A Mixed Method for Axisymmetric Div-Curl Systems

Transcription

1 Portland State University PDXScholar Mathematics and Statistics Faculty Publications and Presentations Fariborz Maseeh Department of Mathematics and Statistics 2008 A Mixed Method for Axisymmetric Div-Curl Systems Dylan M. Copeland Jay Gopalakrishnan Portland State University, gjay@pdx.edu Joseph E. Pasciak Texas A & M University - College Station Let us know how access to this document benefits you. Follow this and additional works at: Part of the Algebra Commons, and the Applied Mathematics Commons Citation Details Copeland, Dylan M.; Gopalakrishnan, Jay; and Pasciak, Joseph E., "A Mixed Method for Axisymmetric Div-Curl Systems" (2008). Mathematics and Statistics Faculty Publications and Presentations. Paper This Post-Print is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. For more information, please contact pdxscholar@pdx.edu.

2 A MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS DYLAN M. COPELAND, JAYADEEP GOPALAKRISHNAN, AND JOSEPH E. PASCIAK Abstract. We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.. Introduction We present a mixed method for solving three-dimensional axisymmetric div-curl systems. We are interested in the case when symmetry with respect to an axis permits a reduction of the computational domain from three to two dimensions via cylindrical coordinates. The dimension reduction results in computationally efficient methods for applications in magnetostatics. However, the necessary use of cylindrical coordinates complicates the mathematical analysis of numerical methods. In particular, the coordinate transformation gives differential operators with singularities on the axis and function spaces weighted by the radial variable are necessary. This note gives a few techniques to overcome the difficulties caused by the degenerate weights. Specifically, we construct simple interpolation operators satisfying a commuting diagram property involving a sequence of weighted spaces. This allows the application of standard techniques in the theory of mixed methods to axisymmetric div-curl systems. An efficient method for the dimension-reduced div-curl system was recently proposed in [9], using a negative-norm least-squares technique. This method was shown to have excellent conditioning independent of jumps in the coefficient, but it allowed only for a piecewise constant coefficient. As an alternative, in this paper we present a mixed method which allows for a piecewise smooth coefficient. Another advantage is that the implementation of the mixed method involves finite element spaces more commonly used than those of the negative-norm least-squares method. The mixed method simply uses standard two-dimensional Nédélec elements [7] and continuous piecewise linear functions. Thus the use of cylindrical coordinates does not necessitate special finite element spaces. As a model problem, we consider Maxwell s system for magnetostatics. Let Ω R 3 be a convex, simply-connected, bounded domain in R 3, with a connected boundary. We assume that Ω is axisymmetric, i.e., it is invariant under rotations about the z-axis. Moreover, 2000 Mathematics Subject Classification. 65F0, 65N30, 78M0, 74G5, 78A30, 35Q60. Key words and phrases. mixed methods, Nedelec elements, axisymmetry, Maxwell s equations, div-curl systems, magnetostatics. This work was supported in part by the National Science Foundation through grants DMS , SCREMS , and DMS

3 2 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK let D be a polygonal domain representing the cross-section, i.e., Ω is the revolution of D about the z-axis. We assume that Ω intersects the z-axis on an interval of positive length. If Ω does not intersect the z-axis, then the analysis of the problem is simple and standard finite element methods can be applied. The magnetostatic system is curlh = J in Ω (.) div(µh) = 0 in Ω µh n = 0 on Ω, where H is the magnetic field, J is the current density, µ is the magnetic permeability, and n is the unit outward normal on Ω. We assume that J is in L 2 (Ω) 3 and satisfies the compatibility condition divj = 0. The coefficient µ is assumed to be piecewise smooth and to satisfy the bounds 0 < µ 0 µ(x) µ for all x Ω, where µ 0 and µ are positive constants. A standard technique to solve the magnetostatic system is via vector potentials. Define B = µh, the magnetic intensity, and note that divb = 0. It is well known (see e.g. [] or [0, Theorem I.3.6]) that our assumptions on Ω imply the existence of a vector potential A in H 0 (curl, Ω) satisfying B = curla and diva = 0. Substituting H = µ B into (.), we obtain the div-curl system curlµ curla = J in Ω (.2) diva = 0 in Ω A n = 0 on Ω. A scalar-valued function is said to be invariant under rotation if it is constant with respect to the angular variable in cylindrical coordinates, and a vector-valued function is said to be axisymmetric if its component functions in cylindrical coordinates are invariant under rotation. We assume that µ is invariant under rotation and J is axisymmetric. Therefore, we seek an axisymmetric solution A (cf. [5]). It is well known that (.2) decouples into two independent systems in the presence of axial symmetry [3]. To exhibit the decoupled system, let D be as before, namely D = {(r, z) : (r, θ, z) Ω}, where we have used cylindrical coordinates (r, θ, z). We assume that D has a polygonal boundary D, which can be expressed as the disjoint union D = Γ 0 Γ, where Γ = {(r, z) D : r > 0}. Writing the system (.2) in cylindrical coordinates and assuming that the three components in A = A r e r + A θ e θ + A z e z are θ-independent, we obtain the two decoupled systems (.3) curl rz µ curl rz (A r, A z ) = (J r, J z ) in D, div rz (A r, A z ) r r (ra r) + A z = 0 in D, z (A r, A z ) ( n z, n r ) = 0 on Γ, and (.4) r ( µ r ) r (ra θ) ( µ A θ z z ) = J θ in D, A θ = 0 on Γ.

