); ten H is a separable Hilbert space. Te proof consists of sowing solvability of te equation F γ [u] = 0 in H by te metod of Browder and Minty, and establising its equivalence wit te HJB equation (2.3). Let te operator A: H H be defined by A(u), v = F γ [u] L λ v dx, u, v H. (3.4) Ω We claim tat A is Lipscitz continuous and strongly monotone. Indeed, let u, v H and set w := u v. Ten, by adding and subtracting L λ w, we get A(u) A(v), u v = L λ w 2 L 2 (Ω) + (F γ [u] F γ [v] L λ w) L λ w dx. (3.5) Lemma 1 and te Caucy Scwarz inequality sow tat Ω A(u) A(v), u v L λ w 2 L 2 (Ω) 1 ε w H 2 (Ω),λ L λ w L 2 (Ω). (3.6) We ten use (3.2a) to obtain A(u) A(v), u v ( 1 1 ε ) L λ w 2 L 2 (Ω), so u v 2 H 2 (Ω) A(u) A(v), u v as a result of (3.2b), tus sowing tat A is strongly monotone. Compactness of Λ and continuity of te data imply tat A is Lipscitz continuous: to see tis, let u, v, z H. Ten, we find tat A(u) A(v), z F γ [u] F γ [v] L2 (Ω) L λ z L2 (Ω) C u v H2 (Ω) z H2 (Ω),

7 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 7 were te constant C depends only on λ and on te supremum norms of a ij, b i, c and γ over Ω Λ, for i, j = 1,..., n. Lipscitz continuity and strong monotonicity imply tat A is bounded, continuous, coercive and strongly monotone, so te Browder Minty teorem [24] sows tat tere exists a unique u H suc tat A(u) = 0. For every g L 2 (Ω), tere is a v H suc tat L λ v = g. Terefore A(u) = 0 implies Ω F γ[u] g dx = 0 for all g L 2 (Ω), tus sowing tat F γ [u] = 0 a.e. in Ω. We claim tat F γ [u] = 0 if and only if u solves (2.3). Since γ α is positive, γ α (L α u f α ) 0 for all α Λ is equivalent to L α u f α 0 for all α Λ; i.e F [u] 0 if and only if F γ [u] 0. Compactness of Λ and continuity of a, b, c, f and γ imply tat at a.e. point of Ω, te suprema in te definitions of F [u] and F γ [u] are attained by an element of Λ, tereby giving F [u] 0 if and only if F γ [u] 0. Terefore, existence and uniqueness of te solution u of F γ [u] = 0 in Ω is equivalent to existence and uniqueness of a solution of (2.3). 4. Finite element spaces. Let {T } be a sequence of sape-regular meses on Ω, consisting of simplices or parallelepipeds. For eac element T, let := diam. It is assumed tat = max T for eac mes T. Let F i denote te set of interior faces of te mes T, i.e. F F i if and only F =, for some elements and T, and F is te closure of a non-empty smoot connected ypersurface tat is open relative to. Since eac element as piecewise flat boundary, it follows tat any interior face is flat. Let F b denote te set of boundary faces of te mes, tat is, F F b if and only if F is te closure of a non-empty smoot connected ypersurface tat is a relatively open subset of Ω, for some T, and F is maximal in te sense tat tere is no smoot connected ypersurface containing F tat is also a relatively open subset of Ω. Boundary faces are terefore also flat. Te set of all faces is F i,b := F i F b. Mes conditions. We sall make te following assumptions on te meses. Te meses are allowed to be irregular, i.e. tere may be anging nodes. We assume tat tere is a uniform upper bound on te number of faces composing te boundary of any given element; in oter words, tere is a c F > 0, independent of, suc tat { } max card F F i,b T : F c F T, > 0. (4.1) It is also assumed tat any two elements saring a face ave commensurate diameters, i.e. tere is a c T 1, independent of, suc tat max(, ) c T min(, ), (4.2) for any and in T tat sare a face. For eac, let p = (p : T ) be a vector of positive integers. In order to let p appear in te denominator of various expressions, we sall assume tat p 1 for all T. We make te assumption tat p as local bounded variation [14]: tere is a c P 1, independent of, suc tat max(p, p ) c P min(p, p ), (4.3) for any and in T tat sare a face. Function spaces. For eac T, let P p () be te space of all polynomials wit eiter total or partial degree less tan or equal to p. Te discontinuous Galerkin finite element space V,p is defined by V,p := { v L 2 (Ω), v P p (), T }. (4.4)

