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hp-Finite Elements for Highly Indeﬁnite Helmholtz Problems Lecture Notes of the Zürich Summerschool 2016. S.A. Sauter∗ August 21, 2016 Abstract These lecture notes comprise the talks of the author given at the Zürich Summerschool 2016. They are based on the papers [14], [17], [5]. 1 Model Helmholtz Problems and their Discretization 1.1 Model Problems The Helmholtz equation describes wave phenomena in the frequency domain which, e.g., arises if electromagnetic or acoustic waves are scattered from or emitted by bounded physical objects. In this light, the computational domain for such wave problems, typically, is the unbounded complement of a bounded domain Ωin ⊂ Rd , d = 1, 2, 3, i.e., Ωout := Rd \Ωin . We assume that Ωin has a Lipschitz boundary Γin := ∂Ωin . The classical Helmholtz problem depends on the constant wavenumber k > 0. 1.1.1 Some Function Spaces In this section, we will introduce some spaces and sets of functions. Let Ω ⊂ Rd be a bounded Lipschitz domain. For s ≥ 0, 1 ≤ p ≤ ∞, let W s,p (Ω) denote the classical (complex-valued) Sobolev spaces. As usual we write Lp (Ω) short for W 0,p (Ω) and H s (Ω) for W s,2 (Ω). The scalar product and norm in L2 (Ω) and L2 (Γ) are denoted by (u, v) := Ω uv̄ and (u, v)Γ := Γ uv̄ and u := (u, u)1/2 in L2 (Ω) , 1/2 u Γ := (u, u)Γ in L2 (Γ) . For parameters −∞ < a < b < ∞, we introduce the subset L∞ (Ω, [a, b]) := w ∈ L∞ (Ω, R) : a ≤ ess inf w (x) ≤ ess supw (x) ≤ b . x∈Ω x∈Ω ∗ ([email protected]), Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland 1 Let the boundary Γ = ∂Ω be partitioned into two disjoint, relatively open subsets ΓD , ΓN such that the decomposition Γ = ΓD ∪ ΓN ∪ Π with Π := ΓD ∩ ΓN (1.1a) forms a Lipschitz dissection of Γ (cf. [11]). For simplicity, we assume for the following that both, ΓD and ΓN , are closed Lipschitz manifolds so that (see Fig. 1) ΓD ∩ ΓN = ∅. (1.1b) For functions u ∈ C 0 Ω , the restriction to the Dirichlet boundary ΓD is well deﬁned and denoted by (γD,0 u) (x) := u (x) for all x ∈ ΓD . This mapping can be extended continuously to γD,0 : H 1 (Ω) → L2 (ΓD ) and allows to deﬁne H 1/2 (ΓD ) := γD,0 (H 1 (Ω)) as the range of γD,0 . We set H := u ∈ H 1 (Ω) | γD,0 u = 0 . (1.2) The trace operator γN,0 : H 1 (Ω) → H 1/2 (ΓN ) is deﬁned in an analogous way. The dual ′ ′ spaces are denoted by H −1/2 (ΓN ) := H 1/2 (ΓN ) and H −1/2 (ΓD ) := H 1/2 (ΓD ) . Since the boundary ∂Ω is Lipschitz, it follows by Rademacher’s theorem that an exterior unit normal vector ﬁeld n : Γ → Sd−1 exists almost everywhere and allows to deﬁne a normal derivative γN,1 for function in C 1 Ω by (γN,1 u) (x) := n (x) , ∇u (x) for almost every x ∈ ΓN . This mapping can be extended continuously to a mapping γN,1 : H 1 (Ω) → H −1/2 (ΓN ) — for details we refer, e.g., to [11], [22]. 1 We will also need the Sobolev space Hloc (Ωc ) for the unbounded exterior Ωc := Rd \Ω. It contains all functions u : Ωc → C with the property that ϕu ∈ H ℓ (Ωc ) for all ∞ ϕ ∈ Ccomp (Ω) := u|Ω : u ∈ C0∞ Rd 1.1.2 . Derivation of the Classical Helmholtz Equation with DtN and Robin Boundary Conditions 1 For a given right-hand side f ∈ L2 (Ωout ), the Helmholtz problem is to seek U ∈ Hloc (Ωout ) such that −∆ − k 2 U = f in Ωout (1.3a) is satisﬁed. Towards inﬁnity, Sommerfeld’s radiation condition is imposed |∂r U − i kU | = o x 1−d 2 for x → ∞, (1.3b) where ∂r denotes diﬀerentiation in radial direction and |·| the Euclidian vector norm. For simplicity we restrict here to homogeneous Dirichlet boundary condition on Γin U|Γin = 0. (1.3c) We assume that f is local in the sense that there exists some bounded, simply connected Lipschitz domain1 Ω⋆ such that a) Ωin ⊂ Ω⋆ and b) supp(f) ⊂ Ω⋆ . The computational domain (cf. Figure 1) will be Ω := Ω⋆ \Ωin (1.4) 2 G W in G W o u t in W o u t W * Figure 1: Scatterer Ωin with boundary Γin and exterior domain Ωout . The