4 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 3 Here, we have used row-vector notation, n r and n z are the r and z-components of n, respectively, and the two curl operators are defined by (.5) curl rz (A r, A z ) z A r r A z and curl rz φ ( z φ, r r(rφ) ), where we have abbreviated / r to r, etc. The first system, namely (.3), forms the so-called meridian problem, while (.4) represents the azimuthal problem. Finite elements for such problems were considered in [8, 4], although no error analysis in the natural weighted spaces was given. The finite element suggested in [4] for the azimuthal problem was independently arrived at in [], where a convergence analysis was also provided. Additionally, in [], the convergence of a discretization of the azimuthal problem using a more standard element (namely, the bilinear finite element), as well as the uniform convergence of a V-cycle multigrid algorithm, was also proved. In this paper, we will analyze finite elements for the meridian problem (.3). The main new ingredient here is an interpolation operator for a weighted H(curl)-space described next. Due to the transformation to cylindrical coordinates, the subsequent analysis of problem (.3) involves function spaces weighted by the radial variable r. We denote by L 2 α (D) the weighted Lebesgue space of measurable functions v on D bounded in the norm v L 2 α (D) = ( D rα v 2 dr dz) /2. Our notation for the natural norm on any Banach space X is simply X, but for convenience in the case of the often used space L 2 (D) (i.e., when α = ), we abbreviate L 2 (D) to r. Additionally, we denote the inner product in L 2 (D) by (u, v) r = r uv dr dz D and abbreviate the norm H (D) to,r. The weighted Sobolev space Hα k (D) for any k is the space of all functions in L 2 α (D) whose distributional derivatives of order at most k are in L 2 α (D). We shall need the following additional weighted spaces: H r (curl, D) = {v L 2 (D)2 : curl rz v L 2 (D)}, H, (D) = {φ H (D) : φ = 0 on Γ }, H (D) = {φ L 2 (D) : grad rz φ L 2 (D) 2 }. Here grad rz φ ( r φ, z φ) and the norm on H r (curl, D) is defined by v r,curl = ( v 2 r + curl rzv 2 r) /2. It can be shown that the trace condition in the definition of H, (D) is well defined (see [5] or Lemma A.). These spaces have occurred in the literature previously, and we shall use their properties established in recent papers studying the axisymmetric Stokes problem [4] or the axisymmetric Maxwell system [3], as well as in earlier books [5, 3]. The remainder of the paper is outlined as follows. Section 2 is devoted to carefully defining the tangential trace operator on the weighted space H r (curl, D). In Section 3, we define and analyze a mixed formulation for (.3) on the continuous level. The analysis proceeds by verifying the standard conditions in the theory of mixed methods [7]. In Sections 4 and 5, we consider discretization and approximation based on rectangular and triangular meshes, respectively. Section 6 contains an analysis of the resulting discrete mixed formulations, simultaneously treating the cases of rectangular and triangular meshes. Numerical results are given in Section 7.

5 4 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK 2. The Tangential Trace Operator The purpose of this section is to define the tangential trace of meridian vector fields in H r (curl, D). This is necessary to introduce a variational formulation incorporating the boundary condition of (.3), namely (A r, A z ) t = 0 on Γ, where t = ( n z, n r ) denotes the unit tangent vector to D in the r-z plane, oriented counterclockwise. We will first show that smooth vector fields are dense in H r (curl, D). This will show that an integration by parts formula that holds for smooth functions also holds for H r (curl, D) functions. The continuity of a trace operator follows. Before proving the density result, we first set some more notations. Let D(R 2 ) denote the space of infinitely smooth real-valued functions with compact support in R 2, and set D(D) = {φ D : φ D(R 2 )}. Also, let R 2 + = {(r, z) : r > 0}. Note that for any w in L 2 (R2 + ) and φ in D(R2 + )2, the action of the distribution curl rz w on rφ, denoted by curl rz w, rφ, satisfies (2.) curl rz w, rφ = r w curl rz φ dr dz, R 2 + which also equals rcurl rz w, φ. Here, products with r are to be considered in the sense of products of distributions with smooth functions. Proposition 2.. The space of smooth vector fields D(D) 2 is dense in H r (curl, D). Proof. Denoting the dual of a Hilbert space H by H, we show that any functional in H r (curl, D) which vanishes on D(D) 2 also vanishes on H r (curl, D). The proposition then follows from Banach space theory. (This technique is also used in [0, Theorem I.2.4].) Hence, we start by considering a linear functional l in H r (curl, D) that vanishes on D(D) 2. Since H r (curl, D) is a Hilbert space, the Riesz Representation Theorem yields some u in H r (curl, D) such that (2.2) l(v) = (u,v) r,curl for all v in H r (curl, D), where (, ) r,curl is the inner product in H r (curl, D) with diagonal 2 r,curl. The remainder of this proof proceeds by observing the properties of the function u in (2.2). Let Z : L 2 (D) L 2 (R 2 +) denote the operator that trivially extends functions by zero into R 2 +. Also for vector functions, let Z(u r, u z ) = (Zu r, Zu z ). Then r ( Zu η + (Zcurl rz u)(curl rz η) ) dr dz = l(η D ) = 0 for all η in D(R 2 +) 2. R 2 + By (2.), this implies that r(zu + curl rz Zcurl rz u) = 0 in the distributional sense. But since rzu is in L 2 (R 2 +) 2, we find that the same equality must hold in L 2 (R 2 +) 2, or equivalently, (2.3) Zu = curl rz (Zcurl rz u) holds in L 2 (R2 ) 2. Next, let v denote the revolution of v, i.e., for any v in L 2 (D), define v a.e. in Ω by v(r, θ, z) = v r (r, z)e r + v z (r, z)e z. It is proved in [3] that for any v in H r (curl, D), the revolved function v is in H(curl, Ω). Hence for the u in (2.2), the function curlŭ is in L 2 (Ω) 3. It is easy to see that if we revolve the functions on either side of (2.3) we obtain (2.4) Z Ω ŭ = curl (Z Ω curlŭ) in L 2 (R 3 ) 3,

6 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 5 where Z Ω : L 2 (Ω) 3 L 2 (R 3 ) 3 is the extension by zero. Thus, Z Ω curlŭ is in H(curl, R 3 ), and hence curlŭ is in H 0 (curl, Ω). We need to show that l(v) = 0 for all v in H r (curl, D). Since curlŭ in H 0 (curl, Ω) and v is in H(curl, Ω), by an integration by parts formula (see e.g. [6, Theorem 3.3]), 2π l(v) = 2π (u,v) r,curl = (ŭ, v) L 2 (Ω) 3 + (curlŭ,curl v) L 2 (Ω) 3 = (ŭ, v) L 2 (Ω) 3 + (curlcurlŭ, v) L 2 (Ω) 3 = 0. Thus l vanishes on H r (curl, D), and the proof is complete. Next, we establish an integration by parts formula. Functions in H (D) are well known to have zero trace on Γ 0 [3, 5, ]. Since H (D) is a subspace of H (D), and since the latter is well known to have traces in L 2 (Γ ) (cf. [, Proposition 2.]), we can define the trace operator γ t : D(D) 2 H (D) by (2.5) γ t (v), φ = r v t φ ds for all φ H (D). Γ Here, and elsewhere later, with slight abuse of notation we continue to use, for the duality pairing in different spaces. For example, in (2.5) it is the duality pairing in H (D), while in (2.) it is the duality pairing in D(R 2 + )2. The space will be clear from context. The next result shows that the domain of γ t can be expanded. Proposition 2.2. The trace operator extends to a continuous linear map from H r (curl, D) into H (D). Moreover, the integration by parts formula (2.6) γ t (v), φ = (v,curl rz φ) r (curl rz v, φ) r holds for all v in H r (curl, D) and φ in H (D). Proof. Equation (2.6) obviously holds for all smooth functions v in D(D) 2 and φ in D(D). By [, Lemma 3.], the set {φ D(D) : φ vanishes in a neighborhood of Γ 0 } is dense in H (D). Hence, for any fixed v in D(D)2, the identity (2.6) holds for all φ in H (D). Note that here we have used the easily verifiable fact that the map φ curl rz φ is continuous from H (D) to L 2 (D) 2. Finally, by the density asserted in Proposition 2., the identity (2.6) holds for all v in H r (curl, D). The continuity of the trace operator follows by standard arguments (see e.g., the proof of [0, Theorem I.2.5]). In the next section, we will present a mixed formulation for (.3). The boundary condition there becomes an essential boundary condition, so we will use the subspace H r, (curl, D) = ker γ t in our variational formulation. Clearly, by Proposition 2.2, H r, (curl, D) is a closed subspace of H r (curl, D). Moreover, from (2.5) we see that smooth functions in ker γ t have tangential components that vanish on Γ. 3. Mixed Formulation We can derive a mixed formulation of the meridian problem by multiplying the first two equations of (.3) by test functions as usual and integrating by parts. The process is similar to the derivation of the standard mixed formulation [2, 6] for the three dimensional magnetostatics equations. The appropriate integration by parts formula in weighted spaces for the curl and divergence operators are given in Proposition 2.2 and [9,