8 8 I. SMEARS AND E. SÜLI Let s = (s : T ) denote a vector of non-negative real numbers, and let r [1, ]. Te broken Sobolev space W s,r (Ω; T ) is defined by W s,r (Ω; T ) := {v L r (Ω), v W s,r (), T }. (4.5) For sortand, define H s (Ω; T ) := W s,2 (Ω; T ), and set W s,r (Ω; T ) := W s,r (Ω; T ), were s = s, s 0, for all T. For v W 1,r (Ω; T ), let v L r (Ω; R n ) denote te broken gradient of v, i.e. ( v) = (v ) for all T. Higer broken derivatives are defined in a similar way. Define a norm on W s,r (Ω; T ) by v r W s,r (Ω;T ) := wit te usual modification wen r =. T v r W s,r (), (4.6) Jump, average, and tangential operators. For eac face F, let n F R n denote a fixed coice of a unit normal vector to F. Since eac face F is flat, te normal n F is constant. For an element T and a face F, let τ F : H s () H s 1/2 (F ), s > 1/2, denote te trace operator from to F. Te trace operator τ F is extended componentwise to vector-valued functions. Define te jump operator over F by φ := {τ F ( φ ext ) τf ( φ int ) τ F ( φ ext ) if F F i, if F F b, (4.7) and define { }, te average operator over F, by { 1 ( ) ( )) {φ} := 2( τf φ ext + τf φ int ( ) τ F φ ext if F F i, if F F b, (4.8) were φ is a sufficiently regular scalar or vector-valued function, and ext and int are te elements to wic F is a face, i.e. F = ext int. Here, te labelling is cosen so tat n F is outward pointing for ext and inward pointing for int. Using tis notation, te jump and average of scalar-valued functions, resp. vectorvalued, are scalar-valued, resp. vector-valued. For two matrices A, B R n n, we set A : B = n i,j=1 A ijb ij. For an element, we define te inner product, by u v dx if u, v L2 (), u, v := u v dx if u, v L2 (; R n ), u : v dx if u, v L2 (; R n n ). (4.9) Te abuse of notation will be resolved by te arguments of te inner product. Te inner products, and, F, F F i,b, are defined in a similar way. For a face F, let T and div T denote respectively te tangential gradient and tangential divergence operators on F ; see [12, 26]. 5. Numerical sceme. Te definition of te numerical sceme requires te following bilinear and nonlinear forms. First, for λ 0 as above, te symmetric

9 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 9 bilinear form B, : V,p V,p R is defined by B, (u, v ) := [ D 2 u, D 2 v + 2λ u, v + λ 2 ] u, v T + [ ] divt T {u }, v n F F + div T T {v }, u n F F λ F F i λ F F i [ T { u n F }, T v F + T { v n F }, T u F ] [ { u n F }, v F + { v n F }, u F ] [ {u }, v n F F + {v }, u n F F ], were u and v will denote functions in V,p trougout tis work. Ten, for positive face-dependent quantities µ F and η F to be specified later, te jump stabilisation bilinear form J : V,p V,p R is defined by J (u, v ) := [ ] µf T u, T v F + η F u, v F + F F i µ F u n F, v n F F. (5.1) For eac θ [0, 1], define te bilinear form B,θ : V,p V,p R by B,θ (u, v ) := θb, (u, v ) + (1 θ) T L λ u, L λ v + J (u, v ). (5.2) Te nonlinear form A : V,p V,p R is defined by A (u ; v ) := F γ [u ], L λ v + B, 1 (u, v 2 ) L λ u, L λ v. (5.3) T T Te form A is linear in its second argument but nonlinear in its first argument. Te numerical sceme for approximating te solution of (2.3) is to find u V,p suc tat A (u ; v ) = 0 v V,p. (5.4) Te coice of nonlinear form in (5.3) is made to mirror te addition substraction step of (3.5) in te proof of Teorem 3. It will be seen below tat te last two terms of (5.3) vanis wen te first argument of te form is smoot, and it is in tis sense tat tis metod relates te residual of te numerical solution to its lack of smootness Consistency. Te next result sows tat te bilinear form B,θ is obtained from a discrete analogue of te identities tat underpin Teorem 2. Lemma 4. Let Ω be a bounded Lipscitz polytopal domain and let T be a simplicial or parallelepipedal mes on Ω. Let te function w belong eiter to C 1 (Ω) H s (Ω; T ) or to H s (Ω), s > 5/2, and suppose tat w F = 0 in te trace sense for every boundary face F F b. Ten, for every v V,p, we ave te identities B, (w, v ) = L λ w, L λ v and J (w, v ) = 0. (5.5) T

