REGULARITY AND MULTIGRID ANALYSIS FOR LAPLACE-TYPE AXISYMMETRIC EQUATIONS

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1 MATHEMATICS OF COMPUTATION Volume 84, Number 93, May 015, Pages S )0879- Article electronically published on September 8, 014 REGULARITY AND MULTIGRID ANALYSIS FOR LAPLACE-TYPE AXISYMMETRIC EQUATIONS HENGGUANG LI Abstract. Consider axisymmetric equations associated with Laplace-type operators. We establish full regularity estimates in high-order Kondrat vetype spaces for possible singular solutions due to the non-smoothness of the domain and to the singular coefficients in the operator. Then, we show suitable graded meshes can be used in high-order finite element methods to achieve the optimal convergence rate, even when the solution is singular. Using these results, we further propose multigrid V-cycle algorithms solving the system from linear finite element discretizations on optimal graded meshes. We prove the multigrid algorithm is a contraction, with the contraction number independent of the mesh level. Numerical tests are provided to verify the theorem. 1. Introduction Let Ω R + := {r, z), r > 0} be a bounded polygonal domain in the rz-plane and the integer β = 0 or 1. Let Γ 0 be the interior part of Ω {r =0} and Γ 1 := Ω\Γ 0. See Figure 1 for example. We consider equations with parameter β, 1) L β u β := r + r 1 r + z βr β )u β = f β in Ω, with the β-dependent boundary conditions ) u 0 =0 on Γ 1, r u 0 =0 on Γ 0, 3) u 1 =0 on Ω. Equation 1) plays an important role in the study of various equations in 3D axisymmetric domains. For instance, denote by R 3 Ω := Ω Γ 0 ) [0, π) the domain obtained by the rotation of Ω meridian domain) about the z-axis Figure 1). Recall the Cartesian coordinates x, y, z) and the cylindrical coordinates r, θ, z)on Ω. For any ṽ L Ω), its Fourier coefficients with respect to θ are defined by v r, z) =F ṽ) := 1 π 4) ṽr, θ, z)e iθ dθ, Z. π π Let w be the solution of the 3D Poisson equation 5) Δ w = f in Ω, w =0 on Ω. Then, the first three Fourier coefficients w, 1, of w are given by equation 1)asfollows: w 0 = u 0,givenf 0 = F 0 f) β =0);w 1 = u 1,givenf 1 = F 1 f) β =1);andw 1 = u 1,givenf 1 = F 1 f) β = 1). In particular, when the 3D data are axisymmetric, u 0 is the meridian trace of w and L 1 u 1 = f 1 is regarded as Received by the editor February 13, 013 and, in revised form, August 8, Mathematics Subject Classification. Primary 65N30, 65N55; Secondary 35J05. The author was supported in part by the NSF grant DMS c 014 American Mathematical Society Reverts to public domain 8 years from publication 1113

2 1114 HENGGUANG LI z z Γ 1 Γ 0 Γ1 o r o Γ 1 r Figure 1. The 3D axisymmetric domain Ω left); the meridian polygonal domain Ω right). the azimuthal Stoes equation [1,]. We also mention that L 1 coincides with the principal part of the azimuthal Maxwell operator [6, 15]. This dimension reduction from 3D to D has the potential for substantial computational savings in numerical approximations of equations on 3D axisymmetric domains. However, the coordinate transformation introduces new differential operators with singular coefficients as in 1) and new function spaces with singular or degenerate weights. The development of robust numerical methods for these equations calls for rigorous numerical analysis, which has recently drawn much attention from the scientific community. For example, [6] provided a comprehensive introduction on spectral methods for various axisymmetric equations. Under the assumption that the solution has sufficient regularity, finite element analysis for axisymmetric problems was discussed in [1, 5, 1, 1, 6]. In the case that the solution possesses singularities, the singular expansion of the solution for equation 1) was studied in [17, 7]. As the most relevant results, we mention the papers [3] and [15], both for equation 1) with β = 0. In [3], a second-order regularity estimate was established in a class of weighted Kondrat ev-type) spaces for singular solutions. Consequently, the author proposed new linear finite element methods approximating singular solutions in the optimal rate. In [15], the multigrid V-cycle algorithm was studied for finite element spaces augmented with non-polynomial functions. The result was derived only for solutions with required regularity. This paper contains our systematic study for equation 1) on its regularity, finite element approximation, and multigrid analysis. Motivated by [18,, 4], we first introduce high-order Kondrat ev-type weighted spaces Definition.) to handle possible singular solutions from the non-smoothness of the domain and from the singular coefficients. Using appropriate isomorphic mappings in weighted spaces, we prove the full-regularity estimates Theorem 3.5) for singular solutions of equation 1). Based on this regularity result and properties of interpolation operators in weighted spaces, we give an explicit construction of nested graded meshes, such that the associated high-order finite element methods achieve the optimal convergence rate for singular solutions Theorem 3.16). Then, we analyze the system from the linear finite element discretization on optimal graded meshes. With the growth rate of the condition numbers for nested subspaces Lemma 4.5), we provide smoothing properties of the Richardson smoother and approximation properties of the numerical solution in various weighted spaces. This leads to our convergence result Theorem 4.1) on the multigrid V-cycle algorithm for singular solutions of equation 1).

3 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 1115 For equation 1), our high-order regularity result and optimal high-order finite element methods approximating singular solutions are new. These results generalize the results in [3] low-order regularity and linear finite element results only for the case β = 0). The uniform convergence of the proposed multigrid method on graded meshes is also new for the axisymmetric equation 1), which recovers the classical multigrid result for elliptic equations with full-regularity solutions [7, 11]. We mention that high-order Fourier coefficients ) of w in 5) are determined by equations similar to equation 1) with β = 1 [6]. Therefore, our analysis shall provide building blocs for approximating general D Fourier coefficients of the 3D problem 5). The multigrid method studied here shall be a crucial ingredient in developing efficient multigrid solves for more complex axisymmetric problems. The rest of the paper is organized as follows. In Section, we introduce weighted Sobolev spaces for our problem. We also summarize estimates for functions in these spaces. In Section 3, we establish high-order regularity results in Kondrat evtype weighted spaces for the possible singular solution of equation 1). Then, we give high-order finite element methods approximating the singular solutions in the optimal rate. In Section 4, we first describe the multigrid V-cycle algorithm for equation 1). Then, we show the proposed multigrid algorithm converges uniformly, independent of the mesh level, even when the solution is singular. In Section 5, we include multiple numerical tests for the multigrid method to verify the theory.. Preliminaries and notation We introduce appropriate weighted spaces to study equation 1). Useful estimates regarding these spaces will be collected in the second part of this section..1. Function spaces. We first recall a class of weighted Sobolev spaces from [6]. Definition.1 Type I weighted spaces). For an integer m 0, define L 1Ω) := {v, v rdrdz < }, H1 m Ω) := {v, c α v L 1Ω), α m}, Ω where the muti-index α =α 1,α ) is a pair of non-negative integers, α = α 1 + α, and c α = α 1 r α z. The norms and the semi-norms for any v H1 m Ω) are v H1 m Ω) := c α v) rdrdz, v H1 m Ω) := c α v) rdrdz. α m Ω α =m Furthermore, we define two spaces H+ m Ω) and H m Ω). For H+ m Ω), if m is not even, 6) H+ m Ω) := {v H1 m Ω), r i 1 v {r=0} =0, 1 i< m }, v Hm + Ω) = v H m 1 Ω); if m is even, besides the condition in 6), we require Ω m 1 r v) r 1 drdz < for any v H+ m Ω), and the corresponding norm is v H m + Ω) = v H1 mω) + r m 1 v) r 1 drdz ) 1/. Ω For H m Ω), if m is not odd, 7) H m Ω) := {v H1 m Ω), r i v {r=0} =0, 0 i< m 1 } v Hm Ω) = v H m 1 Ω); Ω