7 6 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK Lemma ], respectively. The resulting weak formulation seeks u H r, (curl, D) and p H, (D) satisfying (3.) { (µ curl rz u, curl rz v) r + (v,grad rz p) r = (f,v) r for all v H r, (curl, D), (u,grad rz q) r = (g, q) r for all q H, (D), where f = (J r, J z ) and g = div rz (A r, A z ). Although (A r, A z ) is divergence-free in the model problem, we allow for g to be any function in L 2 (D). In the remainder of this section, we quickly verify the conditions of the Babuška-Brezzi theory [7] to conclude the unique solvability of (3.). Defining the linear operator B : H r, (curl, D) H, (D) by Bv(q) = (v,grad rz q) r, we denote W 0 = ker B, i.e. W 0 = {v H r, (curl, D) : (v,grad rz q) r = 0 for all q H, (D)}. Then we have the following theorem. In this theorem and in the remainder of the paper, the letter C will denote a generic constant whose value may vary in different occurrences. Theorem 3.. The following holds for the axisymmetric mixed formulation (3.). () Inf-sup condition: There exists a constant C > 0 such that C p,r (v,grad sup rz p) r, for all p H, v H r, (curl,d) v (D). r,curl (2) Coercivity on ker B: There exists a constant C > 0 such that v r,curl Cµ /2 µ /2 curl rz v r, for all v W 0. (3) Unique solvability: There is a unique u in H r, (curl, D) and a unique p in H, (D) satisfying (3.). Proof. () Denote by H 0(Ω) the functions in H0(Ω) invariant under rotation. The mapping H 0 (Ω) H, (D) defined by v(r, z) = v(r, θ, z) is an isomorphism (see II.4 of [5]). Applying this isomorphism and the Poincaré inequality on H0 (Ω) we get (3.2) 2π p 2,r p 2 H (Ω) C grad p 2 L 2 (Ω) = 2πC grad rzp 2 r, where grad( ) denotes the three dimensional gradient operator. Note that for any p in H, (D), the gradient grad p is in H 0 (curl, Ω) and hence grad rz p is in H r, (curl, D) (cf. [3, Proposition 3.9]). Moreover, curl rz (grad rz p) = 0 implies C p,r (grad rzp,grad rz p) r (v,grad sup rz p) r. grad rz p r,curl v H r, (curl,d) v r,curl (2) Let v W 0 and consider the revolved vector field v = v r e r + v z e z defined in Ω. Clearly v n = 0 on Ω. By [3, Proposition 3.9], v is in H(curl, Ω). Let us now also show that (3.3) ( v,grad q) L 2 (Ω) = 0, for all q H 0(Ω).

8 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 7 The left hand side is ( v,grad q) L 2 (Ω) = where D = D 2π 0 2π 0 = 2π D r v grad q dθ dr dz r (v r r q + v z z q) dθ dr dz r (v r r q 2π + v z z q) dr dz, q(r, z) = q(r, θ, z) dθ 2π 0 and the interchanges in the order of integration and differentiations are justified as usual. Since q is in H0 (Ω), by [5, Theorem II.3.], the averaged function is in H, q (D). Consequently, the above calculation implies that ( v,grad q) L 2 (Ω) = 2π(v, grad rz q ) r = 0, as v is given to be in W 0. By a standard Poincaré-Friedrichs estimate [, 0] (sometimes also called Weber inequality [3, 8]) for functions satisfying (3.3), we have v L 2 (Ω) C curl v L 2 (Ω). Since v is axisymmetric, curl v = (curl rz v)e θ, so the above inequality implies a corresponding estimate on D, namely v r C curl rz v r Cµ /2 µ /2 curl rz v r. This is equivalent to the desired inequality. (3) The final statement follows by applying the Babuška-Brezzi theory for mixed systems, as the previous two statements of the theorem verified the assumptions required to apply the theory. 4. Approximation on Rectangular Meshes In this section, we introduce finite element spaces based on rectangular meshes, which approximate the spaces H r (curl, D) and H (D). Triangular finite elements are given in the next section. Although triangular elements cover more general domains, we include the rectangular case, because its analysis readily reveals the main ideas and they are simple to implement when applicable. The rectangular finite element functions are simple bilinear functions for H (D) and the two-dimensional edge-based Nédélec functions for H r (curl, D). To establish notation for our finite element spaces, first define the polynomial spaces Q, = {p(r, z) = c 0 + c r + c 2 z + c 3 rz : c i R for 0 i 3}, ND, = {(a + bz, c + dr) : a, b, c, d R}.

