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AN ADAPTIVE P1 FINITE ELEMENT METHOD FORTWO-DIMENSIONAL MAXWELL’S EQUATIONS?

S.C. BRENNER, J. GEDICKE, AND L.-Y. SUNG

Abstract. Recently a new numerical approach for two-dimensionalMaxwell’s equations based on the Hodge decomposition for divergence-free vector fields was introduced by Brenner et al. In this paper wepresent an adaptive P1 finite element method for two-dimensionalMaxwell’s equations that is based on this new approach. The reli-ability and efficiency of a posteriori error estimators based on theresidual and the dual weighted-residual are verified numerically.The performance of the new approach is shown to be competitivewith the lowest order edge element of Nédélec’s first family.

1. Introduction

Recently Brenner et al. [8] introduced a new approach for solv-ing two-dimensional Maxwell’s equations that is based on the Hodge(Helmholtz) decomposition for divergence-free vector fields. It reducesthe boundary value problem for Maxwell’s equations to several scalarsecond order elliptic boundary value problems, which can be discretizedby standard P1 finite element methods. In this paper we develop aposteriori error estimators for the new approach. The L2 error for theapproximation of the solution u of the Maxwell equations is shown tobe bounded by a residual type error estimator [1, 6, 13, 27], and weconsider two types of error estimators for the L2 error of the approx-imation of ∇ × u. For convex domains we have an L2-type residualerror estimator. For general domains we propose an error estimatorbased on the dual weighted-residual (DWR) approach of Becker andRannacher [2, 4]. We present numerical results that verify the reliabil-ity of the a posteriori error estimators and demonstrate their efficiency.The convergence of adaptive methods based on these error estimatorsis of optimal order.

Key words and phrases. Adaptivity, Error estimates, Finite element method,Hodge decomposition, Maxwell’s equations,

? This work was supported in part by the National Science Foundation underGrant No. DMS-07-13835 and Grand No. DMS-10-16332. The second author wasadditionally supported by the DFG Research Center MATHEON “Mathematicsfor Key Technologies” and the DFG graduate school BMS “Berlin MathematicalSchool”.

Published in Journal of Scientific Computing, 55:738–754, 2013. The final pub-lication is available at Springer via http://dx.doi.org/10.1007/s10915-012-9658-8.

http://dx.doi.org/10.1007/s10915-012-9658-8

AN AFEM FOR TWO-DIMENSIONAL MAXWELL’S EQUATIONS 2

Compared with the discretization of Maxwell’s equations by the low-est order edge element of Nédélec’s first family [23] which has one degreeof freedom per edge, this new discretization leads to smaller but mul-tiple discrete systems. However, most of these systems can be solvedin parallel and the new approach leads to comparable or better resultsas shown empirically by our numerical experiments. We note that theadaptive finite element method (AFEM) for the lowest order edge ele-ment has been studied in [3, 5, 14, 15, 21, 25].

We now introduce the model problem and briefly recall the Hodgedecomposition approach in [8]. The model problem seeks a solutionu ∈ H0(curl; Ω) ∩H(div0; Ω) such that

(∇× u,∇× v) + α(u,v) = (f,v) ∀v ∈ H0(curl; Ω) ∩H(div0; Ω)(1)

for a bounded polygonal domain Ω ⊂ R2 and f ∈ [L2(Ω)]2. Here (·, ·)denotes the L2 inner product over Ω and the spaceH0(curl; Ω) is definedby

H0(curl; Ω) :={v ∈ [L2(Ω)]2 : ∇× v ∈ L2(Ω) and n× v = 0 on ∂Ω

},

where ∇×v = curl v = (∂v2/∂x1)−(∂v1/∂x2) and n denotes the outerunit normal along ∂Ω. The space H(div0; Ω) is defined by

H(div0; Ω) :={v ∈ [L2(Ω)]2 : ∇ · v = 0

},

where ∇ · v = div v = (∂v1/∂x1) + (∂v2/∂x2).The problem (1) is related to the time-harmonic Maxwell equations

for α ≤ 0 and the time-domain Maxwell equations for α > 0 [7, 9, 10,16, 20, 22]. In the following it is assumed that problem (1) admits aunique solution. Thus −α is not a Maxwell eigenvalue and in particularα 6= 0 if Ω is not simply connected.

Using the Hodge (Helmholtz) decomposition for H(div0; Ω), we canwrite

u = ∇× φ+m∑j=1

cj∇ϕj,(2)

where φ ∈ H1(Ω) satisfies (φ, 1) = 0, ∇ × φ = [∂φ/∂x2,−∂φ/∂x1]t,m ≥ 0 is the Betti number for Ω (m = 0 if Ω is simply connected), andthe functions ϕ1, . . . , ϕm are defined as follows.