support of f is assumed to be contained in the bounded region Ω⋆ . The domain for the weak variational formulation is Ω = Ω⋆ \Ωin . and, next, we will derive appropriate boundary conditions at the outer boundary Γout := ∂Ω⋆ . Problem (1.3) can be reformulated in an equivalent way as a transmission problem by seeking 1 functions u ∈ H 1 (Ω) and uout ∈ Hloc (Rd \Ω⋆ ) such that (−∆ − k 2 ) u = f (−∆ − k 2 ) uout = 0 u=0 out u=u and ∂n u = ∂n uout ∂r uout − i kuout = o |x| 1−d 2 in Ω, in Rd \Ω⋆ , on Γin , on Γout , (1.5) for |x| → ∞. Here, n denotes the normal vector pointing into the exterior domain Rd \Ω⋆ and ∂n denotes diﬀerentiation in normal direction. 1 It can be shown that, for given g ∈ H 1/2 (Γout ), the problem: Find w ∈ Hloc Rd \Ω⋆ such that (−∆ − k 2 ) w = 0 in Rd \Ω⋆ , w=g on Γout , (1.6) |∂r w − i kw| = o |x| 1−d 2 for |x| → ∞ has a unique solution. The mapping g → w is called the Steklov—Poincaré operator and 1 denoted by SP : H 1/2 (Γout ) → Hloc Rd \Ω⋆ . The Dirichlet-to-Neumann (DtN) map is given 1/2 out −1/2 by Tk := γ1 SP : H (Γ ) → H (Γout ), where γ1 := ∂n is the normal derivative operator at Γout . Hence, problem (1.5) can be reformulated as: Find u ∈ H 1 (Ω) such that (−∆ − k 2 ) u = f u=0 ∂n u = Tk u 1 in Ω, on Γin , on Γout . (1.7) Since Ωin is bounded, Ω⋆ always can be chosen as a ball. Other choices of Ω⋆ might be preferable in certain situations. 3 The previous problems are posed in the weak formulation given by: Find u ∈ H (cf. (1.2)) such that ADtN (u, v) := Ω ∇u, ∇v̄ − k 2 uv̄ − (Tk u) v̄ = Γout for all v ∈ H. fv Ω (1.8) In this section we restrict to the case ΓD := Γin , while, in general, the deﬁnition (1.2) corresponds to the Lipschitz disection introduced in (1.1). Since the numerical realization of the nonlocal DtN map Tk is costly, various approaches exist in the literature to approximate this operator by a local operator. The most simple one is the use of Robin boundary conditions leading to (−∆ − k 2 ) u = f in Ω, u=0 on Γin , (1.9) ∂n u = i ku on Γout . The weak formulation of this equation is given by: Find u ∈ H such that ARobin (u, v) := Ω 1.1.3 ∇u, ∇v̄ − k 2 uv̄ − i kuv̄ = Γout fv Ω for all v ∈ H. (1.10) Helmholtz-type Equations with Variable Coeﬃcients In the previous section, we have explained how the classical Helmholtz problem on an unbounded domain can be reformulated and approximated by a Helmholtz problem on a bounded domain. In the following we will restrict to Helmholtz(-type) problems on bounded domains. We deﬁne the precise problem class which depends on parameters βmax , cmin , cmax , ρmin , ρmax , k0 ∈ R satisfying 0 < cmin ≤ cmax < ∞, 0 < ρmin ≤ ρmax < ∞, (1.11) k0 > 0, βmax > 0. For parameters which satisfy (1.11) and coeﬃcients , k ≥ k0 , ρ ∈ L∞ (Ω, [ρmin , ρmax ]), c ∈ L∞ (Ω, [cmin , cmax ]), β ∈ L∞ (Γ, [0, βmax ]), we introduce the sesquilinear form Aρ,c : H 1 (Ω) × H 1 (Ω) → C by Aρ,c (u, v) := 1 ∇u, ∇v − ρ k c 2 u, v − i k (βu, v)ΓN . (1.12) We consider the Helmholtz problem with variable coeﬃcients: For given F ∈ H′ , ﬁnd u ∈ H such that ∀v ∈ H. (1.13) Aρ,c (u, v) = F (v) Remark 1.1 For suﬃciently smooth coeﬃcient function ρ, c, β and F (v) = (f, v) + (g, v)ΓN for some f ∈ L2 (Ω) and g ∈ L2 (Γ) the strong formulation of (1.13) is given by − div 2 − kc u = f in Ω, i kβ = g on ΓN , u=0 on ΓD . 1 ∇u ρ 1 ∂u − ρ ∂n 4 Deﬁnition 1.2 We equip the space H with the norm u H := u H,ρ,c 2 ∇u √ ρ := k u + c 2 (1.14) which is obviously equivalent to the H 1 (Ω)-norm. Remark 1.3 The analysis of the Helmholtz equation with variable coeﬃcients is a topic of vivid research and many questions are still open. In our lectures we will concentrate on the case of the classical Helmholtz equation with ρ = c = 1 and restrict to the case k0 ≥ 1. 