9 8 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK Given a rectangular mesh T h aligned with the coordinate axes, define the finite element spaces Additionally, let (4.) V h = {u H (D) : u K Q, for all K T h }, W h = {v H r (curl, D) : v K ND, for all K T h }, S h = {u L 2 (D) : u K is constant, for all K T h }. V h, = {v V h : v = 0 on Γ }, W h, = {v W h : v t = 0 on Γ }. Our aim in this section is to construct projectors Π V h, ΠW h element spaces such that the following diagram commutes. and ΠS h onto the above finite (4.2) H(D) Π V h grad rz Hr (curl, D) Π W h curl rz L 2 (D) Π S h V h grad rz curl rz Wh Sh. The above is somewhat of an abuse of notation as the first two projectors are defined on sufficiently smooth functions in H(D) and H r (curl, D), respectively. The importance of such a property in the analysis of mixed methods with the usual (unweighted) Sobolev spaces is well known. As in the case of the standard unweighted Sobolev spaces, it is quite technical to construct projectors whose domain is the entire corresponding Sobolev space appearing in (4.2). Hence, the projectors we construct will require further regularity. We first define the interpolation operator Π V h by its action element by element. On every element K whose boundary does not intersect Γ 0, (Π V h v) K is the unique bilinear function whose value at each of the four vertices of K coincides with that of v. The remaining elements have vertices of the form (0, z 0 ), (h r, z 0 ), (h r, z ), and (0, z ), with z = z 0 + h z. On these elements, we define new degrees of freedom by (4.3) m (v) = v(h r, z 0 ), m 2 (v) = m 3 (v) = v(h r, z ), m 4 (v) = hr 0 hr 0 r r v(r, z 0 ) dr, r r v(r, z ) dr. The interpolant Π V h v on such an element is the unique function in Q, satisfying m i (Π V h v) = m i (v) for i =,...,4. Remark 4.. The definition and analysis of the above interpolant are very similar to those of an interpolation operator defined in [], where the weight r /2 was used in place of r in defining m 2 and m 4. We chose the r weight for convenience. A similar analysis could be provided for the r /2 weight. In [], the reason for the introduction of integral degrees of freedom was (incorrectly) given to be the lack of continuity of the functional v v(0, z) in H 2(D). However, this functional is indeed continuous in H2 (D), as shown in [5, Theorem 4.7]. Relying on this continuity, an alternate approach to approximation theory in H 2 (D) is pursued in [5]. However, [5] does not give the other projectors and the commutativity properties in (4.2) needed for the analysis of mixed methods.

10 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 9 Next, we define the projection Π W h onto W h. On rectangles K not intersecting Γ 0, Π W h is the standard Nédélec interpolant defined by the unweighted integrals of the tangential components along edges, i.e., the degrees of freedom are e (ΠW h v) t ds for all edges e of K. On the remaining rectangles, the degrees of freedom are (4.4) M (v) = M 3 (v) = (hr,z 0 ) (0,z 0 ) (0,z0 +h z) r v t ds, M 2 (v) = r v t ds, M 4 (v) = (h r,z 0 +h z) K (hr,z 0 +h z) (h r,z 0 ) v t ds, r curl rz v dr dz. Here a 2 a denotes the straight line integral from the point a to the point a 2 in the r-z plane. For any function v for which the moments M i exist, we define Π W h v K as a function in ND, satisfying M i (Π W h v) = M i(v) for i =,...,4. That Π W h v is uniquely defined this way follows from the next lemma. Finally, for the space S h at the end of (4.2), we define Π S h w for any w in L2 (D) as the unique function in S h such that on a mesh element K, its restriction (Π S h w) K satisfies r (Π S h w) v dr dz = r w v dr dz if K intersects Γ 0, and K K (4.5) (Π S h w) v dr dz = w v dr dz otherwise, K K for all v in S h. With these definitions, we have the following lemma verifying the commutative properties in (4.2). Lemma 4.. The interpolation operators defined above have the following properties. () The degrees of freedom in (4.3) are continuous linear functionals on H 2 (D) and are unisolvent. Hence Π V h is well defined and continuous on H2 (D). (2) The degrees of freedom in (4.4) are continuous linear functionals on H (D)2 and are unisolvent. Hence Π W h is well defined and continuous on H (D) 2. (3) The commutative properties curl rz Π W h v = Π S hcurl rz v, Π W h grad rz u = grad rz Π V h u, hold for any u and v for which all of the above quantities are bounded. (4) Π V h and ΠW h preserve zero boundary conditions, i.e., if u is in H, (D) H 2 (D), then Π V h is in V h,, and if v is in H r, (curl, D) H (D)2, then Π W h v is in W h,. Proof. The continuity of m and m 3 on H 2 (D) follows from the standard Sobolev inequality (because the points involved are away from the z-axis). For m 2 and m 4, the same continuity follows from [, Proposition 2.] or Lemma A.. Similarly, M i is continuous on H (D)2 for i 4. Next, let us prove the unisolvence of {M i }. (We omit the proof of unisolvence of {m i } as it is simpler.) Suppose v h = (a + bz, c + dr) on a rectangle K intersecting Γ 0, with M i (v h ) = 0 for i 4. We need to show that v h vanishes. Since curl rz v h = b d,

11 0 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK M 4 (v h ) = 0 implies b = d. Consequently, v h = grad rz ψ h for some bilinear function ψ h in Q,. It follows from M (v h ) = 0 and M 3 (v h ) = 0 that ψ h / r vanishes on the horizontal edges of K. Thus, ψ h (r, z) = c 0 + c 2 z. Now (hr,z 0 +h z) (h r,z 0 ) grad rz ψ h t ds = M 2 (v h ) = 0 implies ψ h (r, z) = c 0 is constant, i.e., v h = grad rz ψ h = 0. The commutative properties follow in a standard way on elements away from Γ 0. For the remaining elements, the identity curl rz Π W h v = ΠS h curl rzv follows immediately from the definition of M 4, since curl rz w is a scalar constant for any w in ND,. To verify the second identity Π W h grad rzu = grad rz Π V h u, it suffices to show that the degrees of freedom M i agree for grad rz u and grad rz Π V h u, because the gradient of any function in V h is in W h. By the definition (4.3) of Π V h, hr M (grad rz (u Π V h u)) = r ( r u(r, z 0 ) r Π V h u(r, z 0) ) dr = 0, 0 z0 +h z M 2 (grad rz (u Π V h u)) = ( z u(h r, z) z Π V h u(h r, z) ) dz z 0 = u(h r, z 0 + h z ) Π V h u(h r, z 0 + h z ) u(h r, z 0 ) + Π V h u(h r, z 0 ) = 0, 0 M 3 (grad rz (u Π V h u)) = r ( r u(r, z 0 + h z ) r Π V h u(r, z 0 + h z ) ) dr = 0 h r M 4 (grad rz (u Π V h u)) = r curl rz grad rz (u Π V h u) dr dz = 0. K This proves the second commutative identity. Finally, the last assertion on the boundary preservation properties is obvious by construction. We conclude this section by proving approximation properties for the interpolation operators. From now on, whenever we use the generic constant C, its value will always be independent of the mesh size h = max K Th diam(k). Lemma 4.2 (Approximation estimates). There exists a constant C such that (4.6) grad rz (u Π V h u) L 2 (D) Ch u H 2 (D), (4.7) v Π W h v L 2 Ch v (D)2 H (D)2, (4.8) curl rz(v Π W h v) L 2 (D) Ch curl rz v H (D), (4.9) w Π S hw L 2 (D) Ch w H (D), for all u H 2 (D), w H (D), and v H (D)2. For (4.8), we additionally require curl rz v H (D). Proof. These estimates are established by proving analogous local estimates on each element, squaring them, and summing. For elements away from the z-axis, standard arguments prove the estimates. For the remaining elements K intersecting Γ 0, we will now