4 1116 HENGGUANG LI if m is odd, besides the condition in 7), we require Ω m 1 r v) r 1 drdz <, for any v H m Ω), and the corresponding norm is v H m Ω) = v H1 mω) + r m 1 v) r 1 drdz ) 1/. Thus, we denote different subspaces: H+,0Ω) 1 := H+Ω) 1 {v Γ1 =0}, H,0Ω) 1 := H Ω) 1 {v Ω =0}. Ω We now introduce another type of weighted spaces for our analysis on singular solutions of equation 1). Definition. Type II weighted spaces). Let Q i be the ith vertex of Ω and define the vertex set Q := {Q i } I i=1.denotebylthesmallest distance from a vertex to any disjoint edge of Ω. Let Bx, r 0 ) be the ball centered at x with radius r 0.Let ϑ C Ω\Q) be a function, such that ϑ = x Q i in V i := Ω BQ i,l/) and ϑ L/ inω\ I i=1 V i.notethatv i and V j are disjoint if i j. Thus, we define for μ R and for any open set G Ω, Kμ,1G) m :={v, ϑ μ+ α c α v L 1G), α m} with the semi-norm and norm v K m μ,1 G) := ϑ m μ c α v L G), v K 1 m μ,1 G) := m v K l G). μ,1 α =m l=0 In addition, for 0 l m, we define the subspaces of Kμ,1Ω): m 8) Kμ,+Ω) m := {v Kμ,1Ω), m Ω i 1 r ϑ μ+l v) ) r 1 drdz <, 1 i l }, 9) Kμ, Ω) m := {v Kμ,1Ω), m Ω i r ϑ μ+l v) ) r 1 drdz <, 0 i l 1 }. The corresponding norms are v K m μ,+ Ω) = v K m μ,1 Ω) + i 1 r ϑ μ+l v) ) 1/, r drdz) 1 l m 1 i l Ω v K m μ, Ω) = v K m μ,1 Ω) + i r ϑ μ+l v) ) 1/. r drdz) 1 l m Ω 0 i l 1 Recall the Fourier coefficients v of ṽ in 4). We then have the following isomorphism Theorem II.3.1) [6]. Proposition.3. For m 0, ṽ v ) Z defines an isomorphic mapping 10) H m Ω) Π Z H m ) Ω), where the space H) m Ω) is defined as follows. For >m 1, w H m ) Ω) = w H1 mω) + m r m w L Ω). 1 If m 1, for0 j 1, H) m Ω) = Hm + Ω) { rw j Γ0 =0} and w H m ) Ω) = w H m + Ω) even ), H) m Ω) = Hm Ω) { rw j Γ0 =0} and w H m ) Ω) = w H m Ω) odd ).

5 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 1117 Remar.4. For ṽ H m Ω), H) m Ω) are natural spaces to study the traces of its Fourier coefficients on Ω. We shall show that the Kondrat ev-type spaces Kμ,1Ω) m are the appropriate spaces to study the trace when ṽ only has reduced regularity. Note that for m 0, H0) m Ω) = Hm + Ω), H±1) m Ω) = Hm Ω). Thus, we shall consider equation 1) in H+ m Ω) if β =0 and in H m Ω) if β =1. Inheriting from the 3D Poisson equation 5), we have the boundary condition u β Γ1 = 0. The restrictions on u β in type I weighted spaces lead to different boundary conditions on Γ 0 := Ω\Γ 1 in ) and 3). Since the Fourier coefficients may be complex, even though the original function is real, we shall denote the complex extensions of all the spaces above without change of notation. The Fourier transform 4) leads to the variational formulation for equation 1) [6]: Find u 0 H+,0 1 Ω) for any v 0 H+,0 1 Ω) resp. u 1 H,0 1 Ω) for v 1 H,0 1 Ω)), such that 11) a β u β,v β ):= r u β r v β + z u β z v β + β r u βv β )rdrdz = f β,v β, Ω where f 0 H+,0 1 Ω) resp. f 1 H,0 1 Ω) ), the dual space of H+,0 1 Ω) resp. H,0Ω)) 1 with the pivot space L 1Ω). For f β L 1Ω), the right-hand side is the L 1-inner product. Then, we have the well-posedness result. Corollary.5. The variational formulation 11) defines a unique solution u 0 H+,0Ω) 1 resp. u 1 H,0Ω)) 1 and there is a constant C>0, independent of u β and f β, such that 1) u 0 H 1 + Ω) C f 0 H 1 +,0 Ω) and u 1 H 1 Ω) C f 1 H 1,0 Ω). Proof. For β = 0, the well-posedness follows from the isomorphism 10) and the 3D Poincaré inequality; for β = 1, it follows from the fact that a β, ) is coercive and continuous in H,0Ω). 1 The estimates in 1) are immediate consequences of the well-posedness. Throughout the paper, by H, we mean the dual space of H. As in Definition.1, we also use the multi-index α =α 1,α,α 3 ) for a 3D domain, such that α = α 1 + α + α 3 and α = α 1 x α y α 3 z. The generic constant C>0inour analysis below may be different at different occurrences. It will depend on the computational domain, but not on the functions involved in the estimates or the mesh level in the finite element algorithms... Weighted estimates. We now recall several estimates from []. These estimates give relations between various spaces and will be used to formulate our high-order regularity results in the next section. We distinguish the vertices on and off the z-axisasfollows. Avertexonthe z-axis will be denoted by Q z ; A vertex away from the z-axis will be denoted by Q r. Recall the neighborhood V := BQ, L/) ΩofthevertexQ. ForavertexQ z,let Γ V =Γ 0 V and we denote by Ṽ := V Γ V) [0, π) Ω the neighborhood of Q z in Ω. For an integer l 1, V/l := B Q, L/l) ) ΩandṼ/l := V/l Γ V/l) [0, π). The following lemma, derived as Lemmas.8 and.9 in [], contains useful weighted estimates in usual Sobolev spaces in the 3D neighborhood Ṽ of a vertex Q z. We refer to [] for a detailed proof.

6 1118 HENGGUANG LI Lemma.6. Let Ṽ Ω be the neighborhood of a vertex Q z and ρ the distance from x to Q z. Suppose α m ρ a+ α α v <, form 0 and a R. L Ṽ) Then, for any 0 l m, C ρ a+l v H l Ṽ) ρ a+l v H l Ṽ). α m ρ a+ α α v L Ṽ) C l m Recall the multi-index α =α 1,α ) and the notation α c from Definition.1. Then, the following lemma concerns the connection between the two types of weighted spaces in the D neighborhood V of a vertex Q z in the rz-plane Lemmas.10 and.11 in []). Lemma.7. Let V Ω be the neighborhood of a vertex Q z and ρ the distance from x to Q z.supposev K m a,1v). Then,for0 l m and a R, C ρ a+l v H1 l V) v K m a,1 V) C ρ a+l v H l V). 1 l m The following two lemmas concern the local behavior of functions in weighted spaces near the vertices. Lemma.8. Recall ϑ in Definition.. In the neighhood V of a vertex Q r away from the z-axis, for any v H+,0 1 Ω) resp. v H1,0 Ω)), we have ϑ 1 v L 1 V) C v H 1 + V), resp. ϑ 1 v L 1 V) C v H 1 V)). Proof. On V, bothh 1 + and H 1 resp. L 1) are equivalent to the usual Sobolev space H 1 resp. L ), since r is bounded away from 0. Therefore, it suffices to show for any v H 1 V) {v Γ1 =0}, 13) ϑ 1 v L V) C v H1 V). However, the estimate in 13) is well nown based on a local Poincaré inequality. See [4, 19, 0, 5]. Lemma.9. Recall ϑ in Definition.. Let v H 1 0 Ω) and let Ṽ =V Γ V) [0, π) be the 3D neighborhood of a vertex Q z on the z-axis. Then, ϑ 1 v L Ṽ) C v H 1 Ṽ). Proof. Ṽ can be characterized in the spherical coordinates ρ, θ, φ) centered at Q z : Ṽ = {ρ, ω), 0 <ρ<l/, ω ω Q z}, where ω Q z S is the polygonal domain on the unit sphere S. Then, for any v H 1 Ṽ) {v Ω =0}, and v = vx + vy + vz = vρ + v φ ρ + vθ ρ sin φ ω Q z v ds C vφ + v θ ω Q z sin φ )sinφdθdφ,