1.2 (1.15) Abstract Variational Formulation Since ΓN is a Lipschitz manifold , it is well known that the following trace estimates hold (see [2, (1.6.6) Theorem]). Lemma 1.4 Let ΓN be as in (1.1b). There exists a constant Ctr such that ∀u ∈ H 1 (Ω) : u H 1/2 (ΓN ) ≤ Ctr u and ∀u ∈ H 1 (Ω) : u Corollary 1.5 For u ∈ H 1 (Ω), we have √ ku L2 (ΓN ) L2 (ΓN ) ≤ Ctr u ≤ Ctr u H 1/2 u (1.16a) H 1/2 H 1 (Ω) . (1.16b) . Proof. It holds k u 2 L2 (ΓN ) ≤ Ctr2 k u u H 1 (Ω) Ctr2 k 2 u 2 + u 2H 1 (Ω) 2 Ctr2 = 1 + k 2 u 2 + ∇u 2 ≤ (1.15) ≤ Ctr2 ku 2 + ∇u 2 2 (1.17) . Both sesquilinear forms ADtN (1.8) and ARobin (1.10) belong to the following class of forms (see Proposition 1.8). Assumption 1.6 (Variational formulation) Let Ω ⊂ Rd , for d ∈ {2, 3}, be a bounded Lipschitz domain. Then H as in (1.2), equipped with the norm · H , is a closed subspace of H 1 (Ω). We consider a sesquilinear form A : H × H → C that can be decomposed into A = a − b, where a (v, w) := Ω ∇v, ∇w̄ − k 2 v w̄ and the sesquilinear form b satisﬁes the following properties: 5 (a) b : H × H → C is a continuous sesquilinear form with |b(v, w)| ≤ Cb v H w for all v, w ∈ H, H for some positive constant Cb . (b) There exist θ ≥ 0 and γell > 0 such that the following Gårding inequality holds: Re A (v, v) + θ kv 2 2 H ≥ γell v (c) The adjoint problem: Find z ∈ H such that for all v ∈ H. for all v ∈ H A (v, z) = (v, f )L2 (Ω) (1.18) (1.19) (1.20) is uniquely solvable for every f ∈ L2 (Ω) with bounded solution operator Q⋆k : L2 (Ω) → H, f → z, more precisely, the (k-dependent) constant Ckadj := k Q⋆k f f sup f ∈L2 (Ω)\{0} H (1.21) is ﬁnite. Problem 1.7 Let A be a sesquilinear form as in Assumption 1.6. For given f ∈ L2 (Ω), we seek u ∈ H such that f v̄ for all v ∈ H. (1.22) A (u, v) = Ω Proposition 1.8 Both sesquilinear forms ARobin (1.10) and ADtN (1.8) (under the additional condition that Γout is a suﬃciently large sphere) satisfy Assumption 1.6. Proof. See [19] and [15]. Here, we restrict to the sesquilinear form ARobin . Condition (a) for ARobin follows from Corollary 1.5. For condition (b) and Robin boundary conditions, we employ Re (ARobin (v, v)) + 2 kv 2 = v 2 H and (1.19) holds with θ = 2 and γell = 1. For condition (c) we may apply Fredholm’s theory and, hence, it suﬃces to prove the implication (A (u, v) = 0 for all v ∈ H) =⇒ u = 0. (1.23) For Robin boundary conditions we consider the imaginary part of the sesquilinear form (1.10) for the test function v = u Im (ARobin (u, u)) = − u 2 L2 (ΓN ) to see that (1.23) implies u ∈ H01 (Ω). Hence, u solves Ω ∇u, ∇v̄ − k 2 uv̄ = 0 for all v ∈ H. (1.24) Let Ω⋆⋆ be a bounded domain such that Ω ⊂ Ω⋆⋆ and ΓN ⊂ Ω⋆⋆ , ΓD ⊂ ∂Ω∗∗ . The extension of u by zero to Ω⋆⋆ is denoted by u0 . It satisﬁes u ∈ H (Ω⋆⋆ ) := u ∈ H 1 (Ω⋆⋆ ) | u|ΓD = 0 and Ω ∇u0 , ∇v̄ − k 2 u0 v̄ = 0 for all v ∈ H (Ω⋆⋆ ) . Elliptic regularity theory implies that u0 ∈ H 2 (Q) for any compact subset Q ⊂ Ω⋆⋆ , in particular, in an open Ω⋆⋆ neighborhood of ΓN . The unique continuation principle (cf. [10, Ch. 4.3]) implies that u0 = 0 in Ω⋆⋆ so that u = 0 in Ω. 6 A R t a ffin e , h -s c a lin g t d is to rtio n , h -in d e p e n d e n t E le m e n t m a p : F t= R t A t Figure 2: hp-Finite element mesh a pullback to the reference element. 1.3 1.3.1 Discretization Conforming Galerkin Discretization A conforming Galerkin discretization of Problem 1.7 is based on the deﬁnition of a ﬁnite dimensional subspace S ⊂ H and is given by: Find uS ∈ S such that for all v ∈ S. A (uS , v) = (f, v) 1.3.2 (1.25) hp-Finite Elements As an example for S as above, we will deﬁne hp-ﬁnite elements on a ﬁnite element mesh T consisting of simplices with maximal mesh width h and local polynomial degree p. Before formulating the conditions on the mesh in an abstract way, we give an example of a typical construction (cf. Fig. 2). Example 1.9 (Patchwise construction of FE mesh.) Let Ω denote a bounded domain. (a) We assume that a polyhedral (polygonal in 2D) domain Ω along with a bi-Lipschitz mapping χ : Ω → Ω is given. Let T macro = Kimacro : 1 ≤ i ≤ q denote a conforming