SIAM J. NUMER. ANAL.
1991 Society for Industrial and Applied Mathematics
004
Vol. 28, No. 6, pp. 1610-1634, December 1991
A MIXED METHOD FOR APPROXIMATING
MAXWELL’S EQUATIONS*
PETER B. MONKt
Abstract. A semidiscrete mixed finite element approximation to the time dependent Maxwell’s
equations on a bounded smooth domain is analyzed. A variational problem for the electric and
magnetic fields in which the boundary conditions are enforced naturally is derived. Then a general
convergence result for mixed methods is proven, and it is shown how this result may be used to prove
various error estimates when Ndlec’s curl conforming finite elements are used.
Key words. Maxwell’s equations, finite elements, error
AMS(MOS) subject
classifications, primary
estimates
65N30; secondary 35L15
1. Introduction. Let t be a smooth, bounded, simply connected domain in
with connected boundary F and unit outward normal n. Let e(x) and it(x)
denote, respectively, the dielectric constant and magnetic permeability of the medium
occupying gt. Let a(x) denote the conductivity of the medium. Then, if E(x, t) and
H(x, t) denote, respectively, the electric and magnetic fields, Maxwell’s equations [10]
state that
n3
eSt + aE- V H
itH+VE
(1.1)
(1.2)
(0, T),
in(0, T)
J
in
0
where J J(x, t) is a known function specifying the applied current. We shall assume
a perfect conducting boundary condition on so that
(1.3)
n
E -0
(0, T).
F
on
In addition, initial conditions must be specified so that
(1.4)
where
(1.5)
E(x, O)
E0 and H0
Eo(x)
are given functions and
V.(itH0)
The divergence-free condition in
(1.6)
H(x, O)
and
0 in
,
H0
Ho(x)
’fix e
satisfies
Ho.n
0 on F.
(1.5) together with (1.2) implies that
V.(itH)--0
in
x
(0, T),
which is usually included with (1.1) and (1.2) in the statement of Maxwell’s equations
(cf. [10]). In addition, the boundary condition in (1.5) (together with (1.1) and (1.2))
implies
(1.7)
H.n
0
on
F
(0, T).
Received by the editors February 5, 1990; accepted for publication (in revised form) January 8,
1991.
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716. (email: monk(C)vaxl, acs. udel. edu). Research supported in part by grants from the Air Force Office of
Scientific Research and the National Science Foundation.
1610
A MIXED METHOD FOR MAXWELL’S EQUATIONS
1611
The coefficients e, tt and a are L(fl) functions for which there exist constants
emax, amax, #rain, and #max such that
min,
a.e. inf,.
Let C’(0, T; X) denote the space of m times continuously differentiable functions
from
[0, T]
into the Hilbert space
X, and define the standard space of functions on fl
2
with an L weak curl by
H(curl; t)= {v e (L2(f))3[V x v e (L2(t))3}.
We shall assume the existence of a solution (E,H) to (1.1)-(1.4) such that E,H e
C1(0, T; (L2(f)) 3) N C(0, T; H(curl; f)). We remark that existence and uniqueness
results for Maxwell’s equations guaranteeing this smoothness are proved in [16] in
the case when J
0 and a _= 0 using spectral techniques. Similar techniques are
used in [1] to prove existence and uniqueness for Maxwell’s equations in two space
dimensions (with variable J, a 0, but constant e and #). The problem has also been
studied in [10] for general piecewise constant coefficients. Clearly the above regularity
assumption requires that J E C(0, T; (L2())3).
Assuming the existence of a solution to (1.1)-(1.4), we obtain a weak formulation
as follows. We multiply equation (1.1) by a test function b E (L2(t)) 3 and integrate
over
e H(curl;f), integrating over f, and
Similarly, multiplying (1.2) by
integrating the curl term by parts (also using (1.3)) we obtain a weak form for (1.2).
If (-, .) denotes the (L2(f)) 3 inner product, and E(t) E(., t), H(t) H(., t) we find
2
that the solution (E,H) e [CI(0, T; (L2(f)) 3) V C(O,T;H(curl;t))] of (1.1)-(1.4)
satisfies
.
(1.8)
(1.9)
for 0 < t
(eE,, b) + ((rE, )
(#Ht,
(V H, b)
(J, b) Vb e (L2()) 3,
< T with the initial conditions
E(0)- E0
(1.10)
(where H0
(1.5)). Of
and
H(0)- H0
course for the above variational problem to make
E e CI(0, T; (L2(fl)))V C(0, T; (L2(t))3), so the
variational problem might be used to prove existence of a weak solution to Maxwell’s
equations. We shall not discuss this further, since we shall concentrate here on using
(1.8)-(1.10) to construct a numerical scheme.
Notice that the boundary condition (1.3) is now imposed weakly via (1.9). This
is one advantage of the weak form given in (1.8)-(1.10) since the boundary condition
does not have to be imposed on trial and test spaces. Indeed the more general condiwhere is a tangential surface field could also be handled easily by
tion n E
this formulation.
Our goal is to analyze the use of discrete versions of (1.8) and (1.9) to approximate the electric and magnetic fields. To this end, let Uh C (L2(Ft)) 3 and let
Vh C H(curl; t) be finite-dimensional subspaces of the given spaces (we shall define Uh and Vh in 3). Then the semidiscrete Maxwell system we will analyze in this
satisfies
sense we need only require that
-
,
1612
PETER B. MONK
(Eh, H h) E CI(0,T; Uh) CI(0,T; Vh), such that
(j, h) vh e Uh,
(eEh, h) / (hE h, h) (V H h, h)
0 V h e Vh
(#H,h,h)/(Eh, vh)
paper is to find
(1.11)
(1.12)
for 0 < t _< T, subject to the initial conditions
(1.13)
_
Eh(o) pFEo,
Hh(o) pBHo
-
where pE (L2(f))3
Uh and pS H(curl; f) Vh are suitable approximation
operators to be detailed later. Equation (1.11)-(1.12) is a system of linear ordinary
differential equations, and thus existence and uniqueness of a solution are well known.
As stated before, the purpose of this paper is to prove estimates for (E- Eh)(t)
and (H- Hh)(t) in the (L2()) 3 norm. We shall first provide an estimate for a
general class of spaces Uh and Vh. Then we shall show how the general result may be
used to prove error estimates when U h is constructed using discontinuous piecewise
polynomials, and V h is constructed using the curl conforming elements of Ndlec [20].