7 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 1119 whichisjustthepoincaré inequality on ω Q z and ds =sinφdθdφ is the volume element on ω Q z.thus,weobtain v L/ L/ Ṽ ρ dxdydz = v dsdρ C v ρ + v φ 0 ω Q z 0 ω Q z ρ + vθ ) ρ ρ sin dsdρ φ = C v dxdydz, which completes the proof. 3. Regularity and finite element analysis The solution u β for equation 1) is well defined in weighted spaces Corollary.5). When the 3D domain Ω is smooth, using the isomorphism in 10), we will have full regularity estimates in type I weighted spaces. These estimates will not hold when Ω is a general polygonal domain, since the regularity of u β is determined by the smoothness of Ω and by the operator L β. In this section, we show type II weighted spaces are suitable to study singular solutions. We derive high-order regularity estimates in these spaces for singular solutions and propose finite element algorithms approximating singular solutions in the optimal rate Regularity analysis. Before formulating our main regularity result Theorem 3.5), we first need several local regality estimates. All these results are written for different values of the parameter β, namely,0or1. Lemma 3.1. Near a vertex Q r away from the z-axis, there exists η β > 0, such that for any 0 a β <η β,iff β Ka m β 1,1V), m 0, the solution u β of equation 11) satisfies 14) u 0 K m+ 0 +1,1 V/) C f 0 K m a0 1,1 V) + f 0 H 1 +,0 Ω) ), u 1 Ka 15) m+ a 1 +1,1 V/) C f 1 K m a1 1,1 V) + f 1 H 1,0 Ω) ). Proof. We only show 14), since the proof of 15) follows similarly. We apply a localization argument. Let ζ be a smooth cutoff function, such that ζ =1onV/ and ζ = 0 outside V. Thus,ζu 0 has the Dirichlet boundary condition on V. Then, 16) r + r 1 r + z)ζu 0 = ζf 0 + g 0 in V, where g 0 = u 0 r + z)ζ + r ζ r u 0 + z ζ z u 0 )+r 1 u 0 r ζ. Note that the support of g 0 is within V\V/ andkμ,1 m = H m on V\V/. Therefore, using the interior regularity estimate in H m [14] and 1), we have g 0 K m a0 1,1 V) = g 0 K m a0 1,1 V\V/) C g 0 H m V\V/) C u 0 H V\V/) C f 0 Hm V\V/4) + f 0 H 1 +,0 Ω) ) C f 0 K m a0 1,1 V\V/4) + f 0 H 1 +,0 Ω) ). By Corollary.5, equation 16) has a unique solution ζu 0 H 1 +V). Since r is bounded away from 0 on V, ζu 0 H 1 V) and the space K m μ,1v) isthesameasthe Kondrat ev space K m μ V) Definition 1.1 in [3]). In addition, the regularity of ζu 0 is determined by the principle part r z of the operator in 16). Then, based on the D regularity estimates in the Kondrat ev spaces for the Laplace operator Ṽ

8 110 HENGGUANG LI [16, 18, 4], for the solution ζu 0 H 1 V), there exists η 0 > 0, such that for any 0 a 0 <η 0, ζu 0 K m+ a 0 +1,1 V) C ζf 0 + g 0 K m a0 1,1 V) C ζf 0 K m a0 1,1 V) + g 0 K m a0 1,1 V) ) C f 0 K m a0 1,1 V) + f 0 H 1 +,0 Ω) ). The lemma is thus proved due to the definition of the function ζ. Remar 3.. Near the vertex Q r,sincer is away from 0, the principle part of the operator L β coincides with the Laplace operator r z. Therefore, the regularity index η β is the same for β =0andβ =1. Letω be the interior angle at Q r. Then, 17) η 0 = η 1 = π/ω, which is the first eigenvalue of the operator pencil associated with the Laplace operator [16, 18, 19, 4]. Define ũ β x, y, z) := 1 π u β r, z)e iβθ and f β x, y, z) := 1 π f β r, z)e iβθ. Then, we see that, by the change of variables, ũ β satisfies the 3D Poisson equation, 18) Δũ β = f β in Ω ũ β =0 on Ω, β =0, 1. In addition, its Fourier coefficients { uβ if = β, F ũ β )= 0 if β. We now give a regularity estimate for ũ β near a vertex on the z-axis. Lemma 3.3. For a vertex Q z on the z-axis, there is η β > 0 for all m 0, such that for any 0 a β <η β, the solution ũ β of equation 18) satisfies ϑ aβ 1+ α α ũ β L Ṽ/) )1/ α m+ C α m ϑ a β+1+ α α fβ L Ṽ) )1/ + f β H 1 Ω)). Proof. We use a localization augment similar to the one in Lemma 3.1. Let ζ be a smooth cutoff function, such that ζ =1onṼ/ andζ = 0 outside Ṽ. Then, 19) where Δζũ β = ζ f β + g β, g β =ũ β Δζ + x ũ β x ζ + y ũ β y ζ + z ũ β z ζ. Since g β vanishes near Q z, using the well-posedness of the Poisson problem 18), the usual interior regularity estimate [14], and the expression of g β above, we have ϑ aβ+1+ α α g β 1/ L Ṽ)) C ũβ H Ṽ\Ṽ/) ) 0) α m C f β H m 1 Ṽ\Ṽ/4) + f β H 1 Ω) ) C ϑ aβ+1+ α α fβ L Ṽ) )1/ + f β H 1 Ω)). α m

9 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 111 Based on the 3D regularity estimates in the Kondrat ev spaces for equation 19) Theorem 1. in []) and 0), there exists η β > 0, such that for any 0 a β <η β, the solution ζũ β satisfies ϑ aβ 1+ α α ζũ β ) L Ṽ) )1/ α m+ C α m C α m C α m ϑ a β+1+ α α ζ f β + g β ) L Ṽ) )1/ ϑ a β+1+ α α ζ f β ) L Ṽ) )1/ + α m ϑ a β+1+ α α fβ L Ṽ) )1/ + f β H 1 Ω)). The lemma is thus proved due to the definition of ζ. ϑ a β+1+ α α g β L Ṽ) )1/) Remar 3.4. Different from the situation in Lemma 3.1 and Remar 3., near the vertex Q z, the index η β is determined by the operator pencil of the 3D Laplace operator in a bounded region on the unit sphere [16, 18]. To be more precise, in the neighborhood Ṽ of Qz,letω be the projection of Ṽ on the unit sphere. We write equation 18) in the polar coordinations, Δũ β = ρ ρ ρ ) +ρ ρ +Δ )ũ β, where Δ is the Laplace-Beltrami operator on ω, Δ =cotφ φ + φ +sin φ θ. For any vr, z) L 1V), define ṽ β := 1 π ve iβθ. The smallest eigenvalue λ β of Δ with respect to ṽ β is given by λ 0 ṽ 0 = Δ ṽ 0 = cot φ φ + φ )ṽ 0, λ 1 ṽ 1 = Δ ṽ 1 = cot φ φ + φ sin φ)ṽ 1, which is determined by the associated Legendre s differential equation [13, 7]. Then, the explicit formula for the index η β near Q z is 1) η β = λ β +1/4. In contrast to the case near Q r in 17), η 0 and η 1 may be different near Q z,since λ β are eigenvalues associated with different differential operators. Combining the local estimates in the lemmas above, we derive the global regularity estimate for equation 1). Theorem 3.5. Let u 0 H+,0Ω) 1 and u 1 H,0Ω) 1 be the solutions of equation 1) with β =0and 1, respectively. For m 0, thereexistυ 0 > 0 and Υ 1 > 0, such that for any 0 a 0 < Υ 0 and 0 a 1 < Υ 1, we have ) u 0 K m+ a 0 +1,1 Ω) C f 0 K m a0 1,+ Ω), 3) ϑ a 1 r 1 u 1 L 1 Ω) + u 1 K m+ a 1 +1,1 Ω) C f 1 K m a1 1, Ω), given that f 0 Ka 1,+Ω) m and f 1 Ka 1, Ω). m Proof. Let η i β be the upper bound of the parameter a β for the ith vertex Q i in Lemmas 3.1 and 3.3. Define Υ 0 =min i η i 0)andΥ 1 =min i η i 1). We only show 3), since the proof for ) follows similarly.