Let the outer boundary of Ω be denoted by Γ0 and the m componentsof the inner boundary be denoted by Γ1, . . . ,Γm. Then the functionsϕj ∈ H1(Ω) are determined by the scalar problems

(∇ϕj,∇v) = 0 ∀v ∈ H10 (Ω), ϕj|Γ0 = 0 and ϕj|Γk = δjk,(3)

where δjk = 1 if j = k and δjk = 0 if j 6= k for 1 ≤ j, k ≤ m.The function φ in (2) is determined by

(∇× φ,∇× ψ) = (ξ, ψ) ∀ψ ∈ H1(Ω)(4)


with the constraint (φ, 1) = 0, where the function ξ = ∇× u ∈ H1(Ω)is determined by

(∇× ξ,∇× ψ) + α(ξ, ψ) = (f,∇× ψ) ∀ψ ∈ H1(Ω)(5)

when α 6= 0, and by (5) together with the constraint (ξ, 1) = 0 whenΩ is simply connected and α = 0.

Remark 1. Note that

(∇× v,∇× w) = (∇v,∇w) ∀v, w ∈ H1(Ω)and the problems (4) and (5) are standard Neumann boundary valueproblems for the Laplace operator.

In the case m ≥ 1 the coefficients cj in (2) are determined by thesymmetric positive-definite system

m∑j=1

(∇ϕj,∇ϕk)cj =1

α(f,∇ϕk) for 1 ≤ k ≤ m.

Let Vh ⊂ H1(Ω) be a finite element space and V̊h = Vh ∩H10 (Ω). Itis shown in [8] that (1) can be solved by the following procedure.

• First we compute ξh ∈ Vh, which provides an approximation of∇×u,by solving the discrete problem

(∇× ξh,∇× ψh) + α(ξh, ψh) = (f,∇× ψh) ∀ψh ∈ Vh(6)

when α 6= 0, and by solving (6) together with the constraint (ξh, 1) =0 when Ω is simply connected and α = 0.• Then we compute φh ∈ Vh by solving the discrete problem

(∇× φh,∇× ψh) = (ξh, ψh) ∀ψh ∈ Vh(7)

with the constraint (φh, 1) = 0.• If Ω is not simply connected (m ≥ 1), we compute the functionsϕ1,h, . . . , ϕm,h ∈ Vh by solving the discrete problems

(∇ϕj,h,∇vh) = 0 ∀vh ∈ V̊h, ϕj,h|Γ0 = 0 and ϕj,h|Γk = δjk,(8)and we compute the coefficients cj,h for 1 ≤ j ≤ m by

m∑j=1

(∇ϕj,h,∇ϕk,h)cj,h =1

α(f,∇ϕk,h) for 1 ≤ k ≤ m.(9)

• The approximation of u is given by

uh = ∇× φh +m∑j=1

cj,h∇ϕj,h.(10)

The rest of the paper is organized as follows. We construct the er-ror estimators in Section 2 where we also provide some analysis. Theadaptive finite element method is described in Section 3. In Section 4


we present numerical results that demonstrate the reliability and effi-ciency of the proposed error estimators, and we also compare the newapproach with the discretization by the lowest order edge element inNédélec’s first family. We end with some concluding remarks in Sec-tion 5.

2. A posteriori error estimators

In this section we construct an a posteriori error estimator for theerror ‖u− uh‖L2(Ω) and two a posteriori error estimators for the error‖ξ − ξh‖L2(Ω).

Let Th be a triangulation of Ω andVh =

{v ∈ H1(Ω) : v|T ∈ P1(T ) ∀T ∈ Th

},

where P1(T ) denotes the space of affine functions on a triangle T . Wewill denote the set of the edges of the triangles in Th by Eh and the setof the edges interior to Ω by E ih.

Let [[v]] := v|T+−v|T− denote the jump over the edge E = T+∩T− forT± ∈ Th and [[v]] := v|T+ on boundary edges. Let nE = [n1, n2]t denotethe unit normal vector pointing from T+ to T− and tE = [−n2, n1]t de-note the unit tangent vector to E. The length of an edge is denoted byhE and hT := diam T . In the following, the notation x . y abbreviatesthe inequality x ≤ Cy with a constant C > 0 that does not depend onthe mesh size.

Our first observation is that the L2 error of uh is controlled by theH1 error of φh and the H

1 errors of ϕj,h for 1 ≤ j ≤ m.

Lemma 1. Let uh be the approximation of u given by (10). We have

‖u− uh‖L2(Ω) . ‖∇ × (φ− φh)‖L2(Ω) +m∑j=1

‖∇(ϕj − ϕj,h)‖L2(Ω).

