1 Introduction
Let be a bounded, open, convex polytopal domain in , where represent the dimension. In this paper, we are interested in the Hamilton–Jacobi–Bellman (HJB) equations of the following type:
(1) 
where is a compact metric space, and
Here, and denote the Hessian and gradient of realvalued function , respectively. The coefficient is assumed to be uniformly elliptic, i.e., there exist constants such that
(2) 
Further, and .
The HJB equations arise in many applications including stochastic optimal control, game theory, and mathematical finance
[11]. In [20, 31], the HJB equations are shown to admit strong solutions under the following Cordes condition.Definition 1 (Cordes condition for (1))
The coefficients satisfy that there exist and such that
(3) 
In the past decade, several studies have been taken on the finite element approximation of strong solutions of the HJB equations with Cordes coefficients (3). The first discontinuous Galerkin (DG) method was proposed in [31], which has been extended to the parabolic HJB equations in [32]. The interior penalty DG methods were developed in [22]. A mixed method based on the stable finite element Stokes spaces was proposed in [12]. Recently, the (nonLagrange) finite element method with no stabilization parameter was proposed in [34], where the element is required to be continuous at dimensional subsimplex, e.g., Hermite family in 2D and Argyris family [21, 9] in 3D. The above discretizations can be naturally applied to the linear elliptic equations in nondivergence form [30, 17, 22, 12, 34]. Other related topics include the unified analysis of DGFEM and IPDG [18], and the adaptivity of IPDG [7, 19].
For all these discretizations, the discrete wellposedness is analysed under the broken norm with possible jump terms across the boundary. This, after linearization, leads to the illconditioned systems with condition number on quasiuniform meshes, where represents the mesh size. Due to the similar performance to the discrete system for fourthorder problems, it is conceivable that the linearised system from HJB equations can be effectively solved by the solvers for fourthorder problems, e.g., geometric multigrid [25, 5, 3, 33, 8] or domain decomposition [38, 6]. In [29], the nonoverlapping domain decomposition preconditioner was studied for the DGFEM discretization of HJB equations.
Traditional geometric multigrid methods depend crucially on the multilevel structures of underlying grids. On unstructured grids, the more userfriendly option is the algebraic multigrid method (AMG) that have been extensively studied for the secondorder equations. In [24], the first biharmonic equation was converted to a Poisson system based on the boundary operator proposed in [13]. Under the framework of auxiliary space preconditioning [35], Zhang and Xu [37] proposed a class of optimal solvers based on the auxiliary discretization of mixed form for the fourthorder problems. As a generalization of [25, 24, 26], it works for a variety of conforming and nonconforming finite element discretizations on both convex and nonconvex domains with unstructured triangulation.
The propose of this work is to study the auxiliary space preconditioner to the finite element discretization of HJB equations. More specifically, the numerical scheme for fully nonlinear HJB equations leads to a discrete nonlinear problem that can be solved iteratively by a semismooth Newton method [31, 22, 34]. The linear system obtained from the semismooth Newton linearization are generally nonsymmetric but coercive. To handle the nonsymmetry, the existing GMRES theory [10] will lead to a guaranteed minimum convergence rate with a symmetric FOVequivalent preconditioner that satisfies (20). The construction of under the auxiliary preconditioning framework follows two steps:

Construct appropriate auxiliary spaces and corresponding transfer operators mapping functions from original space to the auxiliary spaces;

Devise solvers on auxiliary spaces so that the bounds in (20) are uniform with respect to both and the parameter in the Cordes condition.
Based on the stable decomposition for auxiliary spaces, both additive and multiplicative preconditoners are shown to be efficient and uniform for the linearised system. Further, the precondtioners only involve the Poissonlike solver which can be efficiently solved by AMG with nearly optimal complexity.
In general cases, the auxiliary space preoconditioner is additive [35, 37, 14], which usually leads to a stable but relatively large condition number in practical applications. The first contribution of this work is the construction and analysis of a multiplicative preconditioner based on the specific structure of auxiliary spaces. Having a coarse subspace, the symmetrized twolevel multiplicative precondition was shown to be positive definite provided that the smoother on the fine level has contraction property [16]
. The condition number estimate of multiplicative precondition at the matrix level can be found in
[23]. In this work, we show that the contracted smoother together with the stable decomposition for auxiliary spaces leads to a robust multiplicative precondititoner, which is also numerical verified with better performance than the additive version.The parameter in the Cordes condition balances the diffusion and the constant term. We emphasis that this parameter is not involved in the monotonicity constant (2.1), which makes it possible to consider the preconditioner with uniformity on . In this work, we carefully define the norm on the auxiliary space so that the induced preconditioner is uniform with respect to . Although the preconditioner is designed for the finite element approximation, a similar idea can be applied to other discretizations.