10 11 HENGGUANG LI Recall that the weighted space K m a,1 resp., K m a,+ and K m a, ) isequivalenttothe weighted space H m 1 resp., H m + and H m ) in a subdomain Ω sub Ωthatisaway from the vertex set. Based on the isomorphism in 10), the well-posedness and the usual interior regularity estimate for the 3D Poisson equation 18), we have 4) ϑ a 1 r 1 u 1 L 1 Ω sub ) + u 1 K m+ a 1 +1,1 Ω sub) C f 1 Hm Ω ) + f 1 H 1 Ω) C f 1 K m a1 1, Ω ) + f 1 K 0 a1 1, Ω) ), where Ω sub Ω Ω such that Ω is away from the vertex set, and Ω = Ω Ω {r =0}) ) [0, π) is from the rotation of Ω about the z-axis. Let V be the neighborhood of a vertex Q r away from the z-axis. By Lemma 3.1, the fact that r is bounded away from 0 on V, Lemma.8, and a 1 0, we have ϑ a 1 r 1 u 1 L 1 V/) + u 1 K m+ a 1 +1,1 V/) C f 1 K m a1 1,1 V) + f 1 H 1,0 Ω) ) = C f 1 K m a1 1,1 V) + sup f 1,v) L v H,0 1 Ω),v 0 1 / v H 1,0 Ω)) 5) C f 1 K m a1 1, V) + f 1 K 0 a1 1, Ω) ). Now, let V be the small neighborhood of a vertex Q z on the z-axis. By Lemma.7, we first have for any 0 l m, 6) ϑ 1 a 1+l f 1 H l 1 V) C f 1 K m a1 1,1 V). Then, for f 1 Ka m 1 1, V), 6) and the condition in 9), Ω i r ϑ 1 a 1+l f 1 ) ) r 1 drdz <, 0 i l 1)/ lead to ϑ 1 a1+l f 1 H V). l Then, by Lemma.6, the isomorphism in 10), and the definitions of the weighted spaces in 7) and 9), we have ϑ 1 a1+ α α f1 L Ṽ) C ϑ 1 a 1+l f1 H l Ṽ) 7) α m C 0 l m 0 l m ϑ 1 a 1+l f 1 H l V) C f 1 K m a 1 1, V). Note that by Lemma.7, 10), and Lemma 3.3, we have ϑ a 1 r 1 u 1 L 1 V/) + u 1 K m+ a 1 +1,1 V/) C ϑ a 1 r 1 u 1 L 1 V/) + ϑ a1 1+l u 1 H1 l V/))1/) C l m+ C α m l m+ ϑ a 1 1+lũ 1 H l Ṽ/) )1/ ϑ a 1+1+ α α f1 L Ṽ) )1/ + sup v H 1 0 Ω),v 0 f 1,v) L Ω)/ v H1 Ω)).

11 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 113 Therefore, continuing the estimates above, by Lemma.9, the fact that a 1 0, and 7), we have ϑ a 1 r 1 u 1 L 1 V/) + u 1 K m+ a 1 +1,1 V/) C ϑ a1+1+ α α f1 L Ṽ) )1/ + ϑ a ) 1+1 f1 L Ω) 8) α m C f 1 K m a1 1, V) + f 1 K 0 a1 1, Ω) ). Then, the proof is completed by combining 4), 5), and 8). Remar 3.6. We used the global parameter Υ β in Theorem 3.5 in order to simplify the exposition. The proof in fact shows that Theorem 3.5 holds as long as for the ith vertex Q i, we choose the parameter 0 a β <ηβ i,whereηi β is determined as in Remars 3. and 3.4. This will result in the analogue of Theorem 3.5 in Kondrat ev-type spaces with vector subindices, in which the function ϑ is allowed to have different exponents near different vertices to capture the local behavior of the solution. See [4] for such a presentation for elliptic equations. 3.. Finite element analysis. Let T n = {T i } be a triangulation of Ω with shaperegular triangles T i.denotebyv β n the continuous Lagrange finite element space of order m with the corresponding Dirichlet boundary conditions described in ) and 3), respectively. The index n in T n shall denote the number of refinements that we will specify later. Then, we have [1] 9) V 0 n H+,0Ω) 1 and V 1 n H,0Ω). 1 In turn, the finite element solution u β n V β n for equation 11) is defined by 30) a β u β n,vn)= β f β vnrdrdz, β vn β V β n. Ω We denote a point in the rz-plane by x =r, z). Let Π : C 0 Ω) V β n be the usual nodal interpolation operator, such that Πvx i )=vx i ), where x i is the ith node in the triangulation. Note that we use the same notation for the interpolation operators into different spaces V β n, β = 0 and 1. In the text below, by a triangle T, we mean the closed set including both the interior and boundary of the triangle. We first recall two approximation results from Lemmas 6.1, 6. and 6.3 in [6], which are based on embedding theorems in weighted spaces. Lemma 3.7. Let x i be a node in the triangulation, which is on the z-axis. Let K i be the union of triangles that contain x i.then,forv H1 K i ), we have 31) v Πv H 1 1 K i ) Ch m K i v H 1 K i ) ; for v H1 K i ) H Ω), 1 3) r 1 v Πv) L 1 K i ) Ch m K i v H 1 K i ), where h Ki denotes the diameter of K i and m 1. Lemma 3.8. Let T T n be a triangle that does not intersect the z-axis. Suppose min x T rx) Ch T.Then,forv H1 T ), we have 33) v Πv H 1 1 T ) Ch m T v H 1 T ), 34) r 1 v Πv) L 1 T ) Ch m T v H 1 T ), where h T denotes the diameter of T and m 1.

12 114 HENGGUANG LI Remar 3.9. By Lemma 3.7 and Lemma 3.8, one can derive the convergence result for the finite element solution in 30) on quasi-uniform meshes u 0 u 0 n H 1 + Ω) Ch m u 0 H + Ω) and u 1 u 1 n H 1 Ω) Ch m u 1 H Ω), given that u β is sufficiently regular and h is the mesh size. As discussed above, the regularity of the solution, however, depends on the differential operator L β and the domain Ω. When the solution u β is singular e.g., u β / H1 Ω)), the linear finite element method on the quasi-uniform mesh only gives a suboptimal convergence rate [6]. Based on our regularity estimates in Theorem 3.5 and Remar 3.6, we use the following process to generate a sequence of graded triangulations approximating the singular solutions of equation 1). Definition 3.10 The κ-refinement). Let κ 0, 1/] be the grading parameter and T be a triangulation of Ω such that no two vertices of Ω belong to the same triangle of T. Then the κ-refinement of T, denoted by κt ), is obtained by dividing each edge AB of T into two parts as follows. If neither A nor B is in the vertex set Q, then we divide AB into two equal parts. Otherwise, if A Q, we divide AB into AC and CB such that AC = κ AB. This will divide each triangle of T into four triangles and leads to a finer mesh. Suppose the initial mesh T 0 satisfies the above conditions. Then the nth level mesh with κ-refinement is obtained recursively by T n = κt n 1 ), n =1,,... This graded process was proposed in [3] for the linear finite element method approximating the axisymmetric Poisson equation. Here we generalize it to highorder finite element methods for the new equation 1). We need the following notation to carry out the analysis on graded meshes. Let n be the number of κ-refinements of the domain. Thus, the final triangulation is T n. Let T i,j T j, j n, be the union of triangles in T j that contain the vertex Q i Q. Note that T i,j T i,l for j l and i T i,j occupies the neighborhood of the vertex set Q in the triangulation T j. Recall the regularity estimate for the solution and the parameter Υ β in Theorem 3.5. We choose the grading parameter in the κ-refinement, for different values of β, 35) κ β =min1/, m/a β ), for any 0 <a β < Υ β, where m 1 is the degree of piecewise polynomials in the finite element space V β n. Then, based on analysis on T n \ i T i,0, on i T i,j 1\ i T i,j, andont i,n, our error estimates are summarized in the following lemmas. Lemma For κ β defined in 35), letu T n be the union of triangles that intersect T n \ i T i,0. Then, u 0 Πu 0 n H 1 + T n \ i T i,0) C nm u 0 H 1 U), u 1 Πu 1 n H 1 T n \ i T i,0) C nm u 1 H 1 U). Proof. Assume U is away from the vertices of the domain this is true when n>). Then, based on Definition 3.10, the mesh size on U is O n ). Summing up the estimates in 31), 3), 33), and 34) completes the proof. For the estimates on T i,0, the union of initial triangles containing the vertex Q i, we consider the new coordinate system that is a simple translation of the old