12 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS show that all of the estimates follow from the fact that whenever φ in H(K) satisfies any one of the conditions (4.0) r φ(r, z) dr = 0, for an edge e of K having a vertex in Γ 0, or e (4.) φ(r, z) dr = 0, for the vertical edge e of K not intersecting Γ 0, or e (4.2) r φ(r, z) dr dz = 0, then K (4.3) φ L 2 (K) Ch φ H (K). That (4.3) holds under conditions (4.0) and (4.) is contained in [, Lemma 2.]. Its proof under condition (4.2) is very similar, so we omit it. Consider the last operator Π S h first. Since φ = (w ΠS h w) K satisfies (4.2), the estimate φ L 2 (K) Ch w H (K) follows from (4.3) and Π S h w H (K) = 0. This proves (4.9). To prove estimate (4.7) for Π W h, let (φ r, φ z ) = v Π W h v and observe that by construction, φ r satisfies (4.0) and φ z satisfies (4.). Hence φ r L 2 (K) Ch φ r H (K), φ z L 2 (K) Ch φ z H (K). It is easy to verify by direct calculation that Π W h v H C v (K)2 H (K)2. Using this to bound the right hand sides above, we get (4.7). Estimate (4.8) follows from Lemma 4., because curl rz (v Π W h v) L 2 (D) = curl rz v Π S h curl rzv L 2 (D) by Lemma 4.(3), Ch curl rz v H (D) by (4.9). Proof of the final estimate proceeds similarly using Lemma 4.: grad rz (u Π V h u) L 2 (D) = grad rz u Π W h grad rz u L 2 (D) by Lemma 4.(3), Ch grad rz u H (D) by (4.7), from which (4.6) follows. 5. Approximation on Triangular Meshes We now consider triangular meshes, proving results analogous to those given in the previous section for rectangular meshes. In particular, we want to construct projectors onto triangular finite element spaces satisfying the commuting diagram property (4.2) and possessing optimal approximation properties. With triangular elements, we are now able to mesh more general polygonal domains D. Let T h be a mesh of D consisting of triangles, satisfying the usual finite element assumptions. For simplicity, we assume that T h is quasiuniform. Define the polynomial spaces P = {p(r, z) = c 0 + c r + c 2 z : c i R for 0 i 2}, ND = {(a bz, c + br) : a, b, c R}.

13 2 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK Then for each v in ND (), v t is constant along each edge of. The global finite element spaces are V h = {u H (D) : u P for all T h }, W h = {v H r (curl, D) : v ND for all T h }, S h = {u L 2 (D) : u R is constant, for all T h }. One obvious set of global degrees of freedom for V h consists of the function values at all mesh vertices. However, we cannot consider the nodal interpolant corresponding to these degrees of freedom, as the values on the z-axis are undefined for functions in our weighted spaces. As in the rectangular case, we therefore define new degrees of freedom through weighted edge moments. In the triangular case, there is an additional complication due to multiple interior edges meeting at a vertex on the z-axis. Hence, we associate with each vertex a i in Γ 0 one edge e(a i ) such that a i is an endpoint of e(a i ) and e(a i ) Γ 0. If a i is an endpoint of Γ 0, then we choose e(a i ) to be the mesh edge connected to a i lying on Γ. Then, we set one global degree of freedom per mesh vertex by σi V (u) = u(a i), if a i / Γ 0, (5.) σi V (u) = r t grad rz u ds, if a i Γ 0. e(a i ) Proposition 5.. The linear functionals in (5.) are continuous on H 2 (D) and form unisolvent degrees of freedom for V h. Proof. For a i / Γ 0, by a standard Sobolev inequality, while for a i Γ 0, by Lemma A., σ V i (u) = σ V i (u) = u(a i ) c i u H 2 (D), e(a i ) r t grad rz u ds c i u H 2 (D), for some constants c i independent of u. Hence, the degrees of freedom are continuous linear functionals on H(D). 2 It only remains to show that these degrees of freedom are unisolvent. If v h V h has σi V (v h ) = 0 for all i, then obviously v h vanishes at all vertices a i / Γ 0. It also vanishes on the remaining vertices a i Γ 0 because the zero integral degrees of freedom imply that v h (a i ) must equal the value of v h at the other endpoint of e(a i ). Hence v h 0. Next, we define the set of global degrees of freedom for the Nédélec space W h as σi W (v) = r v t ds for all mesh vertices a i Γ 0, e(a i ) (5.2) σe W (v) = v t ds for all mesh edges e with e Γ 0 =, e σ W (v) = r curl rz v dr dz for all mesh triangles with Γ 0. Note that these differ from the standard Nédélec degrees of freedom [7] in three respects: (i) We use weighted edge moments near Γ 0. (ii) Not all interior edges are used in defining the degrees of freedom. (iii) Elements near Γ 0 have an interior degree of freedom even