Proof. It follows from (10) and the triangle inequality that

‖u− uh‖L2(Ω) ≤ ‖∇× (φ− φh)‖L2(Ω) +m∑j=1

‖cj∇ϕj − cj,h∇ϕj,h‖L2(Ω)

≤ ‖∇× (φ− φh)‖L2(Ω) +m∑j=1

(|cj|‖∇(ϕj − ϕj,h)‖L2(Ω)

+|cj − cj,h|‖∇ϕj‖L2(Ω) + |cj − cj,h|‖∇(ϕj − ϕj,h)‖L2(Ω)),

and the arguments in the proofs of [8, Lemmas 4.6 and 4.7] lead to theestimates

|cj| . ‖f‖L2(Ω)and

|cj − cj,h| . max1≤j≤m

‖∇(ϕj − ϕj,h)‖L2(Ω) ≤m∑j=1

‖∇(ϕj − ϕj,h)‖L2(Ω).�


Next we construct error estimators for ‖∇ × (φ − φh)‖L2(Ω) and‖∇(ϕj − ϕj,h)‖L2(Ω).

Lemma 2. Let ϕj,h ∈ Vh, j = 1, . . . ,m, be the solutions of (8). Wehave

‖∇(ϕj − ϕj,h)‖2L2(Ω) .∑E∈Eih

hE‖[[∇ϕj,h]]·nE‖2L2(E).

Proof. Define the residual Resh(v) := −(∇ϕj,h,∇v) for all v ∈ H10 (Ω)and let ‖Resh‖∗ denote the dual norm of the residual with respect toH10 (Ω). Since ϕj − ϕj,h ∈ H10 (Ω), it follows from (3) that‖∇(ϕj − ϕj,h)‖2L2(Ω) = (∇ϕj −∇ϕj,h,∇(ϕj − ϕj,h))

= −(∇ϕj,h,∇(ϕj − ϕj,h))= Resh(ϕj − ϕj,h) . ‖Resh‖∗‖∇(ϕj − ϕj,h)‖L2(Ω).

Let v ∈ H10 (Ω) be arbitrary and vh ∈ V̊h be its Scott-Zhang interpolant[11, 26] so that ∑

E∈Eh

‖h−1/2E (v − vh)‖2L2(E)

. ‖∇v‖2L2(Ω).(11)

Since Resh|V̊h = 0 by (8), we haveResh(v) = Resh(v − vh) = −(∇ϕj,h,∇(v − vh))

= −∑E∈Eih

∫E

h1/2E ([[∇ϕj,h]]·nE)h

−1/2E (v − vh)ds,

which together with the interpolation estimate (11) and the Cauchy-Schwarz inequality yields

‖Resh‖∗ .

∑E∈Eih

hE‖[[∇ϕj,h]]·nE‖2L2(E)

1/2 . �Lemma 3. Let φh be the solution of (7). We have

‖∇ × (φ− φh)‖2L2(Ω).∑T∈Th

h2T‖ξh‖2L2(T ) +∑E∈Eh

hE‖[[∇× φh]]·tE‖2L2(E) + ‖ξ − ξh‖2L2(Ω)

,

where ξh is the solution of (6).

Proof. It follows from (4) that

‖∇ × (φ− φh)‖2L2(Ω) = −(∇× φh,∇× (φ− φh)) + (ξ, φ− φh)= Resh(φ− φh) + (ξ − ξh, φ− φh),

where the residual is defined by

Resh(v) := (ξh, v)− (∇× φh,∇× v) ∀v ∈ H1(Ω).


Therefore we have

‖∇ × (φ− φh)‖2L2(Ω) . ‖Resh‖∗‖∇ × (φ− φh)‖L2(Ω)+ ‖ξ − ξh‖L2(Ω)‖φ− φh‖L2(Ω),

where ‖Resh‖∗ is the dual norm of the residual with respect to H1(Ω).Note that (φ−φh, 1) = 0 and hence we can apply Friedrichs’ inequalityto obtain

‖∇ × (φ− φh)‖L2(Ω) . ‖Resh‖∗ + ‖ξ − ξh‖L2(Ω).

Let v ∈ H1(Ω) be arbitrary and vh ∈ Vh be its Scott-Zhang interpolant[11, 26] so that∑

T∈Th

‖h−1T (v − vh)‖2L2(T )

+∑E∈Eh

‖h−1/2E (v − vh)‖2L2(E)

. ‖∇v‖2L2(Ω).(12)

It follows from (7) and integration by parts that

Resh(v) = (ξh, v − vh)− (∇× φh,∇× (v − vh))

=∑T∈Th

∫T

hT ξhh−1T (v − vh)dx

+∑E∈Eh

∫E

h1/2E ([[∇× φh]]·tE)h

−1/2E (v − vh)ds,

which together with the Cauchy-Schwarz inequality and the interpola-tion estimates (12) yields

‖Resh‖∗ .