ﬁnite element mesh for Ω consisting of open simplices which are regular in the sense of [3]. T macro is considered as a coarse partition of Ω, i.e., the diameters of the elements in T macro are of order 1. We assume that the restrictions χi := χ|K macro are analytic for i all 1 ≤ i ≤ q. (b) The ﬁnite element mesh with mesh width h is generated by reﬁning the mesh T macro in some standard (conforming) way and denoted by T = Ki : 1 ≤ i ≤ N . The corresponding ﬁnite element mesh for Ω then is deﬁned by T = K = χ K : K ∈ T . Note that, for any K = χ K ∈ T , there exists an aﬃne bijection AK : K → K which d i=1 maps the reference element K := x ∈ Rd>0 : xi < 1 to the simplex K. A parametriza- tion FK : K → K can be chosen by FK := RK ◦ AK , where RK := χ|K is independent of the mesh width h := max {hK : K ∈ T }, where hK := diam (K). 7 Concerning the polynomial degree distribution, it will be convenient (cf.[18]) to assume that the polynomial degrees of neighboring elements are comparable: There exists a constant cp > 0 such that c−1 p (pK + 1) ≤ pK ′ + 1 ≤ cp (pK + 1) for all K, K ′ ∈ T with K ∩ K ′ = ∅. (1.26) To formulate the smoothness and scaling assumptions on RK and AK in an abstract way we have to introduce some notation ﬁrst. For a function v : Ω → R, Ω ⊂ Rd , we write |∇n v(x)|2 = α∈Nd0 :|α|=n n! α |∂ v(x)|2 . α! (1.27) Assumption 1.10 Each element map FK can be written as FK = RK ◦ AK , where AK is an aﬃne map and the maps RK and AK satisfy for constants Caﬃne , Cmetric , γ > 0 independent of hK : A′K L∞ (K) ≤ Caﬃne hK , ′ −1 (RK ) L∞ (K) ≤ Cmetric , (A′K )−1 ∇n RK L∞ (K) ≤ Caﬃne h−1 K L∞ (K) ≤ Cmetric γ n n! for all n ∈ N0 . Here, K = AK (K). Deﬁnition 1.11 (hp-ﬁnite element space) For meshes T with element maps FK as in Assumption 1.10 the hp-ﬁnite element space of piecewise (mapped) polynomials is given by S p,1 (T ) := {v ∈ H : v|K ◦ FK ∈ Pp for all K ∈ T }, (1.28) where Pp denotes the space of polynomials of degree p. For chosen T and p, we may let S = S p,1 (T ). 2 A Priori Analysis This section is devoted to existence, uniqueness, stability, and regularity for the Helmholtz problem (1.7). 2.1 Well-posedness ′ Proposition 2.1 Let Ω be a bounded Lipschitz domain. For all f ∈ (H 1 (Ω)) , a unique solution u of problem (1.10) exists and depends continuously on the data. Proof. The proof follows from Fredholm’s alternative and it suﬃces to prove uniqueness of Problem (1.7). However, the same arguments as in the proof of Proposition 1.8 for the adjoint problem can be applied to prove uniqueness for the forward problem. 8 2.2 Discrete Stability and Quasi-Optimality An essential role for the stability and convergence of the Galerkin discretization is played by the adjoint approximability which has been introduced in [17]; see also [21], [1]. Deﬁnition 2.2 (Adjoint approximability) For a ﬁnite dimensional subspace S ⊂ H, we deﬁne the adjoint approximability of Problem 1.7 by ηk⋆ (S) inf v∈S Q⋆k f − v := k sup f f ∈L2 (Ω)\{0} H , (2.1) where Q⋆k is as in (1.21). Theorem 2.3 (Stability and convergence) Let γell , θ, Cb , Ckadj be as in Assumption 1.6 and S as in Section 1.3.1. Then the condition γell ηk⋆ (S) ≤ , (2.2) 2θ (1 + Cb ) implies the following statements: (a) The discrete inf-sup condition is satisﬁed: inf sup v∈S\{0} w∈S\{0} |A (v, w)| γell ≥ > 0. v H;Ω w H;Ω 2 + γell /(1 + Cb ) + 2θCkadj (2.3) (b) Let S satisfy (2.2). Then, the Galerkin method based on S is quasi-optimal, i.e., for every u ∈ H there exists a unique uS ∈ S with A (uS , v) = (f, v) for all v ∈ S, and there holds 2 (1 + Cb ) inf u − v H , v∈S γell 2 k (u − uS ) ≤ (1 + Cb )2 ηk⋆ (S) inf u − v v∈S γell u − uS H ≤ (2.4) H. (2.5) Proof. Let u ∈ S and set z := θk 2 Q⋆k u. Then, A (u, u + z) = A (u, u) + θ kv 2 = A (u, u) + θ kv 2 + A (u, z) − θ kv 2 . This directly implies (cf. (1.19)) Re A (u, u + z) = γell u 2 H . Let zS ∈ S denote the best approximation of z