13 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 115 rz-coordinate system, now with Q i at the origin. Then, for a subset G T i,0 and 0 <λ<1, we define the dilations of G and of a function as follows, for r, z) G, 36) G λ := G/λ = {r λ,z λ )=λ 1 r, λ 1 z)}, v λ r λ,z λ ):=vr, z). Then, we have the following scaling estimates from Lemma 4.5, []. Lemma 3.1. Suppose G λ is in the neighborhood V of Q i.ifq i is on the z-axis, v λ K m a,1 G λ ) = λ a 3/ v K m a,1 G), r 1 λ v λ L 1 G λ ) = λ 1/ r 1 v L 1 G); if Q i is not on the z-axis, then Cλ a 1 v K m a,1 G) v λ K m a,1 G λ ) Cλ a 1 v K m a,1 G). Now, we are ready to give estimates on the region T i,j 1 \T i,j. Lemma Let U T n be the union of triangles that intersect G := T i,j 1 \T i,j. Suppose G V. Let h bethemeshsizeonu and ξ =sup x G ϑx). Let a 0. Then, for v Ka+1,1 U), 37) v Πv H 1 + G) Cξ a h/ξ) m v K U), a+1,1 and for v H U) 1 Ka+1,1 U), 38) v Πv H 1 G) Cξ a h/ξ) m v K U). a+1,1 Proof. Recall the new coordinate system with Q i as the origin. Recall the dilations in 36) and the parameter L in Definition.. Note that Πv) λ =Πv λ ). Then, we choose λ =ξ/l, such that G λ V. If Q i is on the z-axis, by Lemma 3.1, the definitions of the weighted spaces, 31), and 3), we have r 1 v Πv) L 1 G) + v Πv H 1 1 G) C r 1 v Πv) L 1 G) + v Πv K 1 1,1 G) = λ 1/ r 1 λ v λ Πv λ )) L 1 G λ ) + v λ Πv λ ) K 1 1,1 G λ )) Cλ 1/ r 1 λ v λ Πv λ )) L 1 G λ ) + v λ Πv λ ) H 1 1 G λ )) Cλ 1/ h/λ) m v λ H 1 U λ ) Cλ1/ h/λ) m v λ K 1,1 U λ) Ch/ξ) m v K 1,1 U) Cξa h/ξ) m v K U). a+1,1 If Q i is not on the z-axis, the proof is similar. With the corresponding estimate in Lemma 3.1, the definitions of the weighted spaces, 33), and 34), we have r 1 v Πv) L 1 G) + v Πv H 1 1 G) C r 1 v Πv) L 1 G) + v Πv K 1 1,1 G) C v Πv K 1 1,1 G) C v λ Πv λ ) K 1 1,1 G λ ) C v λ Πv λ ) H 1 1 G λ ) Ch/λ) m v λ H 1 U λ ) Ch/λ) m v λ K 1,1 U λ) Ch/ξ)m v K 1,1 U) Cξa h/ξ) m v K U). a+1,1 Thus, the estimate in 38) is proved. The estimate in 37) follows similarly using Lemma 3.1, the definitions of the weighted spaces, 31), and 33).

14 116 HENGGUANG LI Hence, we obtain the interpolation error estimates for u 0 and u 1 on T i,j 1 \T i,j. Lemma For κ β defined in 35), letu T n be the union of triangles that intersect G := T i,j 1 \T i,j.then,for0 <a β < Υ β, u 0 Πu 0 H 1 + G) C nm u 0 K U), a 0 +1,1 u 1 Πu 1 H 1 G) C nm u 1 K U). a 1 +1,1 Proof. Definition 3.10 shows that the mesh size on U is Oκ j 1 β j 1 n ). Using the notation of Lemma 3.13, we have ξ = Oκ j 1 β )ont i,j 1 \T i,j. Therefore, using Lemma 3.13, we have u 1 Πu 1 H 1 G) Cκ j 1)a 1 1 j 1 n)m u 1 K a 1 +1,1 U) C j 1)m j 1 n)m u 1 K a 1 +1,1 U) = C nm u 1 K U). a 1 +1,1 Then, we have proved the second estimate in this lemma. The first estimate can be proved similarly by Lemma 3.13 and the observation on the mesh size for the regions G and U. The following lemma gives the error bounds on the last patch T i,n of triangles that have the vertex Q i as the common node. Lemma For κ β defined in 35), letu T n be the union of triangles that intersect T i,n.supposet i,0 V near Q i.then, u 0 Πu 0 H 1 + T i,n ) C nm u 0 K U), a 0 +1,1 u 1 Πu 1 H 1 T i,n ) C nm u 1 K a 1 +1,1 U) + ϑ a 1 r 1 u 1 L 1 U)). Proof. Definition 3.10 shows that the mesh on U has the size Oκ n β ). Let ζ be a smooth function, such that ζ = 0 in the small neighborhood of the vertex Q i and ζ = 1 on a region containing all the other nodes in T i,n. Let v = u 1 ζu 1. Note that Πu 1 =Πζu 1 )andt i,0 = T i,n /λ, whereλ = κ n β. If Q i is on the z-axis, by Lemma 3.1, we have r 1 ζu 1 L 1 T i,n ) = λ 1/ r 1 λ ζ λu 1 ) λ L 1 T i,0 ) and Cλ 1/ r 1 λ u 1) λ L 1 T i,0 ) = C r 1 u 1 L 1 T i,n ) ζu 1 K m 1,1 T i,n ) = λ 1/ ζ λ u 1 ) λ K m 1,1 T i,0 ) Cλ 1/ u 1 ) λ K m 1,1 T i,0 ) = C u 1 K m 1,1 T i,n ). Thus, by the estimates above and Lemma 3.1, we have r 1 u 1 Πu 1 ) L 1 T i,n ) + u 1 Πu 1 H 1 1 T i,n ) r 1 v L 1 T i,n ) + r 1 ζu 1 Πu 1 ) L 1 T i,n ) + v H 1 1 T i,n ) + ζu 1 Πu 1 K 1 1,1 T i,n ) C r 1 u 1 L 1 T i,n ) + u 1 K 1 1,1 T i,n ) +λ 1/ r 1 λ ζ λu 1 ) λ Πu 1 ) λ ) L 1 T i,0 ) + ζ λ u 1 ) λ Πu 1 ) λ K 1 1,1 T i,0 )).