(∑T∈Th

h2T‖ξh‖2L2(T )

)1/2+

(∑E∈Eh

hE‖[[∇× φh]]·tE‖2L2(E)

)1/2.�

Combining Lemmas 1, 2 and 3, we obtain the following result.

Theorem 1. Let uh be the approximation of u given by (10). We have

‖u− uh‖L2(Ω) . ηR + ‖ξ − ξh‖L2(Ω),

where

η2R :=∑T∈Th

h2T‖ξh‖2L2(T ) +∑E∈Eh

hE‖[[∇× φh]]·tE‖2L2(E)

+m∑j=1

∑E∈Eih

hE‖[[∇ϕj,h]]·nE‖2L2(E).(13)

Remark 2. If f is a piecewise smooth vector field, then ‖ξ − ξh‖L2(Ω) isof higher order compared with ‖u−uh‖L2(Ω) (cf. [8, Remark 4.4]). Thisis the case for the numerical examples in Section 4, where ‖ξ− ξh‖L2(Ω)can be safely ignored in the evaluation of ηR.


Remark 3. The efficiency of individual terms of the error estimator ηRcan be established by the bubble function techniques of Verfürth (cf.[1, 27]). However it is not clear whether the efficiency of ηR as a wholecan also be established by such techniques.

Finally we consider error estimators for ‖ξ − ξh‖L2(Ω), where we as-sume that f|T ∈ [Hβ(T )]2 ∩ H(curl;T ) for all T ∈ Th and for someβ ∈ (1/2, 1]. Since (ξ − ξh, 1) = 0, there exists a solution ζ of the dualproblem

(∇× ζ,∇× v) + α(ζ, v) = (ξ − ξh, v) ∀v ∈ H1(Ω)(14)that satisfies the constraint (ζ, 1) = 0. Therefore we can write

‖ξ − ξh‖2L2(Ω) = (∇× ζ,∇× (ξ − ξh)) + α(ζ, ξ − ξh)= (∇× (ζ − ζh),∇× (ξ − ξh)) + α(ζ − ζh, ξ − ξh),

for all ζh ∈ Vh, where we have also used the Galerkin orthogonality (cf.(5) and (6))

(∇× (ξ − ξh),∇× vh) + α(ξ − ξh, vh) = 0 ∀vh ∈ Vh.It then follows from (5) and integration by parts that

‖ξ − ξh‖2L2(Ω) = (∇× (ζ − ζh), f−∇× ξh)− α(ζ − ζh, ξh)

=∑T∈Th

∫T

[∇× (ζ − ζh) · (f−∇× ξh)− α(ζ − ζh)ξh] dx

=∑T∈Th

∫T

(∇× f− αξh)(ζ − ζh)dx

−∑E∈Eh

∫E

([[f−∇× ξh]]·tE)(ζ − ζh)ds,

(15)

which is valid for any ζh ∈ Vh.If Ω is convex, then the solution ζ of the dual problem (14) belongs

to H2(Ω) [19] and

‖ζ‖H2(Ω) . ‖ξ − ξh‖L2(Ω).In this case we can take ζh ∈ Vh to be the nodal interpolant of ζ andwe have [11, 17]∑

T∈Th

‖h−2T (ζ−ζh)‖2L2(T )

+∑E∈Eh

‖h−3/2E (ζ−ζh)‖2L2(E)

. ‖ξ−ξh‖L2(Ω).(16)

Combining (15) and (16) with the Cauchy-Schwarz inequality, we find

‖ξ − ξh‖L2(Ω) . ηreg,(17)where

η2reg:=∑T∈Th

h4T‖∇ × f− αξh‖2L2(T )+∑E∈Eh

h3E‖[[f−∇× ξh ]]·tE‖2L2(E).(18)


For a nonconvex domain Ω, motivated by (15) and the ideas of Beckerand Rannacher [2, 4] we propose the following dual weighted-residualerror estimator with ‖ξ − ξh‖L2(Ω) as the quantity of interest:

η2DWR :=∣∣∣ ∑T∈Th

∫T

(∇× f− αξh)(ζ̃ − ζh)dx

−∑E∈Eh

∫E

([[f−∇× ξh]]·tE)(ζ̃ − ζh)ds∣∣∣.(19)

Here ζh ∈ Vh is the P1 finite element solution of (14), where the un-known function ξ is replaced by some higher order interpolant ξ̃ of ξh,and ζ̃ is also a higher order interpolant of ζh. Details of the interpola-tion scheme we use can be found in Section 3.

