Computational multiscale methods for nondivergence-form elliptic partial differential equations

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form.


Introduction
In this work, we consider linear second-order elliptic partial differential equations of the form posed on a bounded convex polyhedral domain Ω ⊂ ℝ n , n ≥ 2, with a right-hand side f ∈ L 2 (Ω), subject to the homogeneous Dirichlet boundary condition u = 0 on ∂Ω, ( where A = (a ij ) 1≤i,j≤n ∈ L ∞ (Ω; ℝ n×n sym ), b = (b k ) 1≤k≤n ∈ L ∞ (Ω; ℝ n ), and c ∈ L ∞ (Ω) are heterogeneous coefficients such that A is uniformly elliptic, c ≥ 0 almost everywhere in Ω, and the triple (A, b, c) satisfies a (generalized) Cordes condition.Our main objective in this paper is to propose and rigorously analyze a novel finite element scheme for the accurate numerical approximation of the solution to the multiscale problem (1.1)-(1.2),a task we will refer to as numerical homogenization, by following the methodology of localized orthogonal decomposition (LOD) [31,32].It is worth mentioning that we are working in a framework beyond periodicity and separation of scales.
The motivation for investigating (1.1)-(1.2) stems from engineering, physics, and mathematical areas such as stochastic analysis.Notably, such equations arise in the linearization of Hamilton-Jacobi-Bellman (HJB) equations from stochastic control theory.A distinguishing feature of nondivergence-form problems such as (1.1)-(1.2) is the absence of a natural variational formulation.However, due to the Cordes condition, there exists a unique strong solution to (1.1)-(1.2) which can be equivalently characterized as the unique solution to the following Lax-Milgram-type problem: seek u ∈ V := H 2 (Ω) ∩ H 1 0 (Ω) such that a(u, υ) = ⟨F, υ⟩ for all υ ∈ V (1.3) with some suitably defined F ∈ V * and bounded coercive bilinear form a : V × V → ℝ.
In the presence of coefficients that vary on a fine scale, e.g., when A(x) = Ã ( x ε ) with some (0, 1) n -periodic Ã ∈ L ∞ (ℝ n ; ℝ n×n sym ) and ε > 0 small, classical finite element methods are being outperformed by multiscale finite element methods such as developed in this paper.For periodic coefficients, periodic homogenization has been proposed for linear elliptic equations in nondivergence form; cf.[3,5,21,22,26,28,39,40]. Numerical homogenization of such problems has not been studied extensively so far.A finite element numerical homogenization scheme for the periodic setting has been proposed and analyzed in [9], which is based on an approximation of the solution to the homogenized problem via a finite element approximation of an invariant measure (see also [39]).Further, there has been some previous study on finite difference approaches for such problems in the periodic setting; see [2,15].Concerning fully nonlinear HJB and Isaacs equations, finite element approaches for the numerical homogenization in the periodic setting have been suggested in [20,27], and some finite difference schemes have been studied in [8,12,13].
The case of arbitrarily rough coefficients has not yet been addressed beyond periodicity and scale separation.For divergence-form PDEs, several numerical homogenization methods have been developed in the last decade, which are based on the construction of operator-adapted basis functions and are applicable without such structural assumptions.We highlight the LOD [25,29,31,33], the Generalized Finite Element Method [4,11,30], Rough Polyharmonic Splines and Gamblets [34,36], as well as the recently proposed Super-Localized Orthogonal Decomposition [7,14,24].
The aim of this paper is to transfer such modern numerical homogenization methods to the case of nondivergence-form problems, and to provide a proof of concept that this framework also applies to this class of equations.The only existing link between numerical homogenization and nondivergence-form problems is the metric-based upscaling proposed in [35] which exploits nondivergence-form problems for a problem-dependent change of metric as part of the numerical homogenization of divergence-form problems.Our construction of a practical finite element method for the nondivergence-form problem (1.1)-(1.2) in the presence of multiscale data follows the abstract LOD framework for numerical homogenization methods for divergence-form problems presented in [1].It is based on problem (1.3) as starting point, a-orthogonal decompositions of the solution space V and the test space V into a fine-scale space (defined as the intersection of the kernels of suitably chosen quantities of interest q 1 , . . ., q N ∈ V * ) and some coarse scale space, and a localization argument.In our exemplary approach, the choice of quantities of interest is inspired by the degrees of freedom of the nonconforming Morley finite element.
The remainder of this work is organized as follows.In Section 2, we present the problem setting as well as the theoretical foundation including the well-posedness of (1.1)-(1.2) based on a Cordes condition.In Section 3, we introduce the numerical homogenization scheme for the approximation of the solution to (1.1)-(1.2) based on LOD theory.The proposed numerical homogenization scheme is rigorously analyzed and error bounds are proved.The numerical implementation is based on a H 2 -conforming Birkhoff-Mansfield element and is introduced in Section 4.1.In Section 4, we illustrate the theoretical findings by several numerical experiments, and finally, in Section 5, we discuss an alternative discretization based on mixed finite element theory.