Our first estimate is very general allowing for general curved or polygonal domains
and allowing the full Maxwell system. However, although the estimate is of optimal
order in h, it is not optimal in the regularity required for the continuous solution.
We then present a refined error analysis under special circumstances. In particular,
we assume that the domain is smooth and convex (so a suitable interpolant can be
defined), that e and # are constant, and that a 0. These assumptions allow the
use of a Helmholtz decomposition of the electric and magnetic field vectors. We shall
discuss each of these assumptions in some detail in 4. The refined error analysis
allows a proof of convergence that is of almost optimal order, and almost optimal in
the regularity assumptions for the continuous fields.
When discussing the problem on a smooth domain, for analytical simplicity, we
ignore the important practical problem of numerical integration on curved elements
(cf. [6]). The curved elements we use are well adapted to the natural boundary
condition considered above, but are not suitable in the form we present for the analysis
of mixed boundary value problems in which essential boundary conditions are imposed
on parts of the boundary.
The proofs presented here use the general techniques of Baker and Bramble [4].
The analysis of the Ndelec elements draws heavily on the work of Girault [12], and
uses similar techniques to those developed in [18].
The idea of using curl conforming elements to discretize Maxwell’s equations is not
new. N(dlec [20] suggests using a curl conforming space for E and a divergence conforming space for H. In Ndlec’s method the boundary condition (1.3) is essential,
and no error estimates are provided. Kikuchi [15] has analyzed the use of Ned(lec
elements for discretizing an eigenvalue problem associated with the time harmonic
analogue of (1.1)-(1.2). No order estimates are proved nor is the time-dependent case
analyzed. Monk [18] has analyzed the use of finite elements on the electric field equations obtained by eliminating H from (1.1). This results in a second-order hyperbolic
problem, with essential boundary conditions. There are a number of works concerned
with variational methods for approximating Maxwell’s equations in two dimensions
(cf. [1], [22]-[24], [26]). None of these works use the elements or variational formulation considered in this paper. To our knowledge, the variational method posed in
(1.11)-(1.12) and the forthcoming analysis are novel.
We should also note the connection, pointed out by Kikuchi [15] between the finite
A MIXED METHOD FOR MAXWELL’S EQUATIONS
1613
element methods using N(!d(!lec elements and the finite difference techniques used in
[29]-[31].
2. A general convergence result. Let us start by defining some notation.
The convergence results are given in terms of weighted (L2(12)) 3 norms. For a given
nonnegative L(fl) function w(x), we define
(u, v)w
L
u.v w dx,
and define
IlUllo,w
V/(U, u)w.
e or is, the functionals I1" II0, and I1" IIo,u are norms equivalent
to the standard (L2(12)) 3 norm. The functional I1" II0, is in general only a seminorm;
however, there are cases (cf. [10]) in which a is bounded away from zero on fl in which
case I1" II0, is also a norm. When w 1, which corresponds to the standard (L2(fl)) 3
weight, we dispense with the function subscript on the inner product and norm. We
also define
Note that when w
1/2
(2.1)
lilulil0,,
(T]U(T)I 2 dx dT
Recall also that
H(curl; 1-1)-- {v e (LU(12))a V
v
e (L2(fl)) 3}
with the graph norm
IIIIH:- (11112, + IIV ull,)/’.
In later sections we shall need to use the general Sobolev space of functions with s
weak derivatives in Lp. Thus we define
with the norm
_
II,llw:.,()---
IO"vl ,> dx
2 we obtain the spaces H 8 (12) and in this case we denote the norm by I1" II,.
We also need to use norms of functions of space and time. Given a Hilbert space
When p
X and and index p, 1
p
< oo we define
LP(O,T; X)= v(t) e X
IIv()ll dm<
with the norm
Ivll.,(O,T;:)
IIv(’)ll. dT
1614
PETER B. MONK
These spaces also make sense for p
oc in the obvious way, and are studied, for
example, in [10].
In order to prove a convergence estimate covering the applications in this paper,
we must generalize the Maxwell initial value problem considered in the Introduction.
Let U and V be subspaces of (L2()) 3 and H(curl; gt), respectively. We assume the
existence of a solution
(E,H) e [C1(0, T; (L2(f)) 3) N C(0, T; U)]
x [C1 (0, T; (L2 (f)) 3) 3 C(0, T; V)]
of the problem
(2.2)
(2.3)
(eEt, r) + (aE, r) (V H, b)
(#Ht,)+(E,V)
(J, b) Vb e U,
0
VCeV
for 0 < t _< T, subject to the initial conditions (1.10). In applications in this paper
Y H(curl; ft) and will take either U V Y or U (L2(t)) 3.
Given subspaces Uh C U and Vh C V. The discrete solution (E h (t), H h (t) in Uh Vh
satisfies (1.11)-(1.13).
In order to prove good error estimates, we need to assume some compatibility
conditions between Uh and Vh. The first assumption is critical to the error analysis
and is as follows.
(H1) There exists an operator Ph U Uh such that
we will always take
(Phu, V v h)
(u,V
-
v h)
Vv h E Vh, Vu E U.
The second assumption will allow an improvement in the norms appearing in the
error estimates and is as follows"
(H2) There exists an operator
(2.4)
(V IIv, u )
IIh V
Yh such that
(V v, u ) Vu
U, Vv
V.
Let us remark that (H1) can be satisfied if
(2.5)
vxvcu
and Ph is taken to be the (L2()) 3 projection of U onto Uh. This is the construction
we shall use later in 3. In order for (Sl) and (H2) to hold, both Ph and YIh must
be constructed carefully, and this case is examined in 4.
There is an obvious resemblance between the hypotheses (Hi) and (H2) above
and the hypotheses of [8] made to analyze mixed methods for Laplace’s equation.
Unfortunately, it does not seem to be possible to apply methods similar to those of
[8] to verify (H2). Indeed a discussion of (H2) is the major content of 4.
Let us now state and prove our general convergence theorem.
THEOREM 2.1. Let E(t),H(t) U V satisfy (2.2)-(2.3) and (1.10) and let
Eh(t),Hh(t) e Uh Vh satisfy (1.11)-(1.13) (with the smoothness in time stated
before the respective equations). Suppose that Uh C U and Vh C V.