14 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 3 though we are only considering the lowest order case. Due to these differences, we must now verify that the new degrees of freedom control W h. Proposition 5.2. The linear functionals in (5.2) are continuous on H (D)2 and form unisolvent degrees of freedom for W h. Proof. To prove unisolvence, suppose that all the degrees of freedom in (5.2) vanish for a function v h in W h. Then, clearly v h vanishes on elements not intersecting Γ 0. It remains to show that v h vanishes on the remaining elements, which we collect into a set S. The elements in S can be classified in two types: for k = or 2, we say that a triangle in S is of type k if has exactly k vertices on Γ 0 (see Figure ). For any S, writing v h = (a bz, c + br) and using the fact that the interior degree of freedom is zero, we have 2b r dr dz = r curl rz (a bz, c + br) dr dz = r curl rz v h dr dz = 0. Hence b = 0 and v h is constant on. Therefore, to prove that v h vanishes on any in S, it suffices to show that the tangential component of v h is zero on any two edges of. Note that around any interior vertex of Γ 0, there are two triangles of type two separated by a number (possibly none) of type one triangles. The case of an end point is similar but only involves one triangle of type two. Now let us prove that v h vanishes on all type one triangles in S. Let be a type one triangle and let a be its vertex on Γ 0 (including the endpoints of Γ 0 ). Then, e(a) cannot be an edge shared by type two triangles only, i.e., there is at least one type one triangle, say, having e(a) as an edge (see Figure ). This triangle need not coincide with. It has an edge not intersecting Γ 0 where the tangential component of v h is given to be zero. Therefore, v h t is zero on two edges of, hence v h vanishes on. We can repeat this argument on any type one triangle adjacent to (since v h t = 0 on all edges on ). Therefore v h vanishes on all type one triangles connected to a, including. Next, consider a type two triangle. It has two edges not on Γ 0. We claim that v h t vanishes on these two edges. Indeed, if e is such an edge, and e is shared by a type one triangle, then by the conclusion of the previous paragraph, v h t = 0 on e. Also, if e is on Γ, then obviously v h t = 0 on e. The only other possibility is that e is shared by another type two triangle. But in this case, e coincides with e(a) where a is the common vertex of and on Γ 0. Obviously, v h t = 0 on e(a) since all degrees of freedom are given to be zero. Thus, v h t vanishes on two edges of each type two triangle, so v h vanishes on type two triangles as well. The continuity of the linear functionals on H (D)2 follows easily from Lemma A. and standard results. Define the projector Π V h onto V h by σ V i (ΠV h v) = σv i (v), for all the degrees of freedom σ V i in (5.). Similarly define the projector Π W h onto W h using all the degrees of freedom in (5.2). By Propositions 5. and 5.2, these projectors are well defined. In particular, Π W h u for any u in H (D)2 automatically satisfies the tangential continuity constraints of W h along all mesh edges even though not all edges are degrees

15 4 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK a j+ a j z e(a j ) type 2 0 ẑ ˆ 2 ˆr ẑ a i type 0 ˆ ˆr 0 r (a) (b) Figure. (a) Illustration of the notations used in some of the proofs in this section. The edges that define degrees of freedom connected to vertices a i, a j and a j+ are marked by thicker lines. (b) The two types of mappings and the two corresponding reference triangles. of freedom, i.e., e rv t ds and e r(π W h v) t ds are not equal in general. Additionally, we have the following local norm estimate. Proposition 5.3. There exists a constant C such that Π W h v H () 2 C v H ()2, for all mesh triangles in T h and all v in H ()2. Proof. The result is standard in the case when does not intersect Γ 0. Writing Π W h v = (a bz, c + br) on the remaining triangles, we have 2b r dr dz = r curl rz (a bz, c + br) dr dz = r curl rz Π W h v dr dz = σ W (Π W h v) = σ W (v) = r curl rz v dr dz and hence Π W h v 2 H = ()2 r b 2 dr dz = ( r curl rzv dr dz ) 2 4 r dr dz C v 2 H ()2. We also need a projector Π S h onto S h. Its definition is similar to the rectangular case (4.5), namely, on elements intersecting Γ 0, we define (Π S h w) to be the L 2 ()- orthogonal projection of w onto the space of constants on. On the remaining elements,

16 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 5 (Π S h w) is the L 2 ()-orthogonal projection of w onto the space of constants on. Then, we have the following analogue of Lemma 4. in the rectangular case. Lemma 5.. Items (3) and (4) of Lemma 4. hold verbatim in the triangular case with the above defined projectors. Proof. Since the commutative properties are standard for the interpolants based on the standard degrees of freedom, it suffices to verify that they hold on elements with nonempty Γ 0. On such elements, because of the last set of degrees of freedom in (5.2), the identity (5.3) curl rz Π W h v = Π S hcurl rz v obviously holds. To prove the remaining commutative identity, it suffices to show that the degrees of freedom in (5.2) vanish when applied to (Π W h grad rzp grad rz Π V h p). We need only check σi W and σ W as the case of σe W is standard. Now, for any vertex a i in Γ 0, the definition of Π V h gives r grad rz Π V h p t ds = r grad rz p t ds = r Π W h grad rz p t ds, e(a i ) e(a i ) so σi W (Π W h grad rzp grad rz Π V h p) = 0. Finally, if is a triangle in T h intersecting Γ 0 nontrivially, then r curl rz grad rz Π V h p dr dz = 0 = r curl rz grad rz p dr dz = r curl rz Π W h grad rz p dr dz, so σ W(ΠW h grad rzp grad rz Π V h p) = 0. Thus ΠW h grad rzp and grad rz Π V h p agree on all the degrees of freedom in W h. Hence the commutativity property (5.4) Π W h grad rzp = grad rz Π V h p follows. Finally, the assertion of the lemma on the boundary conditions is obvious by the construction of the interpolants (where we use the fact that e(a i ) is contained in Γ if a i is an endpoint of Γ 0 ). In proving an approximation estimate for the interpolation operator Π W h, we shall use the next lemma to control the error on edges not having degrees of freedom. In the triangular case, local arguments of the type we used in the rectangular case are not applicable, as the degrees of freedom are defined more globally. Lemma 5.2. Let be a mesh triangle in T h having edges e, e 2, e 3. Set Ij e r (u) = r u t ds and Ij (u) = u t ds. j e j e(a i ) () If the edge e 3 is contained in Γ 0, then u 2 L 2 ()2 C ( h 2 u 2 H ()2 + h I r (u) 2 + h I r 2(u) 2 ) for all u in H () 2.