Framework
For a bounded convex polyhedral domain Ω ⊂ ℝ n in dimension n ≥ 2, and a right-hand side f ∈ L 2 (Ω), we consider the problem where we assume that that A is uniformly elliptic, i.e., there exist constants and that the triple (A, b, c) satisfies the Cordes condition, that is, we make the following assumption.(C1) If |b| = c = 0 a.e. in Ω, we assume that there exists a constant δ ∈ (0, 1) such that Further, in this case we set γ := tr(A) |A| 2 and λ := 0. (C2) Otherwise, we assume that there exist constants δ ∈ (0, 1) and λ ∈ (0, ∞) such that Further, in this case, we set Here, we have used the notation |M| := √ M : M to denote the Frobenius norm of M ∈ ℝ n×n .
(ii) Coercivity of a: There exists a constant α > 0 depending only on diam(Ω), n, δ such that The proofs of assertions (i) and (iii) of Lemma 2.1 are straightforward.A proof of assertion (ii) of Lemma 2.1 can be found in [37,38], relying on the observation that the Cordes condition implies that, for any subdomain ω ⊂ Ω, we have that (see [38,Lemma 1]), and using the Miranda-Talenti-type estimates (see [38,Theorem 2]), for all υ ∈ V, with a constant C MT > 0 depending only on diam(Ω) and n.
It is worth emphasizing that, in the setting of periodic homogenization, i.e., for some small parameter ε > 0 and (0, 1) n -periodic Ã , b , c ∈ L ∞ (ℝ n ) satisfying the Cordes condition in ℝ n , we have that the H 2 (Ω)-norm of the solution to (2.1) is uniformly bounded in ε ∈ (0, 1), while generically the H 2+s (Ω)-norm is unbounded as ε ↘ 0 for any s > 0. Note that this is different to the usual periodic homogenization setting for divergence-form equations where generically the H 2 (Ω)-norm is unbounded as ε ↘ 0.

Numerical Homogenization Scheme
For simplicity, we only work in dimension n = 2 and give some remarks on numerical homogenization in higher dimensions in Section 6.2.

Fine-Scale Space
We start by introducing a triangulation of the bounded convex polygon Ω ⊂ ℝ 2 .Thereafter, we define a certain closed linear subspace W of V (recall the definition of V from (2.5)) which will be referred to as the fine-scale space.

Triangulation
Let T H be a regular quasi-uniform triangulation of Ω into closed triangles with mesh size H > 0 and shaperegularity parameter ρ > 0 given by where ρ T denotes the diameter of the largest ball which can be inscribed in the element T. We introduce the piecewise constant mesh-size function h H the set of interior vertices, N ∂ H the set of boundary vertices, and define We label the edges F 1 , . . ., F N 1 and the interior vertices z 1 , . . ., z N 2 so that For each edge F ∈ F H , we associate a fixed choice of unit normal ν F , where we often drop the subscript and only write ν for simplicity.Finally, for a subset S ⊂ Ω, we define N 0 (S) := S and N ℓ (S)

Quantities of Interest and the Space of Fine-Scale Functions
First, let us note that V ⊂ C( Ω) by Sobolev embeddings.For i ∈ {1, . . ., N}, we define the quantity of interest q i ∈ V * by The quantities of interest {q 1 , . . ., q N } ⊂ V * are linearly independent as can be seen from the fact that there exist functions u 1 , . . ., u N ∈ V such that ⟨q i , u j ⟩ = δ ij for any i, j ∈ {1, . . ., N}; see Section 3.3.1.We define the fine-scale space which is a closed linear subspace of V.