(a) Suppose Uh and Vh satisfy hypothesis (H1) alone and let IIhH be any function
A MIXED METHOD FOR MAXWELL’S EQUATIONS
in
Vh. Then there
exists a constant
1615
C independent of t and h such that
II(H- Hh)(t)llo,, + II(E- Eh)(t)llo, / IIIE- EhlIIo,,
<C
H)(0)l[o,, + II(H- pBH)(O)IIo,,
/ll(nH- H)(t)llo,, / II(PhE- E)(0)llo,
/II(E- pEE)(O)IIo, / II(PhE- E)(t)llo,
+ f IIn,H Hllo,, / IIPhE Ello,
/ IIV (nhH- H)llo d- / IIIE-
(2.6)
{ll(lInH-
PhEIIIo,,.
(b) If Uh and Vh satisfy both (H1) and (H2), the term f [IV (lhH- H)]]o dT
may be dropped from the right-hand side of the above estimate.
Remark. If the initial condition operator pE is taken to be Ph, we may drop the
terms I](PhE- E)(0)ll0, and II(E- pEE)(O)IIo, from the right-hand side of (2.6).
The choice pE
Ph is reasonable in the cases we shall analyze since Uh will be a
discontinuous space allowing the computation of RE element by element.
The choice pB IIh is not likely to be practically useful since IIh will turn out
to be costly to compute. A better choice might be to take pB to be the (L2()) 3
projection of H(curl; ) onto Vh.
The following proof can be compared to one in [11] for mixed methods for the
scalar wave equation. The general technique is similar to that in [4].
Proof. Subtracting (1.11)-(1.12) from (2.2)-(2.3) we find that
((E- E), V ) + ((E- E), V )
-(Vx(H-Hh),h)=0 Vdh e Uh,
(.)
(#(H Hh),
+ (E E V ch) 0 V h e Vh.
Now define e(t) (PhE- Eh)(t) and h(t) (IIhH- Hh)(t); then using (2.7) and
h
(2.8)
(2.8)
with gb
e and
ch
h,
(2.9) (eet, e)+ (he, e)- (V h, e)- (e(PhE- E)t, e)+ (a(PhE- E), e)
+(V (IIH- H), e),
(2.10)
(#h, h) d-(e, V
By virtue of (H1)
((nH- H), h)d-(PhE- E, V h).
h)
we may conclude that
(PhE E, V h)
O.
0 in (2.9). Since the term
(IIhH- H),e)
be handled similarly to the term ((PhE- E), e), for
simplicity we shall only provide a detailed proof when (H1) and (H2) both hold. If
(H1) and (H2) both hold, then adding (2.9) and (2.10) we obtain
Furthermore, if (H2) holds, (V
(V
(IIhH- H), e) may
2
}
IIh($)ll,,, / II()ll o, / I111o,
((nH- H), h) + (e(PhE- E), e)
d-(a(PhE E), e).
1616
PETER B. MONK
Integrating the above equality in time we obtain
Then using the Cauchy-Schwarz and arithmetic geometric mean inequalities we obtain
the estimate that for any tl and 0 _< t _< tl _< T,
2
2
2
2
O<v<t
+IIPhE Ello, dT +
IlPh E Ell ao, dT
sup
0<<t
Since this inequality holds for all t in 0 _< t _< tl it must hold when IIh(t)llo,t,-t-Ile(t)llo,
is maximized. Suppose the maximum occurs at t t* then from the above equation
Using the Cauchy-Schwarz inequality and renaming tl to t we conclude that there is
a constant C independent of t and h such that
(2.11)
If
(H2)
does not hold there is an extra term
IIx7 (II,,.,H- H)IIodT
on the right-hand side of
(2.11).
If the initial condition operators (see (1.13)) are chosen so that R E
e(0) 0 and if pS IIh then h(0) 0. Otherwise we estimate
(2.12)
I1(o)11o, < II(PhE- E)(O)llo, + II(E- pEE)(O)IIo,
Ph then
A MIXED METHOD FOR MAXWELL’S EQUATIONS
1617
]]h(0)[[0,u. Finally, we write
[[(E- Eh)(t)[[o, + [[(H- Hh)(t)[[o, + [[[E- Eh[[[o,,t
<_ [[(E- PhE)(t)[[o, + I[(H- HhH)(t)[[o, + [[IE- PhE[][o,,t
with a similar estimate for
Use of this inequality with (2.11), (2.12), and the analogue of (2.12) for IIh(O)ll0,,,
proves the result in part (b) of the theorem. Part (a) is proved in the same way,
[-I
keeping track of the term in V (IIhH- H).
3. Error estimates for smooth solutions. In this section we will show how
some curl conforming elements due to Ndlec [20] may be used to construct Vh. We
provide optimal error estimates when H and E are sufficiently smooth.
Let {Th } h>0 be a family of meshes of gt. Each element K E Th is assumed to be
a tetrahedron if K has no face or edge on F. Elements that have an edge or face on
F are allowed to have, respectively, one curved edge or one curved face (and hence
three curved edges). These elements are called boundary elements. For any boundary
element K we can obtain a standard tetrahedron K by connecting the four vertices
of K by straight edges. We require that the triangulation be uniformly regular (cf.
[14]), which is defined as follows for curvilinear elements. Let PK be the radius of the
largest sphere contained in K or/ and let hg be the radius of the smallest sphere
containing K. Then the triangulation is uniformly regular if there exist constants
-
> 0 and a such that
9/h <_ hK <_ crpK VK Th, Vh > O.
This is essentially the regularity of Scott [27] together with an
Let -h be the solid formed by
inverse inequality
[7].
We note that one consequence of uniform regularity is that if p is a polynomial of
total degree k, there are positive constants C1 and C2 independent of h, p, and K
(but depending on s, k, and q) such that
(3.1)
for 0 < s, 2 < q. The left-hand inequality is proved by considering a generalized
reference element/ consisting of that portion of the unit sphere in the first octant.
For h small enough, mapping K to the standard reference element maps K to a
domain in/. Then equivalence of norms on/ proves the desired result (cf. [27]).
The right-hand inequality is proved similarly using an intermediate reference domain
g which is the tetrahedron with nodes at (0, 0, 0), (0, 1/2, 0), (0, 0, 1/2), (1/2, 0, 0).
Mapping/ to/ then using the equivalence of norms on/ and K followed by the
mapping of K back to a subdomain of K contained in K (if h is small enough) proves
the right-hand inequality.