with respect to the · |A (u, v)| ≤ Cc u v H H H -norm. Note that ∀u, v ∈ H with Cc := 1 + Cb . Then, Re A (u, u + zS ) ≥ Re A (u, u + z) − |A (u, z − zS )| ≥ γell u ≥ u H (γell u ∗ H 2 H − Cc u ∗ H z − zS − θkCc η (S) u ) ≥ (γell − θCc η (S)) u 9 2 H . H The stability of the continuous adjoint problem (cf. (1.21)) implies u + zS H ≤ u H + z − zS H + z ≤ 1 + θη ∗ (S) + θCkadj so that Re A (u, u + zS ) ≥ H u ≤ u H + θkη ∗ (S) u + θkCkadj u H γell − θCc η ∗ (S) u 1 + θη ∗ (S) + θCkadj H u + zS H . Therefore, in view of the assumption (2.2), we have proved inf sup u∈S v∈S\{0} |A (u, v)| γell ≥ . u H v H 2 + γell / (1 + Cb ) + 2θCkadj The convergence of the ﬁnite element discretization is proved by applying the theory as developed in [21] (see also [23, 15, 1], [2, Sec.5.7]). In the ﬁrst step, we will estimate the L2 -error by the H 1 -error and employ the AubinNitsche technique. The Galerkin error is denoted by e = u − uS . We set ψ := Q⋆k e (cf. Assumption 1.6(c)) and denote by ψS ∈ S the best approximation of ψ with respect to the H-norm. The L2 -error can be estimated by using the Galerkin orthogonality 2 k e = kA (e, ψ) ≤ kA (e, ψ − ψS ) ≤ Cc k e ≤ Cc η ∗ (S) e H e , H ψ − ψS H (2.6) i.e., k e ≤ Cc η ∗ (S) e H = Re (A (e, e)) + γell e 2 H . (2.7) To estimate the H-norm of the error we proceed as follows. For any vS ∈ S it holds γell e 2 H ≤ Re A (e, u − vS ) + θk 2 e ≤ Cc e H u − vS H − Re A (e, e) 2 + θCc η∗ (S) e 2 H. This leads to (γell − θCc η ∗ (S)) e H ≤ Cc u − vS H . Noting that (2.2) implies γell − θCc η ∗ (S) ≥ γell /2, we arrive at the ﬁnal estimate. 2.3 Regularity In this section, we derive some explicit bounds for the solution operator Q∗k in (1.21) under the assumption that the right-hand side is in L2 (Ω). To simplify the anylsis we restrict to the full space case Ω = Rd , d = 1, 2, 3, with Sommerfeld’s radiation condition (1.3b) at inﬁnity (the case of bounded domains has been analysed in [17]). Under this assumption, the exact solution of (1.3) can be written as the acoustic volume potential. Let Gk : Rd \ {0} → C denote the fundamental solution to the operator Lk := −∆ − k 2 , i.e., Gk (z) = gk ( z ), where ei kr d = 1, − 2ik (1) i gk (r) := H (kr) d = 2, 4 0 ei kr d = 3. 4πr 10 Then, the solution of (1.3) is given by ∀x ∈ Rd . Gk (x − y) f (y) dy U (x) := (Qk f ) (x) := Ω (2.8) The key ingredient of the analysis of the hp-FEM in Section 2.4 is the following decomposition result. Lemma 2.4 (decomposition lemma) Let Ω be contained in a ball of radius R > 0. Then there exists a constant C > 0 depending only on R and k0 such that for f ∈ L2 (Ω) the function v given by v(x) = Qk f(x) = Ω Gk (x − y)f (y) dy, x ∈ Ω, satisﬁes k −1 v H 2 (Ω) + v H 1 (Ω) +k v ≤C f L2 (Ω) L2 (Ω) . Furthermore, for every λ > 1, there exists a λ- and k-dependent splitting v = vH 2 + vA with ∇p vH 2 L2 (Ω) ≤C 1+ ∇p vA L2 (Ω) ≤ Cλ λ2 √ dλk 1 −1 (λk)p−2 f p−1 f ∀p ∈ {0, 1, 2}, L2 (Ω) ∀p ∈ N0 . L2 (Ω) (2.9a) (2.9b) Here, ∇p vA stands for a sum over all derivatives of order p (see (1.27) for details). Remark 2.5 For f ∈ L2 (Ω) the function v = Qk (f ) cannot be expected to have more Sobolev regularity than H 2 . The decomposition v = vH 2 + vA of Lemma 2.4 splits v into an H 2 -regular part vH 2 and an analytic part vA . The essential feature of this splitting is that the H 2 -part vH 2 has a better H 2 -regularity constant in terms of k than v itself, namely, (2.9a), (2.9b), and the triangle inequality ∇2 v L2 (Ω) ≤ ∇2 vH 2 L2 (Ω) + ∇2 vA L2 (Ω) imply ∇2 vH 2 L2 (Ω) ≤C f versus L2 (Ω) ∇2 v L2 (Ω) ≤ Ck f L2 (Ω) . The fact that vH 2 H 2 ≤ C f L2 for a C > 0 independent of k is essential for the stability and convergence analysis. The decomposition lemma has been generalized to more smooth and less smooth right-hand sides and domains in [16], [6], [13]. Proof of Lemma 2.4. The estimates for v follow directly from those for vH 2 and vA by ﬁxing a parameter