The rest of the paper is organized as follows. In Section 2, we establish the notation and state some preliminaries results. In Section 3, we apply the FOVequivalence preconditioner for the linear system, which can be used to solve nonsymmetric systems appearing in applications to the HJB equations. In Section 4, we construct both the additive and multiplicative auxiliary space preconditioners. We also show that the condition numbers of the preconditioned systems are uniformly bounded with the stable decomposition assumption, which is verified in Section 5. Several numerical experiments are presented in Section 6 to illustrate the theoretical results.
For convenience, we use to denote a generic positive constant which may depend on , share regularity of mesh and polynomial degree, but is independent of the mesh size . The notation means . means and .
2 Preliminaries
In this section, we first review the strong solutions to the HJB equations (1) under the Cordes condition (3). Then we give a brief statement about the finite element scheme in [34].
Given an integer , let and be the usual Sobolev spaces, and denote the Sobolev norm and seminorm. We also denote . For any Hilbert space , we denote for the dual space of , and for the corresponding dual pair. We also denote
as the Euclidian norm for vectors and the Frobenius norm for matrices.
2.1 strong solutions to the HJB equations
We now invoke the theory of strong solutions of the HJB equations. In view of the Cordes condition (3), for each , define
(4) 
And for as in (3), define a linear operator by
(5) 
Next, we define the operator by
(6) 
Note that the continuity of data implies . As a consequence, it is readily seen that the HJB equation (1) is equivalent to the problem in , and on . The Cordes condition leads to the following lemma; See [31, Lemma 1] for a proof.
Lemma 1 (property of Cordes condition)
Under the Cordes condition (3), for any open set and , , the following inequality holds a.e. in :
(7) 
Another key ingredient for the wellposedness of (1) is the MirandaTalenti estimate stated as follows.
Lemma 2 (MirandaTalenti estimate, [15, 20])
Suppose is a bounded convex domain in . Then, for any ,
(8) 
where the constant depends only on the dimension.
Let the operator be
(9) 
By using MirandaTalenti estimate (8) and Cordes condition (3), one can show the strong monotonicity of ,
where . Together with the Lipschitz continuity of , the compactness of and the BrowderMinty Theorem [27, Theorem 10.49], one can show the existence and uniqueness of the following problem: Find such that
(10) 
We refer to [31, Theorem 3] for a detailed proof.
2.2 finite element approximations of the HJB equations
Let be a conforming shape regular simplicial triangulation of polytope and be the set of all faces of . and . Let be the set of all the nodes of . Here , where is the diameter of . We also denote as the diameter of . For and , we use , respectively , to denote the inner product over , respectively .
Following [34], we adopt the Hermite finite elements () in 2D and Argyris finite elements in 3D to solve the HJB equations (1). Define the finite element spaces as
where denotes set of the polynomials of degree on .
For each , we define the tangential Laplace operator as follows, where . Let be a orthogonal coordinate system on . Then, for define
Next, we define the jump of a vector function on an interior face as follows:
where and is the unit outward normal vector of , respectively. For scaler function we define
The following lemma is critical in the design and analysis of finite element approximation of HJB equations (1).
Lemma 3 (discrete MirandaTalenti identity, [34])
Let be a convex polytopal domain and be a conforming triangulation. For each , it holds that
2.3 Semismooth Newton method
It is shown in [31] that the discretized nonlinear system (11) can be solved by a semismooth Newton method, which leads to a sequence of discretized linear systems. We summarized the main ideas on semismooth Newton here and refer [31] for more detials.
Following the discuss in [31], we define the admissible maximizers set for any ,
where denotes the broken Hessian of . As shown in [31, Lemma 9 & Theorem 10], the set is not empty for any .
The semismooth Newton method is now stated as follows. Start by choosing an initial iterate . Then, for each nonnegative integer , given the previous iterate , choose an . Next the function is defined by ; the functions , , and are defined in a similar way. Then find the solution of the linearised system
(12) 
where the bilinear form is defined by
Following [34], we define inner product on as
and the norm . It is also shown in [34] that the bilinear forms are uniformly coercive and bounded on with norm , with constants independent of iterates. Since the preconditioners in this work take advantage on the coercivity and boundedness of , we summarize the relevant results in the following lemma (see [34, Lemmas 4.1 & 4.2] for a detailed proof).
Lemma 4 (coercivity and boundedness of bilinear form)
For every , we have equationparentequation
Here, the constant depends only on , shape regularity of the grid and polynomial degree .