In order to define the curl conforming space of N6d61ec [20] let Pk denote the
standard space of polynomials of total degree less than or equal to k, and let /3k
denote the space of homogeneous polynomials of order k. Define Sk C (Pk) 3 and
Rk C (Pk) 3 by
Sk
{p e
(/5k)3
p().
0,
-< x,x2,x3 >},
1618
PETER B. MONK
For example, in the case k
Rk has the form
1 a polynomial p E
p()
a +/ x
where c and are constant vectors [20].
Now, following Nd61ec [20], we can define
(3.2)
Vh
{v h e H(curl; )
l e R VK e Th}.
To define an interpolant in Vh on each tetrahedral element, we follow N6d61ec and
define the following moments.
DEFINITION 3.1. Let K be a tetrahedron in T3 with general edge e and face f.
Let t be a unit vector parallel to e. Let u e (WI’8(K)) 3 for some s > 2. We define
the following three sets of moments of u on K
(3.3)
{Lu’tqds VqePk-l(e) forthesixedgeseofK},
{uxn.qdA Vqe(Pk_2(f))
Me(u)
2
(3.4)
for all six faces f of K
{/Ku"
(3.5) MK(U)
q da Vq E
},
(Pk-3(K))3}
N6d61ec [20] shows that the above three sets of degrees of freedom are Rk-unisolvent
and curl conforming.
Using these degrees of freedom we can define an interpolant rhU for any function
U
(Wl’s(g)) 3 when g is a tetrahedron by taking rhUIg Rk and
Me(u- rhU)
MI(u- rhU)
MK(U rhU)
{0}.
wm’p(-) where m and p are such that the moments defined
above make sense (for example m 1, p > 2 or m k + 1 > 1 and p 2). Then to
define the interpolant on the entire domain (including boundary elements), we first
extend u wm’p(t) to all 73 in such a way that
Now suppose that u
IIIIw’,’(T) < Cllullw’,,(a)
(cf. [2]) and then interpolate u on -h (i.e., for a boundary element K we interpolate
u on K in the standard way).
The use of the extension of u and interpolation on -h allows us to prove the
following extension of the standard error estimates for the N6d61ec elements [14], [20]
(see also [6] for a discussion of N6d61ec spaces on curved elements).
THEOREM 3.2. Let u (Wl’s()) 3 for some s > 2, then there is a function
h
u
Vh and a constant C independent of h and u (but depending on s) such that
(3.6)
II- ullo + hllV x (u- uh)llo <_ Chllullw,o();
if in addition u (Hk+l(Ft)) 3 then
(3.7)
llu u’llH <_ Chl[ull+.
1619
A MIXED METHOD FOR MAXWELL’S EQUATIONS
Remark. The above theorem assumes that the tetrahedra exactly tile the domain
Ft. In practice the domain would have to be approximated (see [17], [6], [9] for work
in this direction).
Proof. For simplicity, we shall only give the proof of (3.7). The other estimate
follows in a similar way (cf. [14]). As mentioned previously we first extend u to 73.
Then we define u h
rhU (on h). The estimates are then provided element by
element. For tetrahedral elements the results can be found in [14], [20]. For boundary
elements we use the curved reference element introduced previously. Given a curved
element K, let the affine transformation
BK + hg
FK()
map the standard reference tetrahedron g with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0),
and (0, 0, 1) into the tetrahedron sociated with K. Then if we transform vectors
ccording to Ndlec [20] by the transformation
u
the arguments in
(B)-
[20] show that
]]u rhU]](L(K)) < C
where e is the interpolation operator on
so that
BK
det
h
.
(L2(F(K)))3’
Then if h is small enough
We can again follow N6d61ec [20] and use the fact that
degree k- 1 on to estimate that
F(K) C R
+ preserves polynomials of
det
]]U rhU](L:(K))a < C hBKI ] (H(R)
where ]. ](H(R))a denotes the seminorm involving derivatives of total degree k. Also
V x ( e). Cdx is a linear form in for any e (n2()) 3
using the fact that
which is zero whenever
(p)3 we find that
fR
]V X (U rhU)](L2(K))3 < C
det
h2
[(H+(k))
a"
Now we use the scaling estimate (cf. [7])
(H,(k))
h+
Cdet BK ]U](H’(F(R)))
with
k and
k + 1 to complete the estimate on the curved element. Adding
over estimates for each tetrahedron and curved element in the domain D we find that
I1(- hu)ll + IIV (-- hU)l < Ch2k
I1112(HkTI(FK(R)))3
KVh
1620
PETER B. MONK
Finally, using the fact that the norm of the extended function is bounded by the norm
of the function on f completes the proof.
Next let us define Uh. We just take Uh to be the standard space of discontinuous
piecewise k- 1 degree polynomials,
(3.8)
Uh
{uh uh[g E (Pk_l) 3
VK e Th}.
We define Po to be the standard (L2(f)) 3 projection into Uh so that if u e (L2(f)) 3
then
Pou Uh satisfies
(Pou, h)
(3.9)
(u, h) Vh
Uh.
Error estimates for this projection are well known and
[lu- P0u[10 _< C h[Iu[l,
(3.10)
0
<_ _< k.
Now we can state and prove our first convergence theorem.
THEOIEM 3.3. Let (E,H) satisfy (1.8)-(1.10) and suppose that
C(0, T; (Hk+l(f))3).
Let Vh and Uh be given by (3.2) and (3.8). Suppose that (Eh(t),Hh(t)) e Uh x Vh
Ee
satisfy
CI(0, T; (Hk(f)) 3)
(1.11)-(1.13)
(3.11)
(3.12)
He
and
with the initial condition operators satisfying the estimates
I[Eo pEEol[o <_ C hkl[Eol[k,
[[H0 pSHol[o <_ C hk[[Ho[Ik+l
Then there exists a constant C independent
of h
and t such that
(3.13)
J.I(H- Hh)(t)JJo,, / JJ(E- Eh)(t)lJo, / JiJE- EhJ[Jo,,
<_ C h k {[IH(O)]lk+l + JiE(O)Jlk +
Remark. As remarked previously, suitable choices for pE and pB might be the
L 2 projections onto Uh and Vh, respectively. This theorem also holds for polyhedra.
Proof. We apply Theorem 2.1 with Y- g(curl; f) and U (L2()) 3. Clearly
V x Vh C Uh and so we can satisfy (nl) by taking Ph Po where P0 is the (L2(f)) 3
projection of U onto Uh introduced before the statement of the theorem. We let
IIhH Vh be the function approximating H guaranteed by Theorem 3.2. Then
using estimates (3.7), (3.10), (3.11), and (3.12) in Theorem 2.1, together with the
estimates
(3.14)
_< IlH(O)llk+l +
(3.15)
< IIE(O)II +
(cf. [11]) proves the desired result.