10 10 I. SMEARS AND E. SÜLI Proof. Te second part of (5.5) is obvious. We also note tat all terms in B, (w, v ) tat involve jumps of w or of its first derivates vanis. For te case λ = 0, te stated result reduces to [26, Lemma 5], wic treats te consistency of te second order terms, namely B, (w, v ) = w, v for all v V,p. So, for λ > 0, te identities of (5.5) are deduced from te previous result and from te identities λ w, v = λ w, v λ { w n F }, v F, (5.6) T T λ T w, v = λ T w, v λ {w}, v n F F, (5.7) F F i for all v V,p, were we ave used te fact tat w F = 0 for all F F b in (5.7). If te function w satisfies te ypoteses of Lemma 4, ten (5.5) implies tat B,θ (w, v ) = L λ w, L λ v v V,p, θ [0, 1]. (5.8) T Te following consistency result for te sceme (5.4) follows immediately from Teorem 3, (5.8) and from te definition of A in (5.3). Corollary 5. Let Ω be a bounded convex polytopal domain, let T be a simplicial or parallelepipedal mes on Ω and let u H 2 (Ω) H 1 0 (Ω) be te unique solution of problem (2.3). If u H s (Ω) or if u C 1 (Ω) H s (Ω; T ), s > 5/2, ten u satisfies A (u ; v ) = 0 for every v V,p. 6. Stability. For λ 0 as above, define te seminorms H 2 (),λ, T, and H 2 (Ω;T ),λ on H 2 (Ω; T ) by v 2 H 2 (),λ := D2 v 2 L 2 () + 2λ v 2 L 2 () + λ2 v 2 L 2 (), (6.1) v 2 H 2 (Ω;T ),λ := T v 2 H 2 (),λ. (6.2) For a positive constant c, independent of and to be specified later, and θ [0, 1], define te functionals DG(θ) : V,p R 0 by v 2 DG(θ) := [ ] θ v 2 H 2 (),λ + (1 θ) L λv 2 L 2 () + c J (v, v ). (6.3) T For eac θ [0, 1], DG(θ) is a norm on V,p. Indeed, omogeneity and te triangle inequality are clear. If v DG(θ) = 0, ten v H 2 (Ω) H 1 0 (Ω) since v = 0 for all F F i, and v = 0 for all F F i,b v H 2 (),λ = 0 ), so v 0 as a result of (3.2b). For eac face F F i,b F := { min(, ), if F F i,, if F F b, p F :=. Moreover, L λv 0 (if θ = 1, use, define { max(p, p ), if F F i, p, if F F b, (6.4) were and are suc tat F = if F F i or F Ω if F F b. Te assumptions on te mes and te polynomial degrees, in particular (4.2) and (4.3), sow tat if F is a face of, ten c T F and p F c P p. (6.5)

11 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 11 Lemma 6. Let Ω be a bounded convex polytopal domain and let {T } be a saperegular sequence of simplicial or parallelepipedal meses satisfying (4.1). Ten, for eac constant κ > 1, tere exist positive constants c stab and c, independent of, p and θ, suc tat v 2 DG(θ) κb,θ(v, v ) v V,p, θ [0, 1], (6.6) wenever, for any fixed constant σ 1, µ F = σc stab p 2 F F and η F > σλ c stab p 2 F F. (6.7) Te strict inequality in te second part of (6.7) serves to cover te case λ = 0. Proof. For v V,p, we ave B,θ (v, v ) = θ v 2 H 2 (Ω;T ),λ + (1 θ) L λ v 2 L 2 () + J (v, v ) + θ 4 i=1 I i, T were I 1 := 2 div T T {v }, v n F F, I 3 := 2λ {v }, v n F F, I 2 := 2 F F i F F i T { v n F }, T v F, I 4 := 2λ { v n F }, v F. In [26, Lemma 7], it is sown tat tere is a constant C(n) depending only on n, suc tat for any δ > 0, I 1 δc(n)c Tr c F I 2 δc(n)c Tr c F T D 2 v 2 L 2 () + T D 2 v 2 L 2 () + F F i p 2 F δ F v n F 2 L 2 (F ), (6.8) p 2 F δ F T v 2 L 2 (F ), (6.9) were C Tr is te combined constant of te trace and inverse inequalities, and c F is given by (4.1). Te inverse and trace inequalities also sow tat I 3 2λ F F i δ F p 2 F {v } 2 L 2 (F ) F F i δc(n)c Tr c F λ 2 v 2 L 2 () + T Similarly, it is found tat I 4 δc(n)c Tr c F p 2 F δ F v n F 2 L 2 (F ) F F i T 2λ v 2 L 2 () + p 2 F δ F v n F 2 L 2 (F ). (6.10) λ p 2 F 2δ F v 2 L 2 (F ). (6.11)

12 12 I. SMEARS AND E. SÜLI We may take C(n) to be te same constant in eac of te above estimates. So, B,θ (v, v ) θ(1 δc(n)c Tr c F ) v 2 H 2 (Ω;T ),λ + (1 θ) L λ v 2 L 2 () T + ( ) µ F 2θ p2 F v n F 2 L δ 2 (F ) + ( µ F θ p2 F F δ F F F i + ) T v 2 L 2 (F ) ( ) η F λθ p2 F v 2 L 2δ 2 (F ). F For κ > 1, tere is a δ > 0 suc tat 1 δc(n)c Tr c F > κ 1. Set c stab = 4/δ and c = κ/2 so tat (6.6) olds wen µ F and η F are cosen in accordance wit (6.7). Teorem 7. Let Ω be a bounded convex polytopal domain and let {T } be a sape-regular sequence of simplicial or parallelepipedal meses satisfying (4.1). Let Λ be a compact metric space and let te data satisfy (2.4) and eiter (2.5) or (2.6) wit b 0, c 0, λ = 0. Let c stab, c, η F and µ F be cosen so tat Lemma 6 olds wit κ < (1 ε) 1/2. Ten, for every u, v V,p, we ave u v 2 DG(1) C( A (u ; u v ) A (v ; u v ) ), (6.12) were te constant C := 2κ/ ( 1 κ 2 (1 ε) ). Moreover, tere exists a constant C independent of and p suc tat, for any u, v and z in V,p, A (u ; z ) A (v ; z ) C u v DG(1) z DG(1). (6.13) Terefore, tere exists a unique solution u V,p to te numerical sceme (5.4). We ave te bound u DG(1) 2κ n + 1 γ C(Ω Λ) 1 κ 2 (1 ε) sup f α L 2 (Ω). (6.14) α Λ Proof. Let u and v belong to V,p and set w := u w. Ten, we ave A (u ; w ) A (v ; w ) = B, 1 2 (w, w ) + Note tat Lemma 1 gives T F γ [u ] F γ [v ] L λ w, L λ w. F γ [u ] F γ [v ] L λ w, L λ w 1 ε w H2 (),λ L λ w L2 () T T Tis estimate and Lemma 6 sow tat κ(1 ε) w 2 H 2 2 (Ω;T ),λ + κ 1 2 T L λ w 2 L 2 (). A (u ; w ) A (v ; w ) 1 κ2 (1 ε) w 2 H 2κ 2 (Ω;T ),λ + c κ J (w, w ) 1 κ2 (1 ε) w 2 DG(1) 2κ.