17 6 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK (2) If e 3 does not intersect Γ 0, but e and e 2 have a common endpoint on Γ 0, then u 2 L 2 ()2 + h I r (u) 2 C ( h 2 u 2 H ()2 + h I r 2 (u) 2 + h I 3 (u) 2 ) for all u in H () 2. Proof. The triangle is either a type one or type two triangle (using the terminology in the proof of Proposition 5.2). This proof uses scaling arguments (as in [4]) employing maps from two reference triangles, one for each type of. The reference triangles are ˆ and ˆ 2, defined to be the triangles in the ˆr-ẑ plane with vertices {(0, 0), (, ), (, )} and {(0, ), (, 0), (0, )}, respectively (see Figure ). To prove the first estimate, let F be an affine homeomorphism that maps ˆ 2 onto such that F maps the edge of ˆ 2 on the z-axis onto e 3. Map u(r, z) covariantly to define the function û(ˆr, ẑ) = J(F) t u(r, z) on ˆ 2, where J(F) is the Fréchet derivative of F. If λ 3 is the barycentric coordinate function of that vanishes on e 3 Γ 0, then r = h 3 λ 3 where h 3 is the distance from e 3 to the vertex of opposite to e 3. Using this, it is easy to show that (5.5) (5.6) (5.7) u 2 L 2 ()2 C h û 2 L 2 (ˆ 2) 2, û 2 H (ˆ C h u 2 2) 2 H, ()2 2 ˆr û ˆt i dŝ Ch 2 r u t ds ê i e i 2, for i = and 2, where ê i = F (e i ) and ˆt i denotes the unit tangent on ê i. Therefore, once we show that (5.8) C û 2 L 2 (ˆ 2) 2 û 2 H (ˆ 2) 2 + Îr (û) 2 + Îr 2 (û) 2, for all û H (ˆ 2) 2, the first inequality of the lemma follows using (5.5) (5.7). Here, Îr j (û) = ê j ˆr û ˆt dŝ. To establish (5.8), suppose on the contrary that there exists a sequence {v n } n N in H (ˆ 2) 2 such that (5.9) n v n 2 L 2 (ˆ > v 2) 2 n 2 H (ˆ + 2) 2 Îr (v n ) 2 + Îr 2(v n ) 2. Normalizing, we may assume without loss of generality that v n L 2 (ˆ 2 ) = for all n N. Thus (5.9) implies that lim n v n 2 = 0. In particular, {v H (ˆ 2) 2 n } is a uniformly bounded sequence in H (ˆ 2) 2. Consequently, the compact imbedding of H (ˆ 2) 2 in L 2 (ˆ 2) 2 [5, Lemma 4.2] yields a subsequence {v jn } n N converging in L 2 (ˆ 2 ) 2. This convergence in L 2 (ˆ 2 ) 2 and lim n v jn H (ˆ 2 ) = 0 together imply the existence of a v 2 H (ˆ 2 ) 2 such that lim v j n = v in H(ˆ 2 ) 2. n Moreover, since v H (ˆ 2 ) = 0, we find that v is constant. By Lemma A., 2 Îr ( ) and Îr 2 ( ) are continuous linear functionals on H(ˆ 2 ) 2, so we also have that Îj r (v) = (v ˆt j ) r dŝ = 0 for i =, 2. ê j Since the tangent vectors ˆt and ˆt 2 are linearly independent, v ˆt = v ˆt 2 = 0 implies that v vanishes. But this contradicts v 2 L 2 (ˆ 2) 2 = lim n v n 2 L 2 (ˆ 2) 2 =, so we conclude

18 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 7 that (5.8) holds on the reference element ˆ 2. Mapping to a target element we complete the proof of the first inequality of the lemma. The proof of the second inequality of the lemma is similar. For this case, since is a type one triangle, we must use a mapping from the other reference triangle ˆ. Also, now r is given by a different expression, namely r = r λ + r 2 λ 2, where r i denotes the r-coordinate of the vertex of opposite to e i and λ i is the barycentric coordinate function vanishing on e i. But inequalities like (5.5) (5.7) continue to hold, because r /h and r 2 /h are bounded above and below by fixed constants (independent of mesh size, but depending on element angles and the angle Γ makes with Γ 0 ). The argument on the reference element ˆ by contradiction is analogous to that above. Lemma 5.3 (Approximation estimates). The projectors defined in the triangular case also satisfy the estimates of Lemma 4.2, i.e., there exists a constant C such that (5.0) grad rz (u Π V h u) L 2 (D) Ch u H 2 (D), (5.) v Π W h v L 2 (D) Ch v 2 H (D)2, (5.2) curl rz(v Π W h v) L 2 (D) Ch curl rz v H (D), (5.3) w Π S h w L 2 (D) Ch w H (D), for all u H 2 (D), w H (D), and v H (D) 2. For (5.2), we additionally require curl rz v H (D). Proof. First, (5.3) follows from a result already established in the literature [4, Lemma 5]. As in the proof of Lemma 4.2, (5.2) follows from the commutativity property of Lemma 5. and (5.3). The proof of (5.) is more involved. Recalling the notations in the proof of Proposition 5.2 (see Figure ), we first consider a type one triangle with a vertex a on Γ 0. Then there is a sequence of type one triangles T 0, T,..., T n, all connected to a, with T 0 =, and T n having e(a) as an edge. Let l j = T j T j denote the corresponding sequence of edges connected to a. Then, by Lemma 5.2(2) and Proposition 5.3, 2 ) v Π W h v 2 L C (h 2 v 2H(T0)2 + h r (v Π W 2 ()2 h v) t ds, l where we have used the fact that σe W -type degrees of freedom in (5.2) coincide when applied to v and Π W h v. We use Lemma 5.2(2) again, but this time to bound the integral over l appearing above using a seminorm over the next triangle T. Combining with further applications of Lemma 5.2(2) and Proposition 5.3, we have 2 ) v Π W h v 2 L C (h 2 v 2H(T0 T)2 + h r (v Π W 2 ()2 h v) t ds l 2 2 C (h 2 v 2H(T0 T Tn)2 + h r (v Π W h ). v) t ds e(a) The last integral is zero because of the first type of degrees of freedom in (5.2). Thus, if D a denotes the union of all triangles connected to a, we have (5.4) v Π W h v 2 L 2 (Da)2 Ch 2 v 2 H (Da)2,

19 8 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK where we have tacitly used the fact that the number n above is bounded by a fixed constant due to the shape regularity of the mesh. Next consider a type two triangle with edges e, e 2, e 3 such that e 3 Γ 0. Applying Lemma 5.2() and Proposition 5.3, 2 ) v Π W h v 2 L C (h 2 v 2H()2 + h r (v Π W 2 ()2 h v) t ds. e e 2 The portions of the last integral over e i (for i = or 2) vanish if e i is a degree of freedom. Otherwise, they are bounded as above and we conclude that for any type two triangle, (5.5) v Π W h v 2 L 2 ()2 Ch 2 v 2 H (Dx Dx 2 )2 where x i is the vertex of opposite to e i. For triangles not intersecting Γ 0, the estimate (5.6) v Π W h v 2 L 2 ()2 Ch 2 v Π W h v 2 H ()2 Ch 2 v 2 H ()2 follows by standard arguments. Hence by summing over all triangles, and using (5.4), (5.5), or (5.6) as appropriate, we obtain v Π W h v L 2 (D) 2 Ch v H (D)2. This proves (5.). The estimate (5.0) is proved as in the rectangular case using Lemma 5. and commutativity. This completes the proof of all the lemma. 6. The Discrete Mixed Method With the tools constructed in the previous sections, we are now able to analyze a discrete mixed method suggested by the variational formulation in Section 3. The mixed Galerkin method develops approximate solutions in the discrete subspaces considered in the previous sections by requiring them to satisfy the equations of the variational formulation (3.) in the discrete subspaces. To limit the technicalities in the analysis, we assume throughout this section that the revolution of D, namely Ω, is convex. The mixed Galerkin approximation is the function pair (u h, p h ) W h, V h, satisfying { (µ (6.) curl rz u h, curl rz v h ) r + (v h,grad rz p h ) r = (f,v h ) r for all v h W h,, (u h,grad rz q h ) r = (g, q h ) r for all q h V h,. It follows from the next theorem that such a pair is uniquely determined by the above equations. In (6.), the finite element spaces can be either those defined in Section 4 (for rectangular meshes) or those in Section 5 (for triangular meshes), i.e., V h, and W h, are defined by (4.), where V h and W h can be either as defined in Section 4 or as in Section 5. The analysis of this section treats both cases simultaneously, following a standard approach (cf. [6, 6, 7]). Define W 0 h, as the subspace of W h, consisting of discretely divergence-free functions, i.e., W 0 h, = {φ h W h, : (φ h,grad rz p h ) r = 0 for all p h V h, }. Then we have the following theorem (cf. Theorem 3.). Theorem 6.. The following holds for the axisymmetric mixed method (6.).