IIHt(T)llk+ldT,
IIE(’)IIdT
1621
A MIXED METHOD FOR MAXWELL’S EQUATIONS
4. Stability estimates. The error estimates proved in Theorem 3.3 are optimal
and follow from the smooth data estimate in (3.7). Unfortunately these estimates
require H e (Hk+l()) 3 while E need only be in (Hk()) 3. This imbalance in
the smoothness requirements, as well as the high degree of smoothness needed for the
estimates to hold, suggests the need for a better error estimate. In this section we shall
prove error estimates when (H(t),E(t)) e (Wl’s(’))3 x (HI()) 3. Unfortunately, to
prove these estimates we must severely restrict the problem under consideration. We
hope in future to lift most of these restrictions. In this section we assume that
is convex. This assumption is made so that we can construct an interpolant
in Vh that preserves gradients in the sense of Lemma 4.5 below. It is likely
that this assumption can be relaxed.
a
0, e e0, and # #0 where 0 and #0 are positive constants, which we
can take to be 1. The choice e e0 and a 0 allow us to use the Helmholtz
decomposition to analyze the electric field equation, and constitutes an essential assumption for the analysis. The assumption that # #0 allows us
to analyze the projection IIh proposed below since then V. H -0.
We shall prove the following result.
THEOREM 4.1. Let (E, H) satisfy (1.8)-(1.10) and suppose that
(E,H) e C1(0, T; (H1 ()) 3) el(0, T; (wl’s(’)) 3)
3
Let Vh and Uh be given by (3.2) and
for some s > 2 and g(t) e C0 (0, T; (H (2))).
(3.8), and suppose thatE (Eh(t),Hh(t)) e Uh Vh satisfy (1.11)-(1.13) with the initial
Po (orthogonal L 2 projection). Suppose,
condition operator R
is chosen so that
pB
in addition, that
]lH0 PBH0]10 _< ChllHollw1.8(a).
Then, given with 0 < < 1, there exists a constant C C(5, s) such that
II(H- Hh)(t)l]o,, / I](E- Eh)(t)]lo,
_< C h -a {IIH(0)IIWI,s()-{-IlE(O)lll + IlHt(t)llil(O,t;w,(a))
(4.2)
(4.1)
-
Remark. Apart from the factor h this result shows that the stability estimate
the approximation of Maxwell’s equations. The smoothness requirements on H and E are also better balanced than in Theorem 3.3.
The proof of this result is somewhat lengthy. First we make some observations
regarding a priori estimates. Then we prove some convergence properties of an operator related to the time independent magnetic field problem. Finally, we prove the
desired convergence result. One interesting by-product of our analysis is the proof of
a discrete Helmholtz decomposition for piecewise constant finite element vector fields
(3.7) holds for
in
T3 (see 4.4).
4.1. A priori estimates. First, we collect some known results and discuss the
.
discrete spaces further. The following existence theorem for a vector potential is from
[5].
THEOREM 4.2. Let u E (L2()) 3 satisfy V.u
vector potential e (L2()) 3 such that
V
u
and
.n=O
V.
on F.
O
0 in
in
,
Then u has a unique
1622
PETER B. MONK
In addition, if u e (L
8
())3, 2 _< s < oc,
e (WI’8 (gt))3 and
then
(4.3)
Fuheore,
g u e Hk(n),k O,
Hk+i(n)
we have
(4.4)
and
cl v
We shall make extensive use of the Helmholtz decomposition of vector fields. The
following result also can be found in [5].
LEMMA 4.3. Every function v E (L2()) 3 has the orthogonal decomposition
(4.5)
where q e
.
H()/n
F and V.
0 in
v- Vq +
and
Vxb
e (Hl(gt)) 3, such that V x b e H(div; ), n x b
0 on
Here
H(div; Ft)= (v e (n2(t))3 V.v e (n2(t))3}.
Alternatively, we may choose q H() and e H(curl;). In this case we may
construct so that V. 0 in and dp.n 0 on F.
Lemma 4.3 implies that we may write (cf. also [16]) the following orthogonal
decomposition:
(n:()) a VH (t)/T (V H0(curl; t))
(4.6)
and
{u e H(curl; )
H0(curl; gt)
n
u
0 on 0t}.
For convenience we shall define
(4.7)
M
VH()/n
so that
M +/Alternatively,
V
Ho(curl; ).
(L2(fl)) 3 has the following orthogonal decomposition
(L2(t)) 3 VH()
(4.8)
Both these orthogonal Helmholtz decompositions will be used in the following proofs.
Now we can state the final a priori estimate. Let
and J’.n 0 on F) and define u e H(curl; t) by
(4.9)
(4.10)
(4.11)
THEOREM 4.4. Suppose
U
(U2())
3
V.u
0in
u.n
O on F.
f e (L2()) 3
and
Ilull2
and let u satisfy
(4.9)-(4.11).
Then
1623
A MIXED METHOD FOR MAXWELL’S EQUATIONS
This theorem follows from Theorem 2.4 of
[5] together with the
fact that
(V x
u)n=OonF.
Next let us make a few further observations regarding the finite element space
Vh. Since 12 is convex, ’h C and hence the interpolant rh (defined as/ if K is a
boundary element) is defined uniquely without extending u. Since no extension of u
to 73 is needed, the function u h guaranteed in Theorem 3.2 can be taken to be the
unique interpolant rhU.
Of crucial importance to our analysis is the use of a discrete Helmholtz splitting
for Vh. Let
S
then
{qh e C()/n qhlg e Pk VK e Th}
vshk C Vh [20] and so if we define
(4.12)
.
Mh
Vh Mh Mh Remarkably, Girault and Raviart [14] show that rh
essentially maps M (cf. (4.7)) to Mh (strictly the proof in [14] holds only for polygonal
f but using this result on -h proves the result trivially for curved f).
LEMMA 4.5. If u Vp with p E HI(f)/T and if the interpolant of ’hU Uh is
defined, then
we may write
(4.13)
rhU--
Vph for some ph e
S.
.