15 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 117 Therefore, using Lemma 3.7, Lemma 3.1, and the estimates above, r 1 u 1 Πu 1 ) L 1 T i,n ) + u 1 Πu 1 H 1 1 T i,n ) C r 1 u 1 L 1 T i,n ) + u 1 K 1 1,1 T i,n ) + λ 1/ ζ λ u 1 ) λ H 1 U λ ) ) C r 1 u 1 L 1 T i,n ) + u 1 K 1 1,1 T i,n ) + λ 1/ u 1 ) λ K 1,1 U λ) ) 39) = C r 1 u 1 L 1 T i,n ) + u 1 K U)). 1,1 If Q i is not on the z-axis, r is bounded away from 0, by Lemma 3.1, r 1 ζu 1 L 1 T i,n ) = r 1 ϑ)ϑ 1 ζu 1 L 1 T i,n ) C ζu 1 K 0 1,1 T i,n ) C ζ λ u 1 ) λ K 0 1,1 T i,0 ) C u 1 ) λ K 0 1,1 T i,0 ) C u 1 K 0 1,1 T i,n ); ζu 1 K m 1,1 T i,n ) C ζ λ u 1 ) λ K m 1,1 T i,0 ) C u 1 ) λ K m 1,1 T i,0 ) C u 1 K m 1,1 T i,n ). Thus, by Lemma 3.8, Lemma 3.1, and the estimates above, r 1 u 1 Πu 1 ) L 1 T i,n ) + u 1 Πu 1 H 1 1 T i,n ) 40) Note that maxx Ti,n r 1 v L 1 T i,n ) + r 1 ζu 1 Πu 1 ) L 1 T i,n ) + v H 1 1 T i,n ) + ζu 1 Πu 1 H 1 1 T i,n ) C u 1 K 1 1,1 T i,n ) + ζu 1 Πu 1 K 1 1,1 T i,n )) C u 1 K 1 1,1 T i,n ) + ζ λ u 1 ) λ Πu 1 ) λ K 1 1,1 T i,0 )) C u 1 K 1 1,1 T i,n ) + ζ λ u 1 ) λ K 1,1 T i,0) ) C u 1 K 1,1 T i,n). ϑx) =Oκn β ). Combining 39) and 40), we have r 1 u 1 Πu 1 ) L 1 T i,n ) + u 1 Πu 1 H 1 1 T i,n ) C r 1 u 1 L 1 T i,n ) + u 1 K U)) 1,1 Cκ na 1 1 ϑ a 1 r 1 u 1 L 1 T i,n ) + u 1 K U)) a 1 +1,1 C nm ϑ a 1 r 1 u 1 L 1 T i,n ) + u 1 K U)). a 1 +1,1 This completes the proof of the second estimate of the lemma. The first estimate follows from a similar process using Lemmas 3.1,.8,.9, 3.7, and 3.8. The local estimates above lead to the main result on the convergence rate of our finite element method. Theorem Let u β be the solution of equation 1) and u β n V β n be the finite element solution defined in 30). Recall κ β from 35) for0 <a β < Υ β as defined in Theorem 3.5. Then,forf 0 Ka m 1 0 1,+ Ω) and f 1 Ka m 1 1 1, Ω), u 0 u 0 n H 1 + Ω) CdimV 0 n) m/ f 0 K m 1 Ω), a 0 1,+ u 1 u 1 n H 1 Ω) CdimV 1 n) m/ f 1 K m 1 Ω), a 1 1, where dimv β n)=o4 n ) is the dimension of V β n and m 1. Proof. The proof follows by summing up the estimates in Lemmas 3.11, 3.14, and 3.15, together with the regularity results in Theorem 3.5. Remar The graded meshes from the κ-refinements are nested and consist of shape-regular triangles. Various numerical tests for the linear finite element methods m = 1) approximating singular solutions in the case β = 0 can be found

16 118 HENGGUANG LI in [3], which convincingly verify Theorem As mentioned in Remar 3.6, instead of a uniform α β for all vertices, we can choose different regularity indices a β near different vertices Q i. This will result in different grading parameters κ β close to the vertices. Nevertheless, as long as 0 <a β <η i β for each Q i,weshall have the optimal convergence rate indicated in Theorem This flexibility can help to improve the shape regularity of the triangulation. 4. The multigrid algorithm From now on, we concentrate on the multigrid analysis for the linear finite element approximation of equation 1) on graded meshes given in the last section The multigrid V-cycle algorithm. We denote the β-dependent linear finite element spaces V β 0 Vβ 1 Vβ K that are defined on graded meshes T in the last section, 0 K, wherethe parameter κ β is chosen as in 35) for m =1. Namely,κ β =min1/, 1/a β ), for 0 <a β < Υ β. For simplicity, we assume that there is only one singular vertex Q for the solution. Namely, the solution is in H1 or ηβ i > 1) except in the neighborhood of Q. Therefore, the graded mesh κ β < 0.5) is implemented only for triangles touching Q, while the usual quasi-uniform κ = 0.5) decomposition is performed for other triangles. If we modify the weight function in the definition of the space Kμ,1 m Ω), letting ϑ be the distance to Q in its neighborhood V and be 1 in the neighborhoods of other vertices, then Theorem 3.16 in this case reads 41) u 0 u 0 H 1 + Ω) CdimV 0 ) 1/ f 0 K 0 a0 1,+ Ω), 4) u 1 u 1 H 1 Ω) CdimV 1 ) 1/ f 1 K 0 a1 1, Ω), where u 0 V0 and u1 V1 are the corresponding linear finite element solutions on level. In this section, we use this modified function ϑ and also define := r, z ) t. Let T T be the union of all the triangles in T touching Q. Define the jth layer of T, 43) L j := T j 1 \T j, 0 j, where T 1 := T 0. Then, we define the piecewise-constant function on T, 44) ω Lj := κ β ) j, where κ β = 1/a β < 0.5 is the grading parameter for Q. Then, we define the following mesh-dependent weighted inner product 45) v,w ) := Ω ω v w rdrdz, v,w V β, and the norm induced by the inner product, v := ωv rdrdz, v V β. Ω Let I 1 : Vβ 1 Vβ be the coarse-to-fine operator, which is the natural injection. The fine-to-coarse operator I 1 : V β Vβ 1 is the transpose with respect to the inner product in 45), I 1 v,w 1 ) 1 := v, I 1w 1 ), v V β, w 1 V β 1.

17 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 119 Let A β : Vβ Vβ be the operator associated with equation 30), 46) A β v,w ) := a β v,w ), v,w V β. In addition, let f β Vβ be the function, such that f β,v ) =f β,v ) L 1 Ω), v V β. Thus, we are ready to formulate the abstract multigrid V-cycle algorithm for equation 1) with the Richardson smoother. Algorithm 4.1 The V-cycle algorithm). The th level V-cycle algorithm produces MG, f β,z 0,l) as an approximate solution for A β uβ = f β with initial guess z 0 V β,wherel denotes the number of pre-smoothing and postsmoothing steps. For =0, we define MG0,f β 0,z 0,l)=A β 0 ) 1 f β 0. For 1, the approximate solution MG, f β,z 0,l) is computed recursively in three steps: Pre-smoothing. For 1 j l, compute z j by z j = z j 1 + γ f β Aβ z j 1), where the constant γ is the damping factor that we will specify later. Coarse grid correction. Let g 1 = I 1 f β Aβ z l) be the restriction of the residue on the 1st level. Define q V β 1 by q := MG 1,g 1, 0,l). Then, we compute z l+1 by z l+1 = z l + I 1q. Post-smoothing. For l + j l +1, compute z j by z j = z j 1 + γ f β Aβ z j 1). We then define MG, f β,z 0,l)=z l+1. Remar 4.. Let E : V β Vβ be the error-propagation operator for the th level V-cycle algorithm defined above. Namely, E u β z 0)=u β MG, fβ,z 0,l). The relation below follows from a straightforward calculation: 47) E = R l Id I 1P 1 + I 1E 1 P 1 )R l, where Id is the identity operator on V β, R : V β Vβ measuring the effect of the smoothing step is defined by 48) R := Id γ A β, and P 1 : V β Vβ 1 is the transpose of I 1 with respect to the bilinear form a β, ), i.e., for any v V β, 49) a β P 1 v,w 1 )=a β v, I 1w 1 ), w V β 1.