Remark 4. The reliability and efficiency of the error estimator ηDWRwill be demonstrated numerically in Section 4. We note that the theo-retical justification of dual weighted-residual error estimators has beendiscussed in [12, 24].

3. An Adaptive Finite Element Method

The adaptive finite element method (AFEM) creates a sequence ofnested triangulations T0, T1, T2, . . . with associated P1 finite elementspaces

V0 ( V1 ( V2 ( . . . ( V` ⊂ V = H1(Ω)

and discrete solutions u` and ξ`. It consists of the following loop

Solve→ Estimate→ Mark→ Refine.

3.1. Solve. We apply the following algorithm from [8] (cf. the descrip-

tion in Section 1 where Vh = V` and V̊h = V̊` ∩H10 (Ω)) to compute thediscrete solutions u` and ξ` on the mesh T`.(1) Compute a numerical approximation ξ` ∈ V` of ξ = ∇ × u by

solving (6) when α 6= 0, and by (6) together with the constraint(ξ`, 1) = 0 when Ω is simply connected and α = 0.

(2) Compute a numerical approximation φ` ∈ V` of φ by solving (7)under the constraint (φ`, 1) = 0.

(3) If Ω is not simply connected, compute numerical approximationsϕ1,`, . . . , ϕm,` ∈ V` of ϕ1, . . . , ϕm by solving (8), and compute nu-merical approximations c1,`, . . . , cm,` of c1, . . . , cm by solving (9).

(4) The approximation of u is given by (cf. (10))

u` = ∇× φ` +m∑j=1

cj,`∇ϕj,`.


ωT

Tuu

u

u

u

u

uu

uFigure 1. Example of an element patch. The samplingpoints are all vertices.

3.2. Estimate. Based on the discrete solutions u` and ξ` the errors‖u − u`‖L2(Ω) and ‖ξ − ξ`‖L2(Ω) are estimated with the a posteriorierror estimators of Section 2.

The error ‖u−u`‖L2(Ω) is estimated by ηR (cf. (13)) with the (local)refinement indicators

η2`,R(T ) := h2T‖ξ`‖2L2(T ) +

1

2

∑E⊂∂T

hE‖[[∇× φ`]]·tE‖2L2(E)

+1

2

m∑j=1

∑E⊂∂T\∂Ω

hE‖[[∇ϕj,`]]·nE‖2L2(E),

where the higher order term ‖ξ − ξ`‖L2(Ω) is ignored.The error ‖ξ−ξ`‖L2(Ω) is estimated by ηreg (cf. (18)) with the (local)

refinement indicators

η2`,reg(T ) := h4T‖∇ × f− αξ`‖2L2(T ) +

1

2

∑E⊂∂T

h3E‖[[f−∇× ξ`]]·tE‖2L2(E),

or by ηDWR (cf. (19)) with the (local) refinement indicators

η2`,DWR(T ) :=∣∣∣ ∫

T

(∇× f− αξ`)(ζ̃ − ζ`)dx

− 12

∑E⊂∂T

∫E

([[f−∇× ξ`]]·tE)(ζ̃ − ζ`)ds∣∣∣.

Here ζ` ∈ V` denotes the solution of the discrete dual problem(∇× ζ`,∇× v`) + α(ζ`, v`) = (ξ̃ − ξ`, v`) ∀v` ∈ V`

that satisfies the constraint (ζ`, 1) = 0, and the functions ξ̃ and ζ̃ arehigher order interpolants of ξ` and ζ` constructed by the L2 averagingtechnique of [28].

In the first step of the construction of ξ̃ a higher order interpolationξ̂ ∈ P2(T ) is computed from ξ` for each element T ∈ T` separately.The values for the interpolated function at the vertices and midpoints


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Figure 2. Refinement rules: sub-triangles with corre-sponding reference edges depicted with a second edge.

of the edges for an element T ∈ T` are determined by a least squaresapproximation of a global quadratic polynomial on the element patchωT := ∪K∈T`,T∩K 6=∅K as depicted in Figure 1. The sampling points forthe least squares interpolation are chosen to be all the vertices of theelement patch.

In the second step, a smooth interpolant ξ̃ ∈ P2(T`) is obtained bytaking the average of ξ̂ over the nodal patch ωz := ∪K∈T`,z∈KK for thedegree of freedom at the vertex z:

ξ̃(z) =1

|T ∈ T`, T ⊆ ωz|∑

T∈T`,T⊆ωz

ξ̂(z)|T ,

and by taking the average of ξ̂ over the edge patch ωE := ∪K∈T`,E⊂KKfor the degree of freedom at the midpoint mE of E:

ξ̃(mE) =1

|T ∈ T`, T ⊆ ωE|∑

T∈T`,T⊆ωE

ξ̂(mE)|T .