3 FOVequivalent preconditioners for GMRES methods
The preconditioned GMRES (PGMRES) methods are among the most effective iterative methods for nonsymmetric linear systems arising from discretizations of PDEs. Our study will start by discussing PGMRES methods in an operator form. Let be a linear operator which may be nonsymmetric or indefinite, defined on a finite dimensional space , and be a given functional in its dual space . The linear equation considered here is of the following form
(14) 
Let be an inner product on , and be the preconditioner. The PGMRES method for solving (14) is stated as follows: Begin with an initial gauss and denote the initial residual, the th steps of PGMRES method seeks such that
where is the Krylov subspace of dimension generated by and .
In the semismooth Newton steps, the discrete linear equations (12) have a common form: Find such that
(15) 
where we shall omit to denote independence of the bilinear form and of the righthand side on the iteration number of the semismooth Newton method. Define the operator by
(16) 
then the discrete system (15) can be written in an operator form, namely
(17) 
Moreover, a general operator is used to denote the preconditioner. Given an inner product , we can estimate the convergence rate of the PGMRES method. It is proved in [10, 28] that if is the iteration of PGMRES method and is the exact solution of (17), then
where
(18) 
Therefore, we conclude that as long as we find an operator and a proper inner product such that condition (18) is satisfied with constants and independent of the discretization parameter and the Cordes condition parameter , then is a uniform preconditioner for GMRES method. Such preconditioners are usually referred to as FOVequivalent preconditioners. In what follows, we always take
Next, we give a general principle for constructing . Define an SPD operator by
(19) 
Recalling Lemma 4 (coercivity and boundedness of bilinear form), is coercive and bounded on with the inner product . It is therefore that an efficient preconditioner for can also be used as an FOVpreconditioner for the GMRES algorithm applied to , which is shown in the following lemma.
Lemma 5 (FOVequivalent preconditioner)
4 Fast auxiliary space preconditioners
In this section, we construct both additive and multiplicative auxiliary space preconditioners for SPD operator . From Lemma 5 (FOVequivalent preconditioner), those preconditioners can be applied to the discrete linearised systems (12) arising from each semismooth Newton step of solving the HJB equations.
4.1 Space decomposition
For the purpose of constructing auxiliary space preconditioners, we give the following space decomposition of as
where the auxiliary space denotes the continuous piecewise linear element space on with homogeneous Dirichlet boundary condition, and is a linear injective map which will be defined later. As a result, the induced operator is also SPD, and hence we define on . We also introduce a projection by
A direct calculation shows the following identity
(22) 
Smoother and norm on
Define the discrete Laplacian operator by
(23) 
Then, the smoother on , denoted by , is defined by
(24) 
Note that for any given , can be obtained by solving the following two discrete Poissonlike equations equationparentequation
(25a)  
(25b) 
It can be shown that the above two equations can be solved within operations, where denotes the number of degrees of freedom. We will give a detailed explanation in Remark 4 (computational complexity). The smoother induces a norm on , i.e., . By using (23) and (24), it is straightforward to show that
The relationship between and is shown in the following lemma, whose proof is postponed to Section 5.
Lemma 6 (spectral equivalence of )
Smoother on
Let denote the GaussSeidel smoother for and be the symmetric GaussSeidel smoother, i.e.,
We also define as a norm on . Note that is an SPD operator, thus has the following contraction property
(26) 
Transfer operator
We now give the definition of . To this end, we first give some notation on the degrees of freedom of finite element space . We denote the degrees of freedom as
where is the domain of the integral with respect to the degree of freedom, denotes the integral average on . In general, is a subsimplex of the triangulation. When is a point, the average of the integral is reduced to the evaluation on the point. are identical or different unit vectors to denote the direction of the derivative where . When only one direction is involved for the derivative, and the direction is denoted by . Let be the nodal basis function corresponding to . Define , and . We are now ready to give the definition of as follows: For any
(27)  
where is the unit outer normal vector of . We note that the degrees of freedom corresponding to the secondorder derivative vanish since is piecewise linear.
The general theory of auxiliary space preconditioning simplifies the analysis of preconditioners to the verification of the following two key assumptions.
[stable decomposition] There exists a uniform constant independent of both and , such that for any , there exist and satisfy equationparentequation
(28a)  
(28b) 
[boundedness] There exist uniform constants and independent of both and such that equationparentequation
(29a)  
(29b) 
4.2 Additive preconditioner
Firstly, we introduce the additive preconditioner as
(30) 
The following theorem plays a fundamental role in the theory of auxiliary space preconditioning [35].
Theorem 1 (spectral equivalence of additive preconditioner)
In light of the above theorem and Lemma 5 (FOVequivalent preconditioner), one can see that is a uniform FOVequivalent preconditioner of as long as Assumption 4.1 (stable decomposition) and Assumption 4.1 (boundedness) are verified. We postpone those verifications to Section 5.
Remark 1 (additive preconditioner with Jacobi smoother)
Let be the Jacobi smoother of . From the norm equivalence between and [36, Lemma 4.6], the additive preconditioner
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