13 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 13 Since κ 2 (1 ε) < 1, we obtain (6.12). Now let z V,p. Ten, using linearity of B,θ and inverse inequalities, we find tat tere exists a constant C depending on te constants appearing in te proof of Lemma 6, but not on or p, suc tat B, 1 (u v 2, z ) C u v DG(1) z DG(1). Using Lemma 1 and te above estimates, we deduce tat tere is a constant C depending only on n and ε suc tat F γ [u ] F γ [v ] L λ (u v ), L λ z C u v DG(1) z DG(1). T It ten follows tat A is Lipscitz continuous, as stated in (6.13). Te Browder Minty teorem [24] wit (6.12) and (6.13) imply tat tere exists a unique u V,p suc tat A (u ; v ) = 0 for all v V,p. By taking v = 0 in (6.12), we find tat u 2 DG(1) C A (0 ; u ) C sup [ γ α f α ], L λ u α Λ T C γ C(Ω Λ) sup f α L (Ω) 2 n + 1 u DG(1), α Λ were C = 2κ/ ( 1 κ 2 (1 ε) ), tus sowing te bound (6.14). 7. Error analysis. Te good stability properties of te proposed metod make it possible to obtain te following a priori error bound. Teorem 8. Let Ω be a bounded convex polytopal domain, let te sape-regular sequence of simplicial or parallelepipedal meses {T } satisfy (4.1) and (4.2), wit p satisfying (4.3) for eac. Let Λ be a compact metric space and let te data satisfy (2.4), and eiter (2.5) or (2.6) wen b 0, c 0 and λ = 0. Let u H 2 (Ω) H0 1 (Ω) be te unique solution of (2.3). Assume eiter tat u H s (Ω), s > 5/2, or tat u C 1 (Ω), and assume in addition tat u H s (Ω; T ), wit s 3 for all T. Let c stab, c, µ F and η F be cosen as in Teorem 7 and coose η F p 4 F / 3 F for all F F i,b. Ten, tere exists a positive constant C, independent of, p and u, but depending on max s, suc tat u u 2 DG(1) C 2t 4 p 2s 5 T u 2 H s (), (7.1) were t = min(p + 1, s ) for eac T. Note tat for te special case of quasi-uniform meses and uniform polynomial degrees, if u H s (Ω) wit s 3, te a-priori estimate (7.1) simplifies to u u DG(1) C min(p+1,s) 2 p s 2.5 u H s (Ω). Terefore, te convergence rates are optimal wit respect to te mes size and suboptimal in te polynomial degree only by alf an order. Proof. Since te sequence of meses is sape-regular, tere is a z V,p and a constant C, independent of u, and p, but dependent on max s, suc tat u z H q () C t q p s q D β (u z ) L 2 ( ) C t q 1/2 p s q 1/2 u H s (), 0 q s, (7.2) u H s k (), β = q; 0 q s 1. (7.3)