The evaluation of ζ̃ follows the same procedure. Numerical exam-ples below indicate that the proposed procedure leads to reliable andefficient a posteriori error estimators.

3.3. Mark. Based on the refinement indicators, elements are markedfor refinement in a bulk criterion [18] such that M` ⊆ T` is a minimalset of marked elements with

θ∑T∈T`

η2` (T ) ≤∑T∈M`

η2` (T ),

for a bulk parameter 0 < θ ≤ 1, where η` is either η`,R, η`,reg or η`,DWR.This is carried out by a greedy algorithm which marks elements withlarger contributions.


101 102 103 104 105 106

10 4

10 3

10 2

10 1

100

101

102

N

reg,

DW

R, ||

l||

1

1

reg uniform

DWR uniform|| l|| uniform

reg adaptive|| l|| for reg adaptive

DWR adaptive|| l|| for DWR adaptive

Figure 3. Convergence histories of ‖ξ − ξ`‖L2(Ω), ηregand ηDWR on uniformly and adaptively refined meshesfor the unit square example.

3.4. Refine. The mesh is refined locally using the set M` of markedelements. Once an element is selected for refinement, all of its edgesare marked for refinement. In order to preserve the quality of the mesh,additional edges are marked by the closure algorithm. For each trianglelet one edge be the uniquely defined reference edge E(T ). The closurealgorithm computes a superset of marked edges, such that once an edgeof a triangle T ∈ T` is marked for refinement, its reference edge E(T ) ismarked as well. After the closure algorithm is applied, triangles withmarked edges are refined by one of the refinement rules depicted inFigure 2.

4. Numerical Examples

In this section we present the results of several numerical experi-ments. The reliability and efficiency of the error estimators for uni-formly and adaptively refined meshes are verified for several examplesusing different domains and different values for α. We also comparethe new numerical approach with the discretization by the lowest orderedge element in Nédélec’s first family [23].

4.1. Unit Square Example. As the first example we consider prob-lem (1) for α = 1 on the unit square Ω = (0, 1)2 and with right-hand


101 102 103 104 105

10 4

10 3

10 2

10 1

100

N

||uu l

||, ||

l||, ||

uu lN

d ||, |

|×(

uu lN

d )||

1

1/2

1

1||u ul|| uniform|| l|| uniform

||u ulNd|| uniform

|| ×(u ulNd)|| uniform

Figure 4. Comparison between the new numerical ap-proach with errors ‖u − u`‖L2(Ω) and ‖ξ − ξ`‖L2(Ω) andthe discretization by the lowest order edge element witherrors ‖u− uNd` ‖L2(Ω) and ‖∇ × (u− uNd` )‖L2(Ω).

side f such that the smooth solution is given by

u(x, y) = (sin πy, sin πx).

Note that in this example Ω is simply connected and thus u = ∇× φand u` = ∇× φ`.

The convergence histories of ‖ξ − ξ`‖L2(Ω) using ηreg or ηDWR as therefinement indicator are displayed in Figure 3. Since the domain isconvex, the estimate (17) for the a posteriori error estimator ηreg isvalid. The numerical results in Figure 3 indicate that both error esti-mators ηreg and ηDWR are empirical reliable and efficient for uniformlyand adaptively refined meshes, and that the adaptive meshes with ηregor ηDWR as refinement indicators lead to similar errors for ‖ξ−ξ`‖L2(Ω).Thus the proposed interpolation scheme for ηDWR is numerically stable.The error estimator ηDWR also seems to be almost exact.

Since the solution ξ is smooth, the convergence rate for ‖ξ− ξ`‖L2(Ω)is numerically of optimal order O(N−1` ) as predicted by [8, Lemma 4.1]for both uniform and adaptive meshes, where N` is the dimension of

V`. Note that for uniform meshes O(N−1/2` ) ≈ h`.The numerical results displayed in Figure 4 provide a comparison

between the approximation u` ∈ V` based on the Hodge decomposition


1

0.5

0

0.5

1

10.5

00.5

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 0 11

0

1

Figure 5. Plots of |u`|2 (left) and the vector field u`(right) for the L-shaped domain example.

and the approximation uNd` ∈ V Nd` based on the lowest order edgeelement on uniform meshes, where V Nd` is defined by

V Nd` :=

{v ∈ H(curl; Ω) : (v|T )(x) =

[aT,1aT,2

]+ bT

[ −x2x1

],

aT,1, aT,2, bT ∈ R,∀T ∈ T`}.