λ > 1. In order to construct the splitting v = vH 2 + vA , we start by recalling the deﬁnition of the Fourier transform for functions u with compact support û (ξ) = (2π)−d/2 e− i ξ,x u (x) dx Rd ∀ξ ∈ Rd and the inversion formula u (x) = (2π)−d/2 ei x,ξ û (ξ) dξ Rd 11 ∀x ∈ Rd . Let BΩ ⊂ Rd be a ball of radius R containing Ω. Extend f by zero outside of Ω and denote this extended function again by f . Let µ ∈ C ∞ (R≥0 ) be a cutoﬀ function such that supp µ ⊂ [0, 4R] , µ|[0,2R] = 1, ∀x ∈ R≥0 : 0 ≤ µ (x) ≤ 1, µ|[4R,∞[ |µ|W 1,∞ (R≥0 ) ≤ C , R (2.10) C = 0, |µ|W 2,∞ (R≥0 ) ≤ 2 . R Deﬁne M (z) := µ ( z ) and vµ (x) := BΩ ∀x ∈ Rd . Gk (x − y) M (x − y) f (y) dy The properties of µ guarantee vµ |BΩ = v|BΩ so that we may restrict our attention to the function vµ . Since supp f ⊂ BΩ we may write vµ = (Gk M ) ⋆ f, (2.11) where “⋆” denotes the convolution in Rd . We will deﬁne a decomposition of vµ (which will determine the decomposition of v on BΩ ) by decomposing its Fourier transform, i.e., vµ = vH 2 + vA . (2.12) In order to deﬁne the two terms on the right-hand side of (2.12), we let Bλk (0) denote the ball of radius λk centered at the origin where λ > 1 is the ﬁxed constant (independent of k) selected in the statement of the lemma. The characteristic function of Bλk (0) is denoted by χλk . The Fourier transform of f is then decomposed as f = f χλk + (1 − χλk )f =: fk + fkc . By the inverse Fourier transformation, this decomposition of f entails a decomposition of f into fk and fkc given by ei x,ξ χλk (ξ) fˆ (ξ) dξ fk (x) := (2π)−d/2 Rd and fkc (x) := f − fk . (2.13) Accordingly, we deﬁne the decomposition of vµ by vµ,H 2 := (Gk M ) ⋆ fkc and vµ,A := (Gk M ) ⋆ fk . (2.14) The functions vH 2 and vA in (2.12) are then obtained by setting vH 2 := vµ,H 2 |Ω and vA := vµ,A |Ω . We will obtain the desired estimates by showing the following, stronger estimates: vµ,H 2 H 2 (Rd ) Dα vµ,A L2 (Rd ) ≤C f L2 (Rd ) , |α|−1 ≤ Cλ (λk) (2.15a) f L2 (Rd ) , ∀α ∈ Nd0 . (2.15b) The estimates (2.15) are obtained by Fourier techniques. To that end, we compute the Fourier transform of Gk M : Gk M (ξ) = (2π)−d/2 = (2π)−d/2 e− i ξ,x Gk (x) M (x) dx Rd ∞ gk (r) µ (r) rd−1 e− i r ξ,ζ dSζ Sd−1 0 = (2π)−d/2 I (ξ) . 12 dr The inner integral in I (ξ) can be evaluated analytically2 and I (ξ) = ι ( ξ ) with ∞ 2 gk (r) µ (r) cos (sr) dr d = 1, 0 ∞ 2π gk (r) µ (r) rJ0 (rs) dr d = 2, ι (s) = 0∞ sin (rs) 4π dr d = 3. gk (r) µ (r) r2 (rs) 0 (2.17) Applying the Fourier transform to the convolutions (2.14) leads to vµ,H 2 = (2π)d/2 Gk M fkc = (2π)d/2 Gk M f(1 − χλk ), vµ,A = (2π)d/2 Gk M fk = (2π)d/2 Gk M f χλk . To estimate higher order derivatives of vµ,H 2 and vµ,A we deﬁne for a multi-index α ∈ Nd0 the function Pα : Rd → Rd by Pα (ξ) := ξ α and obtain — by using standard properties of the Fourier transformation and the support properties of χλk — for all |α| ≤ 2 ∂ α vµ,H 2 = (2π)d/2 Pα Gk M (1 − χλk ) f L2 (Rd ) ≤ (2π)d/2 max ξ∈Rd :|ξ|≥λk |Pα I (ξ)| ≤ (2π)d/2 max s|α| ι (s) f s≥λk (2.18) L2 (Rd ) (1 − χλk ) f L2 (Ω) L2 (Rd ) . Lemma 2.6, (iv) implies for |α| ∈ {0, 1, 2} max s|α| ι (s) ≤ C (λk)|α|−2 1 + s≥λk λ2 1 −1 . Thus, ∂ α vH 2 L2 (BΩ ) 1 −1 f L2 (Ω) max s|α| ι (s) f L2 (Ω) ≤ C (λk)|α|−2 1 + and (2.9a) follows. Completely analogously, we derive for all α ∈ Nd0 ∂ α vµ,A L2 (Rd ) ≤ (2π)d/2 0≤s≤λk λ2 . (2.19) We can complete the proof of the lemma using the bounds on the function ι given in Lemma 2.6, (v) below and using (1.27). Lemma 2.6 For the function ι deﬁned in (2.17) the quantity sm ι (s) can be estimated 2 This is trivial for d = 1 and follows for d = 2 from [7, (3.338)(4.)]