14 14 I. SMEARS AND E. SÜLI Note tat te ypotesis s 3 allows te coice of q = 2 in (7.3). Set ψ := u z and ξ := u z. By Corollary 5, we ave A (u ; v ) = 0 for all v V,p. Strong monotonicity of A on V,p, as sown in Teorem 7, yields ψ 2 DG(1) A (u ; ψ ) A (z ; ψ ) = A (u ; ψ ) A (z ; ψ ). (7.4) By applying te Caucy Scwarz inequality to te terms appearing on te rigt-and side of (7.4) and applying inverse inequalities to ψ V,p, we eventually obtain A (u ; ψ ) A (z ; ψ ) were te quantities E i are defined by 10 i=1 E i ψ DG(1), (7.5) E 1 := ξ 2 H 2 (),λ, E 2 := L λ ξ 2 L 2 (), T T E 3 := F γ [u] F γ [z ] 2 L 2 (), E 4 := µ 1 F div T T {ξ } 2 L 2 (F ), T F F i E 5 := µ F ξ n F 2 L 2 (F ), E 6 := µ 1 F T { ξ n F } 2 L 2 (F ), E 7 := E 9 := F F i µ F T ξ 2 L 2 (F ), E 8 := (λµ F + η F ) ξ 2 L 2 (F ), Te estimate (7.2) sows tat E 1 + E 2 E 10 := 2t 4 p 2s 4 T F F i λ 2 η 1 F { ξ n F } 2 L 2 (F ), λ 2 µ 1 F {ξ } 2 L 2 (F ). u 2 H s (). (7.6) By compactness of Λ, continuity of te data and (2.4), F γ is Lipscitz continuous, so E 3 ξ 2 H 2 () 2t 4 p 2s 4 T u 2 H s (). (7.7) We use (4.1), (4.2), (4.3), (6.7) and (7.3) to obtain E 4 + E 6 E 5 + E 7 p 2 T 2t 5 p 2s 5 p2 p 2s 3 T Similarly, we use (6.7) to get E 8 p 2 T u 2 H s () = 2t 3 u 2 H s () = 2t 3 p 2s 3 u 2 H s () = 2t 4 p 2s 3 T 2t 4 p 2s 5 T 2t 2 p 2s 1 T u 2 H s (), (7.8) u 2 H s (). (7.9) u 2 H s (). (7.10)

15 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 15 By ypotesis, η F p 4 F / 3 F, so (4.2) and (4.3) imply tat E 9 Finally, (7.3) yields p4 3 T E 10 p 2 T 2t 1 p 2s 1 2t 1 p 2s 1 u 2 H s () = u 2 H s () = 2t 4 p 2s 5 T T 2t p 2s +1 u 2 H s (). (7.11) u 2 H s (). (7.12) Te a-priori bound (7.1) is obtained from u u DG(1) ψ DG(1) + ξ DG(1) and te above estimates. 8. Semismoot Newton metod. We turn to te analysis of an algoritm for solving te discrete problem (5.4), wic can be interpreted as a Newton metod for non-smoot operator equations [23]. After sowing tat te algoritm is well-posed, we obtain and ten use a semismootness result for HJB operators in function spaces to establis its superlinear convergence. Te semismootness of finite dimensional HJB operators in a different form was studied in [3]. For 1 r, a function u W 2,r (Ω; T ) defines a vector-valued function u L r (Ω; R m ) troug u = ( u, u, D 2u), were u and D 2 u denote te broken gradient and broken Hessian of u, see 4. For a vector u = (z, p, M) R m, define te function F γ : Ω R m R by F γ (x, u) := sup [γ α (a α : M + b α p c α z f α ) x ]. (8.1) α Λ For eac (x, u) Ω R m, we define Λ(x, u) as te set of all α Λ suc tat te supremum in (8.1) is attained. Tis defines a set-valued map (x, u) Λ(x, u). Lemma 9. Let Ω be a bounded open subset of R n, let Λ be a compact metric space, let te data a, b, c and f be continuous on Ω Λ and suppose tat (2.4) olds. Ten, for eac (x, u) Ω R m, Λ(x, u) is a non-empty closed subset of Λ. Te set-valued map (x, u) Λ(x, u) is upper semicontinuous; tat is, for every (x, u) Ω R m, and any open neigbourood U of Λ(x, u), tere exists an open neigbourood V of (x, u) suc tat Λ(y, v) U for every (y, v) V. We remark tat te uniform ellipticity condition (2.4) is only used in Lemma 9 to guarantee tat γ C(Ω Λ). Proof. For every (x, u) Ω R m, u = (z, p, M), compactness of Λ and continuity of a, b, c, f and γ imply te existence of a maximiser in (2.2); so Λ(x, u) is nonempty. Te set Λ(x, u) is closed: if α is in te closure of Λ(x, u), say α j α, wit α j Λ(x, u) for eac j N, ten continuity of te data implies tat γ α (a α : M + b α p c α z f α ) x = lim j γαj (a αj : M + b αj p c αj z f αj ) x. (8.2) Since α j Λ(x, u) for eac j N, te rigt and side of (8.2) equals F (x, u), tus giving α Λ(x, u) and sowing tat Λ(x, u) is closed. We prove upper semicontinuity of (x, u) Λ(x, u) by contradiction. Suppose tat tere exists an (x, u) Ω R m, a neigbourood U of Λ(x, u), and a sequence {(x j, u j )} j=1, u j = (z j, p j, M j ), converging to (x, u), togeter wit α j Λ(x j, u j )\U for all j N. Because Λ is compact and Λ \ U is closed, tere exists a subsequence,