Even after taking into account that the computation of the vectorfield u` requires the solution of two scalar elliptic problems (i.e., theerror ‖u − u`‖L2(Ω) is plotted versus the degrees of freedom 2N`), thenew approach shows a slightly smaller error in the vector field variablethan the edge element. The error ‖ξ − ξ`‖L2(Ω) for the new approachis of one order higher than ‖∇ × (u− uNd` )‖L2(Ω) for the edge elementand therefore much smaller for larger degrees of freedom. Hence theproposed scheme leads empirically to comparable or better results thanthe lowest order edge element discretization.

4.2. L-Shaped Domain Example. The second numerical exampleconcerns the model problem (1) with α = −1 on the L-shaped domainΩ = (−1, 1)2\[0, 1]2. A singular solution in polar coordinates (r, θ) isgiven by

u = ∇×(r2/3 cos

(2

3θ − π

3

)φ(x)

),

with φ(x) = (1 − x22)2(1 − x21)2. The corresponding right-hand sidef ∈ H(div0; Ω) is computed by f = ∇ × (∇ × u) − u. The Euclideannorm of u` on an adaptive mesh based on ηR and the vector fiels u` onan uniform mesh are portrayed in Figure 5. For visualization purposesthe discontinuous values for |u`|2 have been smoothed by the arithmeticmean value at the vertices.

The convergence histories of the errors and the error estimators arereported in Figure 6 and Figure 7. Since the domain is nonconvex, uni-

form refinement results in suboptimal convergence rates of O(N−1/3` )


102 103 104 10510 3

10 2

10 1

100

101

N

R, |

|uu l

||, ||

l||

11/3

1

1/21

1

R uniform||u ul|| uniform

R adaptive||u ul|| for R adaptive|| l|| for R adaptive

Figure 6. Convergence histories of ‖u− u`‖L2(Ω), ‖ξ −ξ`‖L2(Ω) and ηR on uniformly and adaptively refinedmeshes for the L-shaped domain example.

for ‖u − u`‖L2(Ω) and O(N−2/3` ) for ‖ξ − ξ`‖L2(Ω) as predicted by [8,

Theorem 4.9 and Lemma 4.1], while adaptive refinement leads to em-

pirically optimal convergence rates of O(N−1/2` ) and O(N−1` ). Figure 6

indicates that the error estimator ηR is empirically reliable and efficientfor both uniform and adaptive meshes. It is observed that ‖ξ−ξ`‖L2(Ω)is of higher order for smaller h` � 1 even for adaptively refined meshesbased on ηR.

The numerical results displayed in Figure 7 indicate that ηreg is lessreliable than ηDWR on the nonconvex L-shaped domain because of thesignificantly larger error ‖ξ− ξ`‖L2(Ω) for ηreg on adaptive meshes. Theerror estimator ηDWR is shown to be both reliable and efficient. Fig-ure 8 depicts some adaptively refined meshes. All error estimatorsrefine strongly towards the corner singularity at the origin. However,the mesh for ηreg shows much less refinement at the origin which mayproduce larger errors in ‖ξ − ξ`‖L2(Ω).

The numerical results displayed in Figure 9 provide a comparisonbetween the approximations based on the Hodge decomposition andadaptive meshes generated by the error estimators ηR and ηDWR, andthe approximations based on the lowest order edge element and adap-tive meshes generated by the residual-based error estimator in [14].


101 102 103 104 10510 4

10 3

10 2

10 1

100

101

102

N

reg,

DW

R, ||

l||

1

1

12/3

|| l|| uniform

reg adaptive|| l|| for reg adaptive


Figure 7. Convergence histories of ‖ξ − ξ`‖L2(Ω), ηregand ηDWR on uniformly and adaptively refined meshesfor the L-shaped domain example.

1 0 11

0

1

1 0 11

0

1

1 0 11

0

1

Figure 8. Adaptively refined meshes using the errorestimators ηR, ηreg and ηDWR (from left to right) for theL-shaped domain example.

For a fair comparison we have plotted the errors ‖u − u`‖L2(Ω) ver-sus 2N`. The numerical results show that adaptive mesh refinementleads to optimal convergence rates for both methods and that the newmethod leads asymptotically to smaller errors than the lowest orderedge element method.

4.3. Doubly Connected Domain Example. The last example con-cerns the model problem (1) with α = 1 on a doubly connected domainΩ = (0, 4)2\[1, 3]2. Note that in this example m = 1 and therefore

u = ∇× φ+ c∇ϕ.