18 1130 HENGGUANG LI z z 0,1) T z 1,1) 0,0) T 1 1,0) r T r 0,0) T 1,0) r Figure. Reference triangles ˆT 1 and ˆT corresponding to different triangles touching the z-axis. Note that P 1 is in fact the projection on V β 1 with respect to a β, ). Then, for any v 1,w 1 V β 1,wehave a β P 1 I 1v 1,w 1 )=a β I 1v 1, I 1w 1 )=a β v 1,w 1 ). Then, for any v V β 1, w V β, a β I 1 v 1, Id I 1P 1 ) )w = a β I 1v 1,w ) a β I 1v 1, I 1P 1 w ) 50) = a β v 1,P 1 w ) a v 1,P 1 I 1P 1 w ) ) =0. For the sae of economy of notation, we use the same notation for the operators Id, I 1, I 1,E,R,andP 1, even though V 0 and V1 are different subspaces of H1 1 Ω) see 9)). Let {x i } be the set of nodes in the triangulation T and d i, be the diameter of the support of the basis function associated with the node x i. Throughout the text, by A B, we mean there are generic constants C 1,C > 0, such that C 1 B A C B. For a triangle T, r max T ):=max x T rx) andr min T ):=min x T rx). For a triangle T T away from the z-axis, we denote by ˆT its standard reference triangle of diameter one. If T touches the z-axis, its reference triangle ˆT will be either ˆT 1 or ˆT Figure ), to preserve the intersection set between T and the z-axis. For v V β and x T, we define ˆv ˆx) :=v x), where ˆx ˆT is the image of x after the affine mapping. We now investigate necessary properties of the inner products and of the operators in Algorithm 4.1. Lemma 4.3. Let T i T be a triangle and h i, be its diameter. Then, vrdrdz h i, v x l )max dl,,rx l ) ), v V β, T i x l T i where x l, 1 l 3, is the vertex of T i and d l, is defined as above. Proof. Case IT i {r =0} = ). Thus, r max T i )/r min T i ) <M with a uniform constant M > 0 for any T i. We map T i to the reference triangle ˆT. Based on the norm

19 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 1131 equivalence on finite dimensional spaces, we have vrdrdz r max T i ) vdrdz = r max T i ) T i ˆv dˆrdẑ T i T i ˆT r max T i )h i, ˆv ˆx l ) h i, v x l )max dl,,rx l ) ), x l T i ˆx l ˆT whereweusedthefactthatr max T i ) max d l,,rx l ) ) for any triangle away from the z-axis. Case II T i {r =0} ). Let {λ m } m=1,,3 be the barycentric coordinates associated with the three vertices of T i. 1. T i {r =0} is a line segment.) We map T i to the the first reference ˆT 1. Without loss of generality, we assume the first node of T i is away from the z-axis. Let r 1 be the distance from its first node to the z-axis. Note that r 1 h i,.based on the norm equivalence on finite dimensional spaces, we have vrdrdz = r 1 vλ 1 drdz = r 1 T i ˆv ˆλ 1 dˆrdẑ T i T i ˆT 1 h 3 i, ˆv ˆx l ) h i, v x l )max dl,,rx l ) ). ˆx l ˆT 1 x l T i. T i {r =0} is a point.) We map T i to the second reference ˆT. Suppose that its third vertex is on the z-axis. Denote by r 1 and r the distances form the first and second vertices of T i to the z-axis, respectively. Then, for any x T i, rx) =r 1 λ 1 x)+r λ x) with r 1 r h i,. Using the norm equivalence on finite element spaces, we have vrdrdz = vr 1 λ 1 + r λ )drdz h i, T i ˆv ˆλ 1 + ˆλ )dˆrdẑ T i T i ˆT h 3 i, ˆv ˆx l ) h i, v x l )max dl,,rx l ) ). ˆx l ˆT x l T i Hence, the lemma is proved for any T i T. Lemma 4.4. The weighted inner product in 45) satisfies v,v ) dimv β ) 1 i v x i )max d i,,rx i ) ), v V β. Proof. Recall dimv β ) 4. Thus, for any v V β,wehave dimv β ) 1 dimv β ) i=1 dimv β ) i=1 v x i )max d i,,rx i ) ) κ) Ii) Ii) ) κ Ii) )v x i )max d i,,rx i ) ), where the function I : N N {0} is such that Ii) =j if the node x i is in the jth layer L j of T. By Definition 3.10, for a node x i L j, the diameter of a triangle

20 113 HENGGUANG LI T l touching x i satisfies h l, j κ j. Therefore, using Lemma 4.3, we have dimv β ) 1 dimv β ) i=1 v x i )max d i,,rx i ) ) κ) Ii) Ii) ) κ Ii) )v x i )max di,,rx i ) ) T l T x i T l ω Tl h l, v x i )max di,,rx i ) ) T l T x i T l ω Tl vrdrdz = ωv rdrdz =v,v ), T l T T l Ω which completes the proof. Lemma 4.5. The spectral radus of A β defined in 46) satisfies 0 <ρa β ) CdimVβ ). Proof. According to the definition, A β is symmetric positive definite SPD) with respect to the inner product, ). Therefore, it has a discrete spectrum and all its eigenvalues are real and positive. Thus, we only need to show, for any V β v 0, 51) A β v,v ) v,v ) CdimV β ). Recall that near the singular vertex Q, the grading parameter κ β = 1/a β < 0.5. If a triangle T i is in the jth layer L j of T,0 j, its diameter h i, κ j j, and so is the diameter d l, of the support of the basis function associated with the vertex x l of T i. Case IT i {r =0} = ). Recall that r max T i )/r min T i ) M is bounded with M>0for any T i. Then, v v )rdrdz r max T i ) v v drdz Ch i, r maxt i ) vdrdz T i T i T i r max T i ) 5) vx l ) vx l )rx l ), x l T i x l T i where we used the usual H 1 -L inverse inequality in the finite element space and the norm equivalence on finite dimensional spaces. Note r max T i ) 1 Ch 1 i,. By the norm equivalence on finite dimensional spaces, r 1 vdrdz r max T i ) 1 vdrdz = r max T i ) 1 T i ˆv dˆrdẑ T i T i ˆT 53) Ch i, ˆv ˆx l ) C vx l )max d l,,rx l ) ). x l T i ˆx l ˆT Case II T i {r =0} ). Let {λ m } m=1,,3 be the barycentric coordinates associated with the three vertices of T i.

21 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS T i {r =0} is a line segment.) We map T i to ˆT 1. Suppose the first node of T i is away from the z-axis. Let r 1 be the distance from the first node to the z-axis. Then, r 1 h i,. Based on the norm equivalence on finite dimensional spaces, 54) ˆv ˆv )ˆλ 1 dˆrdẑ + ˆv ˆλ 1 dˆrdẑ C ˆv dˆrdẑ, ˆT 1 ˆT 1 ˆT 1 since each side above defines the square of a norm for ˆv. Then, v v )rdrdz = r 1 v v )λ 1 drdz T i T i r 1 T i h i, ˆv ˆv )ˆλ 1 dˆrdẑ Cr 1 ˆv dˆrdẑ ˆT 1 ˆT 1 r 1 ˆv ˆx l) v x 55) l)max dl,,rx l ) ), ˆx l ˆT 1 x l T i where we used the scaling argument, the norm equivalence on finite dimensional spaces, and 54). For v V 1, v Ti is a linear function that vanishes on the z-axis. Therefore, v Ti = ν r for a constant ν. Then, we have r 1 vdrdz = νrdrdz = r 1 T i ν ˆλ 1 dˆrdẑ h 3 i,ν T i T i ˆT 1 h i, vx l ) v 56) x l )max dl,,rx l ) ), x l T i x l T i where we used the norm equivalence on finite dimensional spaces.. T i {r =0} is a point.) We map T i to ˆT. Suppose that its first two vertices are not on the z-axis. Denote by r 1 and r the distances from the first vertex and the second vertex of T i to the z-axis, respectively. For any point x T i, rx) =r 1 λ 1 x)+r λ x) andr 1 r h i,. Using the norm equivalence on finite dimensional spaces, we have 57) ˆv ˆv )ˆλ 1 + ˆλ )dˆrdẑ + ˆT ˆv ˆλ 1 + ˆλ )dˆrdẑ C ˆT ˆv dˆrdẑ, ˆT since each side of 57) defines the square of a norm for ˆv. Similarly, we have v v )rdrdz h i, T i h i, ˆv ˆv )ˆλ 1 + ˆλ )dˆrdẑ T i ˆT Ch i, ˆv dˆrdẑ h i, ˆv ˆx l ) v 58) x l )max dl,,rx l ) ). ˆT ˆx l ˆT x l T i For v V 1, we can write the linear function v Ti = l ν lφ l,whereforl =1,, ν l is a constant and φ l is the linear basis function associated with the lth vertex of T i. It was shown that T i r 1 φ l drdz < in [1]. Therefore, T i r 1 v drdz is bounded. We then have r 1 vdrdz = r 1 λ 1 + r λ ) 1 vdrdz h 1 i, T i ˆλ 1 + ˆλ ) 1ˆv dˆrdẑ T i T i ˆT h i, ˆv ˆx l )=h i, vx l ) v 59) x l )max dl,,rx l ) ), ˆx l ˆT x l T i x l T i