Connection to the Morley Finite Element Space
We consider the Morley finite element space whose local degrees of freedom are the evaluation of the function at each vertex and the evaluation of the normal derivative at the edges' midpoints.Here, the piecewise action of the differential operator D is indicated by the subscript NC, i.e., we define (D NC υ)| T := D(υ| T ) for any T ∈ T H .Then, letting {ϕ 1 , . . ., ϕ N } ⊂ V Mor H denote Morley basis functions satisfying ⟨q i , ϕ j ⟩ = δ ij for all i, j ∈ {1, . . ., N} (note ⟨q i , ϕ j ⟩ is well-defined although ϕ j ∉ V), we have that the Morley interpolation operator is given by and we observe that In particular, using Morley interpolation bounds (see, e.g., [42]), we have the local estimate and the global bound

Projectors onto the Fine-Scale Space
We introduce the maps where, for υ ∈ V, we define Cυ ∈ W to be the unique function in W that satisfies a(Cυ, w) = a(υ, w) for all w ∈ W, and we define C * υ ∈ W to be the unique function in W that satisfies a(w, C * υ) = a(w, υ) for all w ∈ W.
Remark 3.1.In view of Lemma 2.1, we have by the Lax-Milgram theorem that the maps C, C * are well-defined, and we have the bounds Further, the maps C, C * are surjective and continuous projectors onto W, and we have that

a-Orthogonal Decompositions of V
We define the trial space ŨH ⊂ V and the test space Ṽ H ⊂ V by In view of Remark 3.1, we then have the following decompositions of the space V: We state a few observations below.(iii) We can equivalently characterize the spaces ŨH and Ṽ H via (iv) We have that ŨH = span(A −1 q 1 , . . ., A −1 q N ).
Proof.(i) By the Riesz representation theorem, there exist q 1 , . . ., q N ∈ V such that q i = ( ⋅ , q i ) V in V * for i ∈ {1, . . ., N}. Set S := span( q 1 , . . ., q N ) and note that dim(S) = N as the quantities of interest q 1 , . . ., q N are linearly independent.Then, in view of (3.1), we have that W = S ⊥ and there holds V = W ⊥ ⊕ W = S ⊕ W. The claim follows.
(ii) This follows immediately from the definition of the spaces ŨH and Ṽ H from (3.5), and the definitions of the projectors C and C * from Section 3.1.4.

Ideal Numerical Homogenization
The ideal discrete problem is the following: (3.7)

Theorem 3.1 (Analysis of the Ideal Discrete Problem
).There exists a unique solution ũ H ∈ ŨH to the ideal discrete problem (3.7).Further, denoting the unique strong solution to (2.1) by u ∈ V, the following assertions hold true.
for the approximation of the true solution u ∈ V by ũ H ∈ ŨH .
Proof.First, recall the properties of a and F from Lemma 2.1.Next, we note that, for any ũ H ∈ ŨH , we have that sup where we have used Lemma 3.1 (ii), the fact that ‖1 − C * ‖ = ‖C * ‖ (see [41]), and that ‖C * ‖ ≤ α −1 C a by Remark 3.1.
Similarly, for any υ H ∈ Ṽ H , we have that sup By the Babuška-Lax-Milgram theorem, there exists a unique solution ũ H ∈ ŨH to the ideal discrete problem (3.7), and we obtain (i).It only remains to show (ii) and (iii).
(ii) We show that u − ũ H = Cu ∈ W. Observing that we have the Galerkin orthogonality (recall we find that u − ũ H ∈ W by Lemma 3.1 (ii) and (3.6).Finally, as u − ũ H ∈ W, we have that Here, we have used that C ũ H = 0 by the definition of ŨH from (3.5) and the properties of C from Remark 3.1.
(iii) First, we note that, by Remarks 3.1 and 2.3, we have the bound . In view of the fact that u − ũ H = Cu ∈ W (see (ii)) and using the bound (3.4), we deduce that which concludes the proof.