20 MIXED METHOD FOR AXISYMMETRIC DIV-CURL SYSTEMS 9 () Discrete inf-sup condition: For all p h in V h,, C p h,r (v h,grad sup rz p h ) r. v h W h, v h r,curl (2) Coercivity on the discrete kernel: For all v h in W 0 h,, v h r,curl Cµ µ /2 curl rz v h r. (3) Unique solvability: There is a unique u h in W h, and a unique p h in V h, satisfying (6.). (4) Optimal error estimates: If (u, p) in H r, (curl, D) H, (D) is the exact solution satisfying (3.) and (u h, p h ) in W h, V h, is the discrete solution satisfying (6.), then u u h Hr(curl,D) + p p h H (D) Ch ( u H (D) + curl rz u H (D) + p H 2 (D)). Here, we have implicitly assumed that u and p are smooth enough so that the above norms make sense. Proof. () The proof of the inf-sup condition is similar to the proof of the exact inf-sup condition in Theorem 3.(). The only change is that we use the discrete version of grad rz H, (D) H r, (curl, D), namely that for all p h in V h,, the gradient grad rz p h is in W h,. (2) The proof of this inequality is an adaptation of the proof of Proposition 5.(III) of [0]. Let p H, (D) be the unique solution (cf. []) to and put (grad rz p,grad rz q) r = (v h,grad rz q) r w := v h grad rz p. for all q H, (D), Define the revolved vector field w = w r e r + w z e z on Ω. By the same argument given in the proof of Theorem 3.(2), w is in H 0 (curl,ω) and satisfies ( w,grad q) L 2 (Ω) = 0 for all q H 0(Ω), i.e., w is divergence free. Now, a well known imbedding of H 0 (curl, Ω) H(div, Ω) into H (Ω) 3 for convex Ω [0, Proposition I.3.] yields (6.2) w H (Ω) 3 C curl w L 2 (Ω) 3 = C curl v h L 2 (Ω) 3, where, in the last step, we used curl w = curl v h. By [3, Proposition 3.7], w = (w r, w z ) is in H (D) H (D) (and is bounded by an accompanying norm estimate). Hence, (6.3) w H (D) 2 w H (D) H (D) C w H (Ω) 3 C curl v h L 2 (Ω) 3. To prove the required inequality for v h, we use the above estimates for w. Since w is in H(D) 2, the projection Π W h w is well defined by Lemma 4.(2) (rectangular case) or Proposition 5.2 (triangular case). Obviously, grad rz p = v h w implies that on each element, (grad rz p) is in H ()2, so it possesses enough regularity for application of Π W h. Furthermore, since ΠW h v h = v h, v h = Π W h w + Π W h grad rz p = Π W h w + grad rz Π V h p,

21 20 D. COPELAND, J. GOPALAKRISHNAN, AND J. PASCIAK using the commutativity properties of Lemma 4. (rectangular case) or Lemma 5. (triangular case). Now since v h is L 2 (D)-orthogonal to grad rzv h,, the above equation implies (v h,v h ) r = (Π W h w,v h ) r, which, by the Cauchy-Schwarz inequality, gives v h L 2 (D) 2 ΠW h w L 2 (D) 2. Combining with the approximation properties of Lemma 4.2 or Lemma 5.3, v h L 2 (D) 2 ΠW h w L 2 (D) 2 ΠW h w w L 2 (D) 2 + w L 2 (D)2 Ch w H (D) + w L 2 (D) 2 C curl rz v h L 2 (D), where in the last step we have used (6.3). This proves the coercivity on the discrete kernel. (3) The unique solvability is a well known consequence of () and (2) (cf. [7, 0]). (4) The error estimate now follows from the Babuška-Brezzi theory [7, 0]. Indeed, () and (2) imply quasioptimality, i.e., ( ) u u h Hr(curl,D) + p p h H (D) C inf u v h Hr(curl,D) + inf p q h H v h W h, q h V (D) h, ) C ( u Π Wh u Hr(curl,D) + p Π Vh p H(D). Here we used the previously noted fact that the projectors Π W h and Π V h preserve zero boundary conditions (see Lemma 4.(4) or Lemma 5.), so that Π W h u and ΠV h p are indeed in W h, and V h,, respectively. Estimating the right hand side above by Lemma 4.2 (rectangular case) or Lemma 5.3 (triangular case) and applying a Poincaré inequality completes the proof of the theorem. 7. Numerical results In this section, we verify that the finite element convergence rates predicted by our analysis are achieved in a numerical example. We also make suggestions on how to iteratively solve the discrete matrix system. As in the previous section, the finite element spaces V h, and W h, can be defined on rectangular or triangular meshes. Define the operators A h : W h, (W h, ) and B h : W h, (V h, ) by A h v h (w h ) = (µ curl rz v h, curl rz w h ) r for all v h,w h W h,, B h v h (q h ) = (v h,grad rz q h ) r for all v h W h,, q h V h,. Let A h and B h denote the matrix representations of the operators A h and B h, respectively, in the standard bases for V h, and W h,. Then (6.) is rewritten as the linear system (7.) C h x h = b h, where C h = [ Ah B t h B h 0 ] [ ] [ ] uh fh, x h =, and b p h =. h g h Here, u h and p h denote the vectors of coefficients in the expansions of u h and p h in the standard bases, respectively, and the right hand side vector b h is computed from the right hand side of (6.) as usual. Note that the matrix C h is symmetric but indefinite.