This lemma is the reason for requiring f to be convex. If an operator ’h could
be found such that Theorem 3.2 and the previous lemma hold for general f, then all
the results in this section would extend to an arbitrary smooth bounded domain
4.2. Analysis of the operator IIh. Now we can define and analyze the operator IIh required by Theorem 2.1. For v H(curl; f), we define IIhv
C Vh to
satisfy
M
(4.14)
(V (IIhv- v), V Ch)= 0 V h e M-.
In view of the orthogonal decomposition of Vh and the definition of Mh in (4.12),
(4.14)
(4.15)
is equivalent to requiring that
(VXIIhv, Vh)
Vq
IIhv
Yh satisfies
(Vv,Vh) V heVh,
0
Vq e
This projection operator is the same projection as the one used in [12] except that
the boundary condition here is weakly enforced and corresponds to the continuous
condition v.n 0 on F. The projection in [12], which is extended to smooth domains
in [9], assumes the essential boundary condition n v 0 on F.
THEOREM 4.6. For any v H(curl; f),IIhv is uniquely determined by (4.14).
and v.n
0 in
0 on F then there exists a constant C
Furthermore, if V.v
independent of h and v such that
(a) If v e (Hk+l(f)) 3,
(4.17)
IIv
IIhVl[gc <_ C hkllVllk+.
1624
PETER B. MONK
(b) /f v e (Wl’S(f)) 3 for some s > 2
then for each 5
> O, there
is a
C
C(5, s)
such that
IIv IIhvll o + hl-]]V x (v- IIhv)llO <_
Remark. This theorem provides error estimates for the problem of constructing
an approximate vector potential in the case of the boundary condition v. n 0 on F
(for the case n x v 0 see [9], [18]). Suppose it is desired to approximate the function
v that satisfies
,
Vxv
by finding
Vh
E
Yh such that
(Vxv h,V x h)
(4.19)
(4.20)
V.v=O inf,,
0 onF
v.n
(vh, Vq h)
(j,,V x h)
0 Vq h e
vh E yh,
Shh
In this case v h
are
IIhv and the above theorem provides an error estimate for v v h.
Proof. Estimates for projections of this type (but different boundary conditions)
given in [12], [13], [9], and [18]. The proofs in [12], [13] are explicitly given
-
for the case of essential boundary conditions, but an extension to the case of natural
conditions is straightforward. However, since Theorem 4.6 is essential to the estimates
later, and in addition is the source of the nonoptimal factor h in Theorem 4.1, we
present a sketch of the proof here.
First let us prove uniqueness (which implies existence). If v 0 then IIhv satisfies
V XIIhv--0
inf,.
Vp for some p HI(f)/T4, but since IIhv Vh, p ph
ph O.
implies
IIhv Mh
To prove convergence estimates, we note that (4.15) implies that
Thus
Ilhv
d-
IIv x
Selecting w h
(4.21)
n )llo <_
IIv x
(3.7) shows that
IIV x (v- Ilhv)ll0 <_ C hkllVllk+,
0 implies the bound on V x (v IIhv) in (4.18).
rhV and using
(4.22)
rhV
H(curl; f)
Ilhv
w + Vp
satisfies
VX(rhV--IIhv), V.w=0
onF,
in
0
and p
Then
)11o w e
and selecting w h
To analyze the L 2 error, we use an idea due to Girault and Raviart
only prove (4.18) since (4.17) is proved similarly. We let
where w
S.
H(f)/T4 satisfies
(Vp, Vq)
(rhV Ilhv, Vq) Vq e H(f)/T4.
[14]. We shall
1625
A MIXED METHOD FOI:t MAXWELL’S EQUATIONS
Since rhV
IIhv E (L(f)) 3 for any t > 2, Theorem 4.2 implies w E (WI’()) 3 and
Vph for some ph
Thus rhW is defined, and so rhVp
(4.24)
rhV
hV
rhW
S by Lemma 4.5. Hence
+ Vph.
Now using the approximation roperties of rh well a scaling argument via the
curvilinear reference element g (along the lines of the proof of (3.1)) we have
(4.25)
[[w- rhw[]o Ch[Iw[[w,,,() Chl+3/f-3/2[[V x (rhV- nhV)[]0
C C(t) and t > 2. Then we set
3/2-3# and note that
where
can be
made arbitrarily close to zero by ting t close to 2. Adding and subtracting v to the
right-hand side of (4.25) and using (4.21) and Theorem 2.1 we have
(4.26)
C hl-llvllwx,,()
Ilw- rhW[[0
if
V
e (wl’s()) 3.
Now we write
(v nv, v nv)
(v n, v rv) + ( nv,
+(- n., )+ (- n., v,).
(4.7)
The first two terms can be estimated using the Cauchy-Schwarz inequality together
with Theorem 2.1 and (4.26). The lt term is zero since v M and HhV
Thus we must estimate (v- HhV, w), which we do using a duality argument. Let
M solve
M.
,v
(v
The existence of
)
(, )
v e H(cu; ).
and its smoothness are guaranteed vi Theorem 4.4. Then
(- n, )
(v (- n), v (- ))
c$v ( n.)o h
o.
(.es)
where we have used the a priori estimate of Theorem 4.4. The term
h already been estimated (see (4.21) and discussion there). But expanding w in
(4.28) we have
( n., )
(a.e)
Ch]V ( n)o {
+-
o
]0 +
no + vo}.
The first two terms in { } may be estimated from (4.26) and Theorem 3.2. It remains
to estimate Vph, but using (4.24) and the fact that w M
]]v (
, v) + (
n, v) + (
, v).
The second term on the right-hand side is zero since v M and HhV
first term is estimated using Theorem 3.2 and the lt is estimated using
conclude that
llv]o Ch-]v,,
U. (a.0), (a.ee), nd (a.) v (a.S).
(.30)
i. e
(W*’()) ".
M.
The
(4.26). We
1626
PETER B. MONK
4.3. Proof of Theorem 4.1. We prove Theorem 4.1 in two special cases which
combine to prove the desired result.
THEOREM 4.7. Suppose J Vj for some j E C(O,T;H()), Ho 0 and
E0 Vp0Efor some Po hH(gt). Suppose also that E CI(0, T; (Hk())3), k _> 1.
Then if R
Po and H (0) 0 there is a constant C independent of h and E such
that
(4.31)
_
II(E Eh)(t)llo <_ C hkllE(t)llk.
Remark. In this case H(t) Hh(t) 0 at all times.