102 103 104 10510 3

10 2

10 1

100

101

N

||uu l

||, ||

l||, ||

uu lN

d ||, |

|×(

uu lN

d )||

1

1/2

1

1

||u ul|| for R adaptive|| l|| for DWR adaptive

||u ulNd|| adaptive

|| ×(u ulNd)|| adaptive

Figure 9. Comparison on adaptive meshes betweenthe new numerical approach with errors ‖u−u`‖L2(Ω) and‖ξ − ξ`‖L2(Ω) and the discretization by the lowest orderedge element with errors ‖u− uNd` ‖L2(Ω) and ‖∇× (u−uNd` )‖L2(Ω).

0

1

2

3

4

00.511.5

22.533.5

40

1

2

3

4

5

6

0 1 2 3 40

1

2

3

4

Figure 10. Plots of |u`|2 (left) and the vector field u`(right) for the doubly connected domain example.

The discontinuous right-hand side f is defined by

f =

{[1 + x1, 0] if x1 < x2 and 3 < x1 < 4,[0, 1 + x2] otherwise.

The norm of u` on an adaptive mesh based on ηR and the vector fieldu` on an uniform mesh are displayed in Figure 10, and the convergence


101 102 103 104 10510 5

10 4

10 3

10 2

10 1

100

101

N

R,

DW

R, ||

uu l

||, ||

l||

1

1

11/2

12/3

11/3

||u ul|| uniform

R adaptive||u ul|| for R adaptive|| l|| for R adaptive|| l|| uniform


Figure 11. Convergence histories of ‖u − u`‖L2(Ω),‖ξ − ξ`‖L2(Ω), ηR and ηDWR on uniformly and adaptivelyrefined meshes for the doubly connected domain exam-ple.

0 1 2 3 40

1

2

3

4

0 1 2 3 40

1

2

3

4

Figure 12. Adaptively refined meshes using the errorestimators ηR (left) and ηDWR (right) for the doubly con-nected domain example.

histories of the errors and the error estimators are reported in Figure 11.Since the exact solution is not known, the errors are approximated bythe difference between the discrete solution and the final grid solutions.

Note that the results displayed in Figure 11 indicate that this ap-proximation of the error is sufficient except for the last few levels of


refinement. Since the domain is nonconvex, uniform refinement re-sults in suboptimal convergence rates while adaptive refinement leads

to empirically optimal convergence rates of O(N−1/2` ) and O(N−1` ) for

the errors ‖u − u`‖L2(Ω) and ‖ξ − ξ`‖L2(Ω). The error estimators ηRand ηDWR are empirically reliable and efficient for adaptively refinedmeshes. The error ‖ξ− ξ`‖L2(Ω) is of higher order for ηR. Note that theerror estimator ηDWR underestimates the error ‖ξ − ξ`‖L2(Ω).

Figure 12 displays two adaptively refined meshes which are stronglyrefined at the four inner corners. The mesh for ηDWR is additionallymore refined in the area where the right-hand side f is discontinuous.

5. Concluding Remarks

We have developed an adaptive P1 finite element method for two di-mensional Maxwell’s equations that is based on the Hodge/Helmholtzdecomposition. In our experience the errors ‖ξ − ξ`‖L2(Ω) associatedwith the adaptive meshes based on the dual weighted-residual error es-timator ηDWR are slightly better than those associated with the resid-ual error estimator ηR, while the errors ‖u − u`‖L2(Ω) behave in theopposite way. The overall performance of the two error estimators arecomparable. Therefore between the two choices the adaptive methodbased on ηR is recommended if one wants to approximate both u andξ = ∇× u, since the computational cost of ηDWR is much higher.

For simplicity we set the electric permittivity and magnetic per-meability to be 1. But the results in this paper can be extended toMaxwell’s equations with general electric permittivity and magneticpermeability (cf. [8]).

Acknowledgements

The bulk of this paper was written while the second author en-joyed the hospitality of the Center for Computation and Technology atLouisiana State University.

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(S.C. Brenner) Department of Mathematics and Center for Computa-tion and Technology, Louisiana State University, Baton Rouge, LA70803, USA

E-mail address: [email protected]

(J. Gedicke) Institut für Mathematik, Humboldt-Universität zu Berlin,Unter den Linden 6, 10099 Berlin, Germany


(L.-Y. Sung) Department of Mathematics and Center for Computa-tion and Technology, Louisiana State University, Baton Rouge, LA70803, USA


1. Introduction2. A posteriori error estimators3. An Adaptive Finite Element Method3.1. Solve3.2. Estimate3.3. Mark3.4. Refine

4. Numerical Examples4.1. Unit Square Example4.2. L-Shaped Domain Example4.3. Doubly Connected Domain Example

5. Concluding RemarksAcknowledgementsReferences

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