Construction of a Local Basis
We are going to construct functions u 1 , . . ., u N ∈ V with local support that satisfy ⟨q i , u j ⟩ = δ ij for any i, j ∈ {1, . . ., N}.
To this end, we introduce the Hsieh-Clough-Tocher (HCT) finite element space where K H (T) denotes the triangulation of the triangle T into three sub-triangles with shared vertex mid(T), and we make use of the HCT enrichment operator E H : V Mor H → V HCT H defined in [16, Proposition 2.5].We then define the operator and supp(ζ F i ) ⊂ ω F i , where ω F i denotes the closure of the union of the two elements that share the edge F i .For any υ Mor H ∈ V Mor H , we have that i.e., Ẽ H preserves the quantities of interest q 1 , . . ., q N .Further, we have the bound where the subscript NC indicates the piecewise action of a differential operator with respect to the triangulation T H , and we have that The proofs of [16, Propositions 2.5-2.6]furthermore show the quasi-local bound for any T ∈ T H .We define the functions where ϕ 1 , . . ., ϕ N ∈ V Mor H are the Morley basis functions from Section 3.1.3,and we set By (3.8) and the definition of the Morley basis functions, there holds ⟨q i , u j ⟩ = δ ij for all i, j ∈ {1, . . ., N}, (3.12) and we have that Ω i := supp(u i ) ⊂ N 1 (supp(ϕ i )) for any i ∈ {1, . . ., N}.Further, we have the following result.

Lemma 3.2 (Stability of Basis Representation). For any u H
Then, by the definition (3.11) of u i , the bound (3.9) for Ẽ H , and inverse estimates for Morley functions, we have that where we have used the notation H 2 (Ω; T H ) := {ϕ ∈ L 2 (Ω) | ϕ| T ∈ H 2 (T) for all T ∈ T H } to denote the broken H 2 -space, and ‖ ⋅ ‖ H 2 (Ω;T H ) := √∑ T∈T H ‖ ⋅ ‖ 2 H 2 (T) to denote the broken H 2 -norm.We deduce that In the final step, we have used that Π Mor u H = (Π Mor − 1)u H + u H and a Morley interpolation estimate; see [42].

Projector onto U H
We introduce the projector Remark 3.2.We can equivalently characterize P H as follows.
(i) By (3.11) and (3.2), we have that that is, P H = Ẽ H ∘ Π Mor .(ii) In view of (i) and introducing I H := E H ∘ Π Mor , we have that We list some stability properties of the projector P H below.

Connection of Ũ H and Ṽ H to the Space U H
First, let us note that, in view of (3.12), we have that dim(U H ) = N and U H ∩ W = {0}.Recalling that Further, there holds for all i, j ∈ {1, . . ., N}.

Exponential Decay of Correctors
The following lemma sets the foundation for the construction of a practical/localized numerical homogenization scheme.

Lemma 3.4 (Exponential Decay of Correctors).
There exists a constant β > 0 such that, for any υ ∈ V and any ℓ ∈ ℕ 0 , we have where Proof.First, let us note that supp(P H υ) ⊂ N 1 (S) for any υ ∈ V with supp(υ) ⊂ S, where S is an element patch in T H . Let υ ∈ V and let ℓ ∈ ℕ with ℓ ≥ 5. Let η ∈ W 2,∞ (Ω) be a cut-off function with and let η := 1 − η ∈ W 2,∞ (Ω).We introduce and note that w, w ∈ W, there holds supp(w) ⊂ Ω\N ℓ−2 (Ω υ ), supp( w) ⊂ N ℓ+1 (Ω υ ), and we have that where we have successively used that P H [Cυ] = 0 as ker(P H ) = W, the definition (3.13) of the functions w and w, the fact that supp( w) ⊂ N ℓ+1 (Ω υ ), and coercivity of a from Lemma 2.1 (ii).Next, we observe that where we have used bilinearity of a, the fact that P H [Cυ] = 0, the definition of C, and the observation that, in view of Lemma 2.1 (i), there holds a(υ, w) = 0 as supp(υ) ⊂ Ω υ and supp(w) ⊂ Ω\N ℓ−2 (Ω υ ).Combining (3.14) and (3.15), and using Lemma 2.1 (i), we find that where We proceed by noting that, by Lemma 3.3, we have that Here, the final bound in (3.17) follows from the fact that, for any T ∈ N 1 (R), there holds where we have used the properties of the cut-off function η and the bound (3.3) for the function Cυ ∈ W.
Similarly to (3.17), we find that Combining (3.17)-(3.18)with (3.16), we obtain that there exists a constant C > 0 such that and hence, setting Setting k := ⌊ ℓ 5 ⌋ and recalling that ℓ ≥ 5, a repeated application of this bound yields proving the claim for the case ℓ ≥ 5. Finally, note that, for ℓ ∈ ℕ 0 with ℓ < 5, we have which concludes the proof.
Using similar arguments, one obtains an analogous exponential decay result for the corrector C * .