. For d = 3, we use the formula e−i S2 x,x̂ x̂ x̂ Yℓm (x) dx = gℓ ( x̂ ) Yℓm with gℓ (r) = (−i)ℓ 4πjℓ (r) (which follows by a comparison of [20, Section 3.2.4, formula (3.2.44) and (3.2.54)]) for m = ℓ = 0, where Y00 = const and g0 (r) = 4π sin (r) /r. 13 i. for m = 0 by |ι (s)| ≤ C R , k ii. for m = 1 by 1 + (Rk)−1 |log kR| |sι (s)| ≤ CR 1 1 d = 1, d = 2 and 4Rk ≤ 1, d = 2 and 4Rk > 1, d = 3, iii. and for m = 2 by 1 Rk + Rk |log(kR)| s2 |ι (s)| ≤ C Rk 1 + kR d = 1, d = 2 and 4Rk ≤ 1, d = 2 and 4Rk > 1, d = 3. iv. For ﬁxed R0 , R1 > 0 there exists C > 0 (depending only on R0 , R1 , k0 , d, and the constant appearing in (2.10)) such that for any R ∈ [R0 , R1 ] and any λ > 1 sup s2 |ι (s)| ≤ C 1 + |s|≥λk λ2 1 −1 v. For any λ > 0 and all m ∈ N0 we have sup |s|m |ι (s)| ≤ CλR (λk)m−1 . |s|≤λk The proof of this lemma is quite technical and requires some uniform estimates for some Bessel and exponential functions. We refer to [19, Theorem 3.7] for the details. 2.4 Approximation Properties On meshes Th satisfying Assumption 1.10 with element maps FK we consider the space of piecewise (mapped) polynomials S p,1 (Th ) as in Deﬁnition 1.11. It is desirable to construct an approximant Iu ∈ S p,1 (Th ) of a given (suﬃciently smooth) function u in an elementwise fashion and we employ the following concept: Deﬁnition 2.7 (element-by-element construction) Let K be the reference simplex in Rd , d ∈ {2, 3}. A polynomial π is said to permit an element-by-element construction of polynomial degree p for u ∈ H s (K), s > d/2, if: i. π(V ) = u(V ) for all d + 1 vertices V of K, ii. for every edge e of K, the restriction π|e ∈ Pp is the unique minimizer of π → p1/2 u − π L2 (e) + u−π 1/2 H00 (e) under the constraint that π satisﬁes the condition in Case i. 14 (2.20) iii. (for d = 3) for every face f of K, the restriction π|f ∈ Pp is the unique minimizer of π →p u−π L2 (f ) + u−π (2.21) H 1 (f ) under the constraint that π satisﬁes the conditions in Case i and ii for all vertices and edges of the face f . Remark 2.8 The conditions of Deﬁnition 2.7 are a variation of similar proposals in the literature, e.g., [4] and [8]. For example, the eﬀective diﬀerence between the projection-based interpolation of [4] and the present construction lies in the choice of the norms employed in the minimization process in Deﬁnition 2.7. Our motivation for formulating the conditions in Deﬁnition 2.7 is that they permit us to construct approximation operators with optimal simultaneous approximation properties in L2 and H 1 . Previously, the literature had focused on H 1 -approximation alone. Lemma 2.9 Let d ∈ {1, 2, 3}, and let K ⊂ Rd be the reference simplex. Let γ, C > 0 be given. Then there exist constants C, σ > 0 that depend solely on γ and C such that the following is true: For any function u that satisﬁes for some Cu , h, R > 0, κ ≥ 1 the conditions ∇n u L2 (K) ≤ Cu (γh)n max{n/R, κ}n ∀n ∈ N, n ≥ 2, (2.22) and for any polynomial degree p ∈ N that satisﬁes h/R + κh/p ≤ C (2.23) there holds for m = 1, 2 inf u − π π∈Pp W m,∞ (K) h/R σ + h/R ≤ CCu p+1 + κh σp p+1 (2.24) and additionally admits an element-by-element construction as deﬁned in Deﬁnition 2.7. The approximation of functions with ﬁnite regularity is considered in the following theorem. Theorem 2.10 Let K ⊂ Rd be the reference triangle or the reference tetrahedron. Let s > d/2. Then there exists C > 0 (depending only on s and d) and for every p a linear operator π : H s (K) → Pp that permits an element-by-element construction in the sense of Deﬁnition 2.7 such that p u − πu 2.5 L2 (K) + u − πu H 1 (K) ≤ Cp−(s−1) |u|H s (K) ∀p ≥ s − 1. (2.25) Convergence Rates We are now in position to show that the solution v = Nk⋆ f can be approximated well by the FEM space S p,1 (Th ) provided that kh/p is suﬃciently small and p ≥ c ln k. Theorem 2.11 Let d ∈ {1, 2, 3} and Ω ⊂ Rd be a bounded domain. Then there exist constants C, σ > 0 that depend solely on the constants appearing in Assumption 1.10 such that for every f ∈ L2 (Ω) the function v := Nk⋆ f satisﬁes inf w∈S p,1 (Th ) k v−w H ≤C f L2 (Ω) 15 kh 1+ p kh +k p kh σp p . Proof. We will only prove the cases d ∈ {2, 3}. The case d = 1 follows by similar arguments where the