Proof. We first show that H(t) Hh(t) O, 0 t. By Lemma 4.1 we may write
E(t)
V
A(t) + Vp(t)
for some p(t) e H() and A(t) e H(curl;gt). Substituting this in (1.8)-(1.9) and
H in (1.9), then adding, we conclude that
V A in (1.8) and
taking
(4.32)
(V At, V A) + (Ht, H)
0,
t
But H(0)
H0 0 and (V A)(0) 0 and hence (V
Similarly, in the finite-dimensional case we may write
(4.33)
E h (t)
V
> 0.
A)(t)
H(t)
O, 0 <_ t.
A h (t) + G h
Ah(t) Vh and G h (V Vh) +/- C Uh. Of course if G h (V Vh) +/-,
(4.34)
(G h, V C h) 0 VC h e Yh.
Using (1.11)-(1.12) with b h V A h and ch H h proves the analogue of (4.32)
for A h and H h. But Hh(o) 0 and Eh(o) PoEo so that
(V A h (0) + G h (0) E0, @h) 0 V h e Uh.
Recall that E0 Vp0 (V H(curl; t)) +/-, and so taking h V Ah(o) above we
conclude that V Ah(o) --O. Thus
(4.35)
E(t) Vp(t), p(t) 6 H(), and H(t)= 0
Eh(t) Gh(t) 6 (V X Vh) 1, and Hh(t)= 0
(4.36)
where
Furthermore, using (1.8) and (1.11)we have
(4.37)
(4.38)
(Vpt, Vq)
(Gth,.h)
HI
(Vj, Vq) Vq e (t),
(Vj, h) v./h e (V Yh) +/-.
But the orthogonality of Vp and Vj with V
A for any A
H(curl; gt) implies that
(L2(Ft)) 3,
and similarly, since G h and Vj are orthogonal to V A h, for any A h Yh we have
(G, u h) (Vj, u h) Vu h e Vh.
(4.40)
(4.39)
(Vpt, u)
(Vj, u) Vu e
1627
A MIXED METHOD FOR MAXWELL’S EQUATIONS
Subtracting
(4.40) from (4.39) and using (4.35)-(4.36)
(( E), ) 0 W
proves that
Hence, using the L2 projection P0, we conclude that
((POE Eh)t, PoE E h) O.
Since Eh(o) PoE(O), we conclude that Eh(t) PoE(t), t > 0 and
II(E- En)(t)llo I1(I- Po)E(t)llo.
1-1
An application of the estimate in (3.10) completes the proof.
THEOREM 4.8. Suppose J V x j for some j E C(O, T; (H2(D))3) and suppose
H e CI(0,T; (W1’8(2)) 3) for some s > 2. Suppose also that E e CI(0, T; (H1(2))3),
that
Eo (VH(D)) A-,
and that pE
Po.
Finally, suppose that
(4.41)
Then for any
> 0 there
exists a constant
C
C(, s) such that
(4.42)
Proof. By Lemma (4.3) we know that
E0
and A may be chosen so that V.A 0 in fl, and A.n 0 on F. But using the test
function Vq, q e H0(2) in (1.8), we conclude that E(t) e (VH(fl)) 0 _< t _< T
and hence
-c,
(4.43)
with V.A
t _> 0
V x A(t), A(t) e H(curl; 2),
0 on F. Also, by Theorem 4.2,
0 in 2 and A.n
E(t)
we know that
A e (H2()) 3 with
(4.44)
Unfortunately, in this case
Eh(t) does not satisfy a discrete analogue of (4.43), but
we may write
E h (t)
(4.45)
where
h
h(t)
Gh in (1.11)
,h (t) + G h (t)
Vh and Gh(t) (V x Vh) +/-. It is easy to estimate G h. Taking
and using the fact that G h (V x Vh) +/- we have
X
,
(, ) (v x ) (v x ( ), ().
Hence, by integrating the above inequality and using Theorem 3.2,
(4.46)
II(h(t)llo< Ilah(0)llo / II--rhllH&
< IIGh(0)llo / c hllllL(O,;(H(m)"
1628
PETER B. MONK
To estimate the initial data, note that if R E
Po,
((0) Eo, (0))
0,
and hence
(Gh(o)
V
(A(0) rhA(O)), Gh(o))
O.
An application of the Cauchy-Schwarz inequality together with Theorem 3.2 and
(4.44) shows that
(4.47)
Combining
(4.47) and (4.46) shows that
(4.48)
To estimate E- h, we use Theorem 2.1. Let
U (1)
(VH(n))
.
Then by virtue of (4.43), we may write
H(t)) e U (1) x H(curl; fl) such that
(1.8)-(1.9)
(E(t),
(J, ) V e V (),
0 V e g(u; n).
(E, ) (V x H, )
(H, ) + (E, V x )
(4.49)
(4.50)
the problem of finding
U)= V x Vh we have that (h(t),Hh(t)) e (U), Vh)satisfies
h
gh ,)
(E,)-(V
(J, h Ve vl)
Similarly, defining
(4.51)
(a.2)
(H, )+
Now we apply Theorem 2.1 using the spaces U U (1) and V H(curl; ) with finitedimensional subspaces ) V Vh and Vh. We can take Ph to be the orthogonal
projection of (/(n)) a onto ) and Hh to be the projection analyzed in 4.2.
Let us first check hypothesis (Ul). If u (/:(n)) a
U
U
(Pu- u, V
v)
0 Vv
V,
-
since this is just the definition of Ph. Moreover,-since E
) we
may write
Ph" (L()) a
U
0
for some C h
(P(V A)-V A,V
Vh and
we may choose
v)
Ch
(V C
and similarly
(4.54)
PE- Eo
A(t)
and since
A,V v )
HhA. Hence from Theorem
(4.44)
(4.53)
V
V
Ch]E
4.6 and
1629
A MIXED METHOD FOR MAXWELL’S EQUATIONS
Next we turn to (H2). u h E U(h 1) implies u h
if v e H(curl; t) by (4.15) we conclude that
C h for some C h e Vh and hence
V
(rI- ), )= (v (I- v), v c )
(v
0
is satisfied. Since H is such that V.H 0 in gt and H.n 0 on F we can
use Theorem 4.6 to estimate H- IIhH and Ht- IIhHt.
It remains to analyze the initial condition for
A little care is needed since
and
(H2)
pE Uh
--+
U. Since pE
h.
Po,
( (0) + ( (0)
Picking
_
Hence
Uh
V
Vh
for some v h
E0, u )
V.