Localized Correctors
Motivated by the fact that, for any u H ∈ U H , we have that we define for ℓ ∈ ℕ 0 the localized correctors Here, for i ∈ {1, . . ., N}, the functions φ ℓ i , ψ ℓ i are defined as the unique where we write Note that C ℓ and C * ℓ are well-defined by the properties of a from Lemma 2.1.

Localization Error
We can quantify the error committed in approximating the true correctors C, C * by their localized counterparts C ℓ , C * ℓ .

Theorem 3.2 (Localization Error for Corrector).
There exists a constant s > 0 such that, for any u H ∈ U H and ℓ ∈ ℕ 0 , there holds Proof.First, suppose ℓ ≥ 4. Note that the functions φ i = Cu i and φ ℓ i are uniquely characterized as solutions to the following problems: Therefore, as W(N ℓ (Ω i )) ⊂ W, we can use the properties of a from Lemma 2.1 and Galerkin orthogonality to find that Then, setting where we have used (3.19), the fact that ker(P H ) = W, and an argument analogous to (3.18) for the final bound.By the exponential decay property for C from Lemma 3.4, we obtain that for some constant s > 0, where we have used Remark 3.1 in the final step.Using the triangle inequality, the bound (3.20), and the Cauchy-Schwarz inequality, we find that and hence, by Lemma 3.2, Finally, in the case ℓ < 4, we have from (3.19) and Remark 3.1 that and we can conclude as before.
Using similar arguments, one obtains an analogous result for the corrector C * and its localized counterpart C * ℓ .

Localized Numerical Homogenization Scheme
We are now in a position to state and analyze the practical numerical homogenization method.

The Localized Numerical Homogenization Scheme
We define the N-dimensional spaces Then the numerical homogenization scheme reads as follows: (3.21)

Analysis of the Localized Numerical Homogenization Scheme
The following theorem provides well-posedness of (3.21) as well as error bounds.
for the error in the quantities of interest.(iii) We have the error bound Proof.The well-posedness of (3.21) and the bounds from (i)-(ii) can be shown using identical arguments as in [1].It remains to prove assertion (iii).To this end, we first use the triangle inequality, Theorem 3.1, and (i) to find that for some constant s > 0, where ẽ ℓ H := ũ H − ũ ℓ H . Next, using the triangle inequality, Remark 3.2 (i), the bound (3.9), and a Morley interpolation estimate, we obtain that Finally, by the triangle inequality, Theorem 3.1 (ii), Remark 3.1, quasi-optimality of the Petrov-Galerkin scheme (3.21), and Remark 2.3, we have that

Numerical Experiments
In this section, we numerically investigate the proposed numerical homogenization scheme for nondivergenceform PDEs, which we abbreviate as LOD, due to its origin.We compare it to a conforming Birkhoff-Mansfield finite element method on the respective coarse mesh with mesh size H, denoted as FEM in the convergence plots.To simplify the presentation, H and h denote the minimal side lengths of the elements instead of their diameters in the remainder of this section.