appeal to Theorem 2.11 and Lemma 2.9 is replaced with that to [24, Thm. 3.17]. We note v = Nk⋆ f = Nk f, ﬁx λ > 1 in Lemma 2.4, and split with its aid v = vH 2 + vA with vH 2 ∈ H 2 (Ω) and vA analytic; we have the following bounds vH 2 H 2 (Ω) ≤C f ∇p vA L2 (Ω) , L2 (Ω) ≤ C(λk)p−1 f ∀p ∈ N0 . L2 (Ω) We approximate vH 2 and vA separately. Theorem 2.10 and a scaling argument provides an approximant wH 2 ∈ S p,1 (Th ) such that for every K ∈Th we have, for q = 0, 1, vH 2 − wH 2 H q (K) h p ≤C 2−q vH 2 ∀K ∈ Th . H 2 (K) Hence, by summation over all elements, we arrive at k vH 2 − wH 2 H kh + p ≤C 2 kh p f L2 (Ω) . We now turn to the approximation of vA . Again, we construct the approximation wA ∈ S p,1 (Th ) in an element-by-element fashion. We start by deﬁning for each element K ∈Th the constant CK by ∇p vA 2L2 (K) 2 (2.26) CK := 2p (2λk) p∈N 0 and we note ∇p vA L2 (K) K∈Th ≤ (2λk)p CK 2 CK ≤ 4 3 C λk ∀p ∈ N0 , 2 f (2.27) 2 L2 (Ω) . (2.28) Let the element map for K be FK = RK ◦ AK . Then the function ṽ := vA |K ◦ RK satisﬁes, for suitable constants C̃, C (which depend additionally on the constants describing the analyticity of the element maps RK )3 ∇p ṽ L2 (K) ≤ C C̃ p max{p, k}p CK ∀p ∈ N0 . Since AK is aﬃne, the function v̂ := vA |K ◦ FK = ṽ ◦ AK therefore satisﬁes ∇p v̂ L2 (K) ≤ Ch−d/2 C̃ p hp max{p, k}p CK ∀p ∈ N0 . Hence, the assumptions of Lemma 2.9 (with R = 1 there) are satisﬁed, and we get an approximation w on the element K by lifting an element-by-element construction on K to K via FK which satisﬁes for q ∈ {0, 1} vA − w H q (K) ≤ Chd/2−q h−d/2 CK 3 h h+σ p+1 + kh σp p+1 . This is essentially proved in [12, Lemma 4.3.1]. Speciﬁcally, [12, Lemma 4.3.1] analyzes the case d = 2 and states that C, γ1 depends on the function g. Inspection of the proof shows that the case d = 3 can be handled analogously and shows that the dependence on the function g can be reduced to a dependence on Cg , γg , and γf . 16 Summation over all elements K ∈Th gives vA −w 2 H ≤ h h+σ 2p +k 2 h h+σ 2p+2 k2 + 2 p 2p kh σp +k 2 kh σp 2p+2 2 CK . (2.29) K∈Th The combination of (2.29) and (2.28) yields k vA − w H h h+σ ≤C p hk 1+ h+σ +k kh σp p 1 kh + p σp ! f L2 (Ω) . Furthermore, we estimate using h ≤ diamΩ and σ > 0 (independent of h) h h+σ p 1+ kh σ+h ≤ Ch(1 + kh) p−1 h σ+h ≤ Ch(1 + kh)p−2 ≤ C h p 1 kh + . p p We therefore arrive at k vA − w H ≤C 1 kh + p p kh +k p kh σp p! f L2 (Ω) , which completes the proof of the theorem. Combining Theorems 2.11, 2.3 produces the condition kh +k p p kh σp ≤C for suﬃciently small C independent from k, h, and p required for quasi-optimality of the hp-FEM. We extract from Theorem 2.11 that quasi-optimality of the h-version FEM can be achieved under the side condition that p ≥ C log k: Corollary 2.12 Let Ω = B1 (0) with the additional condition k0 ≥ 1 in the case d = 2. Let Assumption 1.10 be valid. Then there exist constants c1 , c2 > 0 independent of k, h, and p such that (2.2) is implied by the following condition: kh ≤ c1 p p ≥ c2 ln k. together with (2.30) Alternatively, the discrete stability follows from p = O (1) ﬁxed independent of k and kh + k (kh)p ≤ C (2.31) which is understood as a condition on the maximal step size h. Proof. Theorem 2.11 implies kh kη(S) ≤ C 1 + p kh +k p kh σp p . The right-hand side needs to be bounded by 1/Cc . It is now easy to see that we can select c1 , c2 such that this can be ensured. An easy consequence of the stability result Corollary 2.12 is: 17 Corollary 2.13 Let the assumptions of Corollary 2.12 be satisﬁed and let (2.30) or (2.31) hold. Then, the Galerkin solution uS exists and satisﬁes the error estimate u − uS H ≤ Cc h + p kh σp p f L2 (Ω) . Remark 2.14 To the best of the authors’ knowledge, discrete stability in 2D and 3D has only been shown under much more restrictive conditions than (2.30), e.g., the condition k 2 h 1. 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