16 16 I. SMEARS AND E. SÜLI to wic we pass witout cange of notation, suc tat α j α Λ \ U. On te one and, Λ(x, u) is non-empty so tere is β Λ(x, u). Ten, by definition of F, γ α (a α : M + b α p c α z f α ) x F (x, u). (8.3) On te oter and, α j Λ(x j, u j ) implies tat we ave, for eac j N, γ αj (a αj : M j + b αj p j c αj z j f αj ) xj γ β ( a β : M j + b β p j c β z j f β) xj. Taking te limit j in te above inequality sows tat equality olds in (8.3) because β Λ(x, u). Hence, α Λ(x, u); owever, U is an open neigbourood of Λ(x, u) and α Λ \ U, so we ave a contradiction. Te following selection teorem, due to uratowski and Ryll-Nardzewski [17], is required for te analysis of te algoritm for solving (5.4). Teorem 10. Let Ω R n be a bounded open set, let Λ be a compact metric space, and let (x, u) Λ(x, u) be an upper semicontinuous set-valued function from Ω R m to te subsets of Λ, suc tat Λ(x, u) is non-empty and closed for every (x, u) Ω R m. Ten, for any finite a.e. Lebesgue measurable function u: Ω R m, tere exists a Lebesgue measurable selection α: Ω Λ suc tat α(x) Λ ( x, u(x) ) for a.e. x Ω. For u W 2,r (Ω; T ), let Λ[u] be te set of all Lebesgue measurable functions α: Ω Λ suc tat α(x) Λ(x, u(x)) for a.e. x Ω, were u = ( u, u, D 2u). Lemma 9 and Teorem 10 sow tat Λ[u] is non-empty for eac u W 2,r (Ω; T ). For measurable α: Ω Λ, we define γ α : Ω R >0 troug γ α (x) = γ(x, α(x)), were γ : Ω Λ R >0 was defined by (2.9) or (2.10). It follows from uniform continuity of γ over Ω Λ tat γ α L (Ω), wit γ α L (Ω) γ C(Ω Λ). Te functions a α, b α, c α and f α and te operator L α are defined in a similar way and are likewise bounded. It is clear tat if α Λ[u], ten F γ [u] = γ α (L α u f α ) a.e. in Ω Algoritm. We now present te definition of te semismoot Newton metod for solving (5.4) and state te main result concerning its convergence rate. Coose u 0 V,p. Given u k V,p, k N, coose α k Λ[u k ]. Ten, obtain V,p satisfying u k+1 A k (u k+1, v ) = γ αk f αk, L λ v v V,p, (8.4) T were te bilinear form A k : V,p V,p R is defined by A k (w, v ) := T (γ αk L αk w, L λ v + B, 1 2 (w, v ) T L λ w, L λ v. Te fact tat α k : Ω Λ is measurable ensures tat A k is well-defined. As in te proof of Teorem 7, it is found tat te bilinear forms A k, k N, are coercive on V,p. In fact, for eac k N, we ave v 2 DG(1) 2κ 1 κ 2 (1 ε) Ak (v, v ) v V,p. (8.5) Terefore, te sequence of iterates {u k } k=1 is well-defined by (8.4) and remains bounded in V,p. Te main result of tis section is te following.

17 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 17 Teorem 11. Under te ypoteses of Teorem 7, tere exists a constant R > 0, possibly depending on and p, suc tat if u u 0 DG(1) < R, were u solves (5.4), ten te sequence {u k } k=1 converges to u wit a superlinear convergence rate. Te proof of tis teorem will be given in te next section. Despite te possible dependence of R on and p in te above teorem, it is seen from te numerical experiments in 9, in particular in Figure 2 below, tat in practice, te convergence rates of te algoritm depend only weakly on te discretisation parameters Semismootness of HJB operators. Te proof of Teorem 11 rests upon te notion of semismootness, as defined in [27]. We recall te definition below. For sets X and Y, we write G: X Y if G is a set-valued map tat maps X into te subsets of Y. Definition 12. Let X and Y be Banac spaces, and let F : U X Y be a map defined on a non-empty open set U of X. Let DF : U L(X, Y ) be a set-valued map wit non-empty images. For x U, te map F is called DF -semismoot at x if lim e X 0 1 e X sup F [x + e] F [x] De Y = 0. (8.6) D DF [x+e] Te map F is called DF -semismoot on U if F is DF -semismoot at x, for every x U. Te set-valued map DF is ten called a generalised differential of F on U. For 1 q < r, te map DF γ : W 2,r (Ω; T ) L ( W 2,r (Ω; T ), L q (Ω) ) is defined by DF γ [u] := { γ α L α := γ α ( a α : D 2 + b α c α) : α Λ[u] }. (8.7) Teorem 13. Let Ω be a bounded open subset of R n, let Λ be a compact metric space, let te data a, b, c and f be continuous on Ω Λ and suppose tat (2.4) olds. Let T be a mes on Ω. Ten, for any 1 q < r, te operator F γ : W 2,r (Ω; T ) L q (Ω) defined by F γ [u] = F γ (, u, u, D 2 u) is DF γ-semismoot on W 2,r (Ω; T ). Proof. Supposing te claim to be false, tere exist a function u W 2,r (Ω; T ), a constant ρ > 0, and a sequence {e j } j=0 W 2,r (Ω; T ), wit e j W 2,r (Ω;T ) 0, and α j Λ[u + e j ] suc tat, for eac j N, 1 F γ [u + e j ] F γ [u] γ αj L αj e j Lq (Ω) > ρ. (8.8) e j W 2,r (Ω;T ) We will sow tat tere is a subsequence for wic (8.8) is violated, and tus obtain a contradiction. Since e j W 2,r (Ω;T ) 0, by passing to a subsequence witout cange of notation, we may assume tat e j and its first and second broken derivatives tend to 0 pointwise a.e. in Ω. Te following inequality will elp to simplify te argument: F γ [u + e j ] F γ [u] γ αj L αj e j G j ( ej + e j + D 2 e j ), (8.9) were G j : Ω R 0 is defined by G j := inf α Λ(,u( )) γα a α γ αj a αj + γ α b α γ αj b αj + γ α c α γ αj c αj. (8.10) It can be deduced from Lemma 9 tat G j is measurable, since it is te composition of a lower semi-continuous function wit a measurable function; compactness of Λ and continuity of te data imply tat G j L (Ω) is uniformly bounded for all j N.