22 1134 HENGGUANG LI where the ey step is to use the norm equivalence between ˆT ˆλ 1 +ˆλ ) 1ˆv dˆrdẑ) 1/ and ˆx l ˆT ˆv ˆx l) ) 1/. This can be done because the first term is finite for all linear functions that vanish on the z-axis. Hence, based on 5), 55), and 58), we have for any T i T, 60) v v )rdrdz C v x l )max dl,,rx l ) ), v V β. T i x l T i Summing up 53), 56), and 59), we obtain that for any v V 1 and T i T, r 1 vdrdz C v 61) x l )max dl,,rx l ) ). T i x l T i Consequently, using 60), 61), and Lemma 4.4, for any v V β,wehave A β v,v ) = a β v,v )= v v + βr v)rdrdz T i T T i C v x l )max dl,,rx l ) )) T i T x l T i v x i )max di,,rx i ) ) dimv β )v,v ). 1 i dimv β ) Remar 4.6. Based on Lemma 4.5, we can choose the damping factor 6) γ = O dimv β ) 1) for the multigrid V-cycle, such that the spectral radius 63) 0 <ργ A β ) The multigrid analysis. We now concentrate on the convergence analysis of the multigrid V-cycle defined in Algorithm 4.1 for the axisymmetric equation 1). We first have the smoothing property for the Richardson smoother. Lemma 4.7. Recall the operator R from 48). Then, a β Id R )R l ) 1 v,v l a β Id R l ) )v,v, v V β. Proof. Since R = Id γ A β, by 63), for any 0 j i, wehave a β Id R )Rv i ),v = γ RA i β v,a β v ) γ R j Aβ v,a β v ) = a β Id R )R j v ),v. Then, the lemma is proved by the following telescopic cancellation l)a β Id R )R l v,v ) l 1 j=0 a β Id R )R j v,v ) a β Id R l )v,v ). Lemma 4.8. The error-propagation operator E in 47) is symmetric positive semi-definite with respect to a β, ) for 0.

23 REGULARITY AND MG METHODS FOR AXISYMMETRIC EQUATIONS 1135 Proof. We prove it by induction. For = 0, the lemma holds. Assume the statement holds for 1, 1. We first show the symmetry of E. By 47), 48), 49), and the assumption for E 1,wehave a β E v,w )=a β R l Id P 1 + E 1 P 1 )R l v,w ) = a β R l v, Id P 1 )Rw l ) + aβ E 1 P 1 Rv l,rw l ) = a β R l v, Id P 1 )Rw l ) + aβ R l v,e 1 P 1 Rw l ) = a β v,r l Id P 1 + E 1 P 1 )R l w ) = aβ v,e w ). We now show that E is positive and semi-definite. By 47), 49), and the assumption on E 1, a β E v,v ) = a β Rv l,rv l ) a β Id E 1 )P 1 Rv l,p 1 Rv l ) = a β Id P 1 )Rv l, Id P 1 )Rv l ) +a β E 1 P 1 Rv l,p 1 Rv l ) 0. In the next two lemmas, we give approximation results needed in the analysis. Lemma 4.9. Let E be H 1 +Ω) for β =0and be H 1 Ω) for β =1. For any v V β and 1, we have Id I 1P 1 )v CdimV β ) 1/ Id I 1P 1 )v E. Proof. We now give the proof for the case β = 1. Consider the auxiliary variational problem: Find ζ H,0Ω), 1 such that 64) a 1 ζ,ξ)= ωid I 1P 1 )v ξrdrdz, ξ H,0Ω), 1 Ω where a 1 ζ,ξ) =L 1 ζ,ξ) L 1 Ω) is the bilinear form in 11) and ω is the function defined in 44). Let ζ 1 V β 1 be the finite element solution of 64) on the 1st level mesh. Recall the layer L j of the triangulation T in 43). Then, since the regularity index 0 <a 1 < 1, by Theorem 3.5, 4), 44), and 64), we have ζ I 1ζ 1 H 1 Ω) = ζ ζ 1 H 1 Ω) CdimV1 1) 1 L 1 ζ K 0 a 1 1, Ω) dimv 1 ) 1 L 1 ζ K 0 a 1 1, Ω) = dimv1 ) 1 CdimV 1 ) 1 j=0 ϑ 1 a 1 L 1 ζ L 1 L j) j=0 κ 1 a 1)j 1 L 1 ζ L 1 L j) = CdimV1 ) 1 65) = CdimV 1 ) 1 Id I 1P 1 )v. Then, by 50), 65), and Lemma 4.4, we have Id I 1P 1 )v = a 1 ζ,id I 1P 1 ) )v = a 1 ζ I 1 ζ 1, Id I 1P 1 ) )v ζ I 1ζ 1 H 1 Ω) Id I 1P 1 )v H 1 Ω) CdimV 1 ) 1/ Id I 1 P 1 κ j 1 j L 1 ζ L 1 L j) j=0 )v Id I 1 P 1 )v H 1 Ω).

24 1136 HENGGUANG LI Thus, we obtain for β =1, Id I 1P 1 )v CdimV 1 ) 1/ Id I 1P 1 )v H 1 Ω). The proof for β = 0 follows similarly. Lemma Recall the space E from Lemma 4.9. For any v V β and 1, Id I 1P 1 )v E CdimV β ) 1/ A β v. Proof. Using the Cauchy-Schwarz inequality, 50), and Lemma 4.9, we have Id I 1P 1 )v E = a β Id I 1P 1 )v, Id I 1P 1 ) )v = a β Id I 1P 1 ) )v,v =Id I 1P 1 )v,a β v ) Id I 1P 1 )v A β v CdimV β ) 1/ Id I 1P 1 )v E A β v, which completes the proof. Then, we have the estimate for the error-propagation operator E in 47). Theorem Recall the number of smoothing steps l in Algorithm 4.1. Then, 66) a β E v,v ) C l + C a βv.v ), v V β, where C>0 is independent of. Proof. Recall the space E defined in Lemma 4.9. By Lemma 4.10, 6), and Lemma 4.7, we first have a β Id P 1 )Rv l, Id P 1 )Rv l ) 67) = Id P 1 )Rv l ) E CdimV β ) 1 a β A β Rl v,rv l ) = CdimV β ) 1 γ 1 a β Id R )R l v,r l v ) Ca β Id R )R l v,v ) C l a β Id R l )v,v ). We now give the proof by induction. For = 0, 66) holds since E 0 = 0. Assume 66) holds for 1, 1. Let c = Cl + C) 1 where C is the constant in 67). Note that c =1 c)cl 1. Then, by 47), 49), the assumption on E 1,and67), a β E v,v )=a β Id P 1 )Rv l, Id P 1 )Rv l ) +a β E 1 P 1 Rv l,p 1 Rv l ) a β Id P 1 )Rv l, Id P 1 )Rv l ) + caβ P 1 R l v,p 1 R l v ) =1 c)a β Id P 1 )R l v, I P 1 )R l v ) + caβ R l v,r l v ) 1 c)cl 1 a β Id R l ) )v,v + caβ R l v,r l v ) = ca β Id R l ) )v,v + caβ R l v,r l v )=ca β v,v ), which completes the proof. Hence, the multigrid V-cycle algorithm is a contraction with contraction number strictly less than 1, independent of the mesh level.