Conforming Discretization
The method presented in Section 3 is not yet discrete as it relies on the solution of the localized version of the saddle-point problem presented in Remark 3.4, which is still in continuous form.For a finite element discretization, we choose a finite-dimensional subspace V h ⊂ V.Here we choose V h to be the H 2 -conforming reduced Birkhoff-Mansfield element space [6,10].Given a triangle T, we define the space X(T) to be the sum of the space of tricubic polynomials over T (the polynomials that are cubic when restricted to any line parallel to one of the triangle's edges) and the three rational functions λ 2 1 λ 2 /(1 − λ 3 ) (with cyclic permutation of the indices 1, 2, 3).The local shape function space BM(T) is then the nine-dimensional subspace of X(T) consisting of those functions whose normal derivative on any of the three edges is affine.The space V h is defined as the subspace of H 2 (Ω) ∩ H 1 0 (Ω) of functions that belong to BM(T) when restricted to any triangle T. The nine local degrees of freedom are the point evaluation of the function and of its gradient in the three vertices of any triangle.Fore more details on the method and its variants, we refer to [6].

Implementation
In all our experiments, we consider the computational domain and use a mesh T h for the fine-scale discretization that resolves all small oscillations of the coefficients.We evaluate relative errors in the L 2 -norm, the H 1 -seminorm, and the H 2 -seminorm with respect to a reference solution originating from the fine mesh T h with h = 2 −8 .For the implementation, we used Matlab and extended the code provided in [32].

Periodic Coefficient Example
We begin by considering a configuration with periodic coefficients.The coefficient A is chosen as A := A (1) , where A (1) ) We perform two numerical experiments with this periodic coefficient A = A (1) .The righthand side is chosen as f := f (1) .

Experiment 1: Vanishing Lower-Order Terms
Figure 1 shows the corresponding errors for the case of vanishing lower-order terms (|b| = c = 0 a.e. in Ω).Note that the Cordes condition (2.3) is satisfied by Remark 2.1.

Experiment 2: Non-vanishing Lower-Order Terms
For the case of non-vanishing lower-order terms, we choose b := b (1) and c := c (1) , where b The corresponding errors are depicted in Figure 2.
Note that a full error analysis of this scheme is not contained in this work but follows along the same lines of what was presented in this work, with modifications for the mixed formulation and the above-mentioned quasi-interpolation.

Review
In this work, we presented a novel numerical homogenization scheme for linear second-order elliptic PDEs in nondivergence form with coefficients that satisfy a (generalized) Cordes condition.Motivated by the degrees of freedom of the nonconforming Morley finite element, our approach using the LOD framework provides a proof of concept that this method is also applicable to the class of nondivergence-form PDEs.The error analysis revealed that numerical homogenization is applicable to problems with coefficients that do not exhibit any scale separation, even beyond periodicity.Moreover, the favorable accuracy properties of the classical LOD for divergence-form PDEs are preserved.Various numerical experiments have been performed that support the theoretical findings.

Extensions and Future Work
Finally, we give some remarks regarding extensions of our results and we address future work.

The Case n ≥ 3
In Section 3, we assumed that n = 2 for simplicity of the presentation of the methodology.It is straightforward to adapt this to dimensions n ≥ 3 by defining the quantities of interest corresponding to the degrees of freedom of the Morley element in dimension n; see [42].

H 2 -Convergence of the Numerical Homogenization Scheme
The numerical experiments suggest that the numerical homogenization scheme presented in this work converges not only in the H 1 -norm, but also in the H 2 -norm.A proof of an H 2 -error bound is subject of future work.

Improved Localization
We emphasize that, using an improved localization technique proposed in [23], increasing errors for refinements in H for fixed ℓ can be cured.A potential application of the improved localization using the Super-Localized Orthogonal Decomposition [7,14,24] might also be possible.Moreover, we see the potential to use the proposed scheme as a preconditioner.

Different Problem Classes
The method presented in this paper can be applied to any Lax-Milgram-type problem over V = H 2 (Ω) ∩ H 1 0 (Ω) of the form (1.3) with F ∈ V * and a locally bounded and coercive bilinear form a : V × V → ℝ.
Theorem 3.3 (Analysis of the Localized Numerical Homogenization Scheme).Assume that ℓ ≳ log(H −2 √ N) is sufficiently large.Then there exists a unique solution ũ ℓ H ∈ Ũ ℓ H to (3.21).Further, denoting the unique strong solution to (2.1) by u ∈ V and the unique solution to the ideal discrete problem (3.7) by ũ H ∈ ŨH , there exists a constant s > 0 such that the following assertions hold true.(i) There holds ‖P H