18 18 I. SMEARS AND E. SÜLI We prove (8.9): since α j Λ[u + e j ], we ave a.e. in Ω: F γ [u + e j ] F γ [u] γ αj L αj e j = γ αj (L αj u f αj ) F γ [u] 0. (8.11) Now, for a.e. x Ω, and arbitrary α Λ(x, u(x)), we ave 0 F γ [u + e j ] γ α (L α (u + e j ) f α ) = γ αj (L αj u f αj ) F γ [u] + (γ αj L αj γ α L α ) e j = F γ [u + e j ] F γ [u] γ αj L αj e j + (γ αj L αj γ α L α ) e j, (8.12) were it is understood tat te above expressions are evaluated at x. Rearranging (8.11) and (8.12) gives (γ α L α γ αj L αj ) e j F γ [u + e j ] F γ [u] γ αj L αj e j 0, so F γ [u + e j ] F γ [u] γ αj L αj e j (γ α L α γ αj L αj ) e j. (8.13) Since (8.13) olds for arbitrary α Λ(x, u(x)), we readily obtain (8.9). We claim tat G j 0 pointwise a.e. in Ω. Recall tat e j := (e j, e j, D 2 e j) tends to zero pointwise a.e. in Ω. Let ϱ > 0 and x Ω be suc tat e j (x) 0. Ten, by continuity of te data on te compact metric space Ω Λ, tere is a δ > 0 suc tat, for any α, β Λ wit dist(α, β) < δ, γ α a α γ β a β + γ α b α γ β b β + γ α c α γ β c β < ϱ at x Ω. Since (x, u) Λ(x, u) is upper-semicontinuous by Lemma 9, tere is an N N suc tat for eac j N, tere is an α Λ(x, u(x)) wit dist(α, α j (x)) < δ. Terefore 0 G j (x) < ϱ for all j N, and ence G j 0 pointwise a.e. in Ω. Because 1 q < r, setting s = r/q > 1 and s suc tat 1/s + 1/s = 1, we ave 1 s <. Inequality (8.9) followed by an application of Hölder s inequality sows tat 1 F γ [u + e j ] F γ [u] γ αj L αj e j Lq (Ω) G j e j L qs (Ω), (8.14) W 2,r (Ω;T ) Since G j 0 pointwise a.e. and {G j } j=0 is uniformly bounded in L (Ω), te dominated convergence teorem implies tat G j L qs (Ω) 0. Terefore, (8.14) contradicts (8.8), and F γ is DF γ -semismoot at u, tus completing te proof. Remark 1. Te restriction q < r in Teorem 13 cannot be relaxed in general, as evidenced by te counter-example in [13] involving a special case of te class of operators considered ere. Proof of Teorem 11. Since α k Λ[u k ] for eac k, we ave F γ[u k ] = γα k L α k u k γ α k f α k. Terefore, (8.4) is equivalent to A k (u k+1, v ) = γ αk L αk u k F γ [u k ], L λ v v V,p. (8.15) T Te definition of te numerical sceme (5.4) implies tat u satisfies A k (u, v ) = T γ αk L αk u F γ [u ], L λ v v V,p. (8.16) After subtracting (8.16) from (8.15), te bound (8.5) ten sows tat u k+1 u DG(1) C 1 F γ [u k ] F γ [u ] γ α k L α k (u k u ) L 2 (Ω), (8.17)

19 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS u u H 2 (Ω;T ) Mes size p = 2 p = 3 p = 4 p = Fig. 1. Te errors in approximating te solution of te problem of 9.1 for various mes sizes and polynomial degrees. Te optimal convergence rates u u H 2 (Ω;T ) = O ( p 1) are observed. were te constant C 1 depends only on κ, ε, γ, and n as in (6.14), but not on k. Fix r > 2; since V,p is finite-dimensional, tere is a constant C 2 depending on and p suc tat v W 2,r (Ω;T ) C 2 v DG(1) for all v V,p. Teorem 13 sows tat for eac ρ (0, 1), tere is a R ρ > 0 suc tat if w u DG(1) < R ρ, ten, for any α Λ[w ], F γ [w ] F γ [u ] γ α L α (w u ) L 2 (Ω) ρ C 1 C 2 w u W 2,r (Ω;T ). (8.18) If u 0 u DG(1) < R ρ for some ρ < 1, ten we use (8.17) and (8.18) to obtain u k+1 u DG(1) ρ u k u DG(1) k 0, wic yields convergence of u k to u. For eac ρ < 1, u k u DG(1) < R ρ is ten eventually satisfied, tus implying a superlinear convergence rate. 9. Numerical experiments. We provide te results of two tests of te sceme on problems wit strongly anisotropic diffusion coefficients First experiment. We consider once again Example 1 for testing te accuracy of te sceme and te performance of te semismoot Newton metod. Recalling tat Λ = [0, π/3] SO(2) and a α = σ α (σ α ) T /2, wit σ α given by (2.7), let Ω = (0, 1) 2, let b α 0, c α π 2 and coose f α 3 sin 2 θ/π 2 + g, g independent of α, so tat te exact solution of te HJB equation (2.3) is u(x, y) = exp(xy) sin(πx) sin(πy). Tese coices are made so tat te optimal controls vary significantly trougout te domain, and to ensure tat te corresponding diffusion coefficient is not diagonally dominant in parts of Ω. Te numerical sceme (5.4) is applied wit meses obtained by regular subdivision of Ω into uniform quadrilateral elements of size = 2 k, 1 k 6. Te finite element spaces V,p are defined by employing te space of polynomials of fixed total degree p

20 20 I. SMEARS AND E. SÜLI on eac element, wit 2 p 5. Te penalty parameters are set to c stab = 10 and η F = c stab p 4 F / 3 F. Figure 1 confirms te optimal convergence rates wit respect to mes refinement tat are predicted by Teorem u u k H 2 (Ω;T) = 1/4 = 1/ = 1/16 = 1/32 = 1/64 Converged Iteration number k Fig. 2. Convergence istories of te semismoot Newton metod applied to te problem of 9.1 on successively refined meses, wit p = 4. Te predicted superlinear convergence rate is observed, and te number of iterations required for convergence sows little variation under refinement. Te numerical solutions were obtained by te semismoot Newton metod of 8, for wic we use a strict convergence criterion by requiring a relative residual below and a step-increment L 2 -norm below Te initial guess used for eac computation was u 0 0. Te convergence istories sown in Figure 2 demonstrate te fast convergence of te algoritm Second experiment. We investigate te robustness of te sceme against a combination of near-degenerate diffusions, non-smoot solutions and boundary layers. Tis example is to our knowledge te first fully nonlinear second-order problem solved wit an exponentially accurate sceme. Let Ω = (0, 1) 2, b α (0, 1), c α 10 and define ( ) a α := α T 20 1 α, α Λ := SO(2). (9.1) For λ = 1/2, te Cordès condition (2.5) olds wit ε We coose f α so tat te solution of te corresponding HJB equation is ( ) ( u(x, y) = (2x 1) e 1 2x 1 1 y + 1 ) ey/δ, δ > 0. (9.2) e 1/δ 1 Note tat u C 1 (Ω) and u / H 3 (Ω). We coose δ = to be of same order as ε, tus leading to a sarp boundary layer in a neigbourood of { (x, y) Ω: y = 1 }. Te results of [5] sow tat a very large stencil would be necessary to obtain a consistent monotone FD discretisation of tis problem. On uniform grids, tese loworder metods would require a fine grid to resolve te boundary layer, wilst te use of locally refined grids is complicated by consistency and monotonicity requirements. Our metod features no suc constraints, so we are free to take advantage of prefinement tecniques tat are capable of delivering igly accurate approximations

21 DGFEM FOR HJB EQUATIONS WITH CORDÈS COEFFICIENTS 21 Fig. 3. Mes on Ω used for te approximation of (9.2). Te origin is at te bottom left corner. Te mes as 8 geometrically refined layers wit grading factor 1/2. for a smaller computational cost. Following a suggestion in [20], we perform a sequence of computations by increasing te uniform polynomial degrees p from 2 to 10 on a fixed mes sown in Figure 3. Te number of degrees of freedom ranges from 100 to 1320 and te following results were obtained wit c stab = 10, as in 9.1. Figure 4 sows tat te error converges wit a rate of O ( exp( c 3 DoF) ), wic leads to ig accuracy wit few degrees of freedom Relative Error Broken H 2 norm Broken H 1 norm DoF Fig. 4. Exponential convergence in te broken H 1 and H 2 norms of te approximations to te solution defined by (9.2). Te relative errors u u / u are plotted against te cube root of te number of degrees of freedom, wit eac data point corresponding to a computation using a total polynomial degree p = 2,..., Conclusion. We ave considered te PDE analysis and numerical analysis of HJB equations tat satisfy te Cordès condition. Our contributions include an existence and uniqueness result for strong solutions to te fully nonlinear problem, te construction of a consistent and stable p-version DGFEM wit proven convergence rates, and a study of te semismootness of HJB operators. Te numerical experiments demonstrated te ig efficiency and accuracy of te sceme and te fast convergence of te semismoot Newton metod, wilst also igligting te wide applicability of te results of tis work.

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