0 Vu
Yh, we have
((0)- E0, V
v)
0 Vv e
V.
h(0)
PhEo. Now we may apply part (b) of Theorem 2.1, the estimates
for Ph in (4.53)-(4.54), and the estimates for IIh in (4.18), together with (4.41) to
conclude the estimate
(4.55)
II(E- h )(t)[Io + [I(HCh 1-5
- -
{I]EOlll --IIHollw1,8(ft) IIEtllL(O,t;(H(ft))3) I]Htl]L(O,t;(w,8(a))3) }.
Here we have also used estimates like (3.14)-(3.15) to simplify the right-hand side.
The proof is completed by noting that
II(E- Eh)(t)llo <_ IIGh(t)llo + II(E- h)(t)ll0
and so adding (4.48) and (4.55) gives the desired result.
Proof of Theorem 4.1. Now we can prove Theorem 4.1. By Lemma 4.3,
J=Vj+Vj,
H
j e H(curl; ft) and j e
(). Furthermore, by Lemma 4.2 and Theorem 1.10 of
w know that e H(a) nd I111 CIIJII.
In the same way we may write
E0
V
A0 + Vp0,
[14]
-<
where
A0 e H(curl; t),
P0
e
H(t).
Hence, for the purposes of analysis the problem can be split by linearity into two
subproblems. The first problem has current J(1) Vj, and initial data H(0) 0,
and E(0)
Vp0. Estimates for the problem are provided by Theorem 4.7. The
second problem has data j(2) V j, and initial data E0 V A0 and H (0) H0.
Estimates for this problem are provided in Theorem 4.8. In both cases the necessary
smoothness is guaranteed by the a priori assumptions of Theorem 4.1 (using arguments
similar to those sketched above for obtaining the smoothness of j). After estimates
for each subproblem have been obtained, the use of the triangle inequality then proves
the desired result.
1630
PETER B. MONK
In the previous section we made ex-
4.4. The discrete Helmholtz splitting.
tensive use of the discrete Helmholtz splitting
(v x
(v x
-L.
In this short section we remark that it is possible to characterize (V Vh) +/- in some
cases. In this section we shall assume that gt is polygonal and k
1. It is hoped in
the future to extend our result to curved domains and higher-order spaces.
Define the space Nh as follows:
(Ph PhlK E P1, VK E Th
in addition Ph is continuous
at the centroid of any face f common to two elements in Th
and vanishes at the centroid of faces f on
Nh is a three-dimensional analogue of the two-dimensional nonconforming space introduced in [25]. We denote by VNh the space of piecewise constant functions that
are piecewise gradients of functions in Nh. Thus
VNh
{Uh e Uh
3p h e Nh such that uhlg
V(phlK) VK e Th}.
The following theorem and proof are motivated by the work of Arnold and Falk [3].
THEOREM 4.9. If k
1 and 12 is polygonal, Uh has the following orthogonal
decomposition:
U
(4.56)
VxV
VN.
Remark. On a polygonal domain with k
1 this theorem shows exactly the
sense in which E h approximates E in Theorem 4.7. In that case E
Vp(t) and
Eh
Vph(t) where ph Nh. Our proof of Theorem 4.7 (which also holds for
polygonal domains) is essentially a novel proof of convergence of a nonconforming
method for Laplace’s equation.
From the proof of Theorem 4.8 we saw that if J V j and E0 V A0 then
E(t) V x A(t) but in general E h (t) 7 A h + Vph, where A h Vh and ph Nh.
However, if it is desired that Eh(t) be entirely divergence free, we could compute
_h V A h at each time by computing ph, which would require the solution of a
nonconforming Dirichlet problem.
Proof. Following Arnold and Falk [3], we first show the orthogonality of functions
in V x Vh and TNh. Let U h V Vh and G h VNh; then U h V V h for some
v h Vh and G h Vph some ph Nh. But
u h ( h dx
E (V x v h, Vph)g
KETh
-(v.(v
+
x
gETh
where nK is the unit outward normal to K. Of course V.(V x v h) 0 (trivially, since
vhlg (P1)3!). Furthermore, if f is a face common to two elements K1 and K2 then
(nK1.V vh,ph)oKnf--(nK2.V vh,ph)oK2nf
x
ph
1631
A MIXED METHOD FOR MAXWELL’S EQUATIONS
where we have used the fact that up to a minus sign, n.V v h is continuous at
interelement boundaries (since n.V v h is nothing more than the surface curl on f
[20]). But n.V v h is constant on f, and hence if c, represents the centroid of f,
simple quadrature on f [7] shows that
0.
Similarly if f C F, < n.V vh,ph >f-- 0 since ph is zero at the centroid of f. We
have thus shown that
uh.G h dc O.
It remains to show the equality in (4.56). Let NI be the number of faces in the
triangulation and Nt be the number of tetrahedra. Then Ndlec [21] shows that
fa
dim
V x Vh
NI,
-N
and of course
dim
Finally, we note that if ph E Nh with
dim
3Nt.
Uh
Tphlg
V Nh
dim
dimVNh+ dimVxVh
0 for all K
4Nt
Nh
Th then Ph
O. Thus
NI
4N-Nf + Nf-N
3Nt--dimUh,
and the result is proved.
5. The fully discrete problem. In this section we shall make some observations regarding the implementation of a fully discrete version of (1.11)-(1.12). We
shall ignore the important problem of how to approximate the curved domain. For
such vital aspects as quadrature on curved elements, the approximation of curved
domains in 3, and isoparametric methods, consult [17], [6], [9].
In this paper, the .choice of the Ndlec space Vh and the discontinuous space
has
been shown to be a stable and accurate pair of spaces for the semidiscrete
Uh
problem. In particular, the linear Ndlec space (k
1) for Vh and the piecewise
constant space for Uh appear to be an attractive pair for computation. The choice of
time discretization is also extremely important.
There are many possible timestepping methods that can be applied to (1.11)(1.12). For example, the use of Newmark’s method is discussed in [1]. Another
timestepping method used in computational electromagnetics [31] is the leapfrog
scheme. In this scheme we approximate Eh(t) at times
nat, 0
Let {n} be the vector of degrees of freedom of E h at time tn. Hh(t) is approximated at time tn+l/2 with vector of unknowns {7"tn+i/2}n= o. The initial value 7"t 1/2
can be computed using, for example, a second order in time Taylor series method.
Given (t n, 7-tn+1/2), the new approximation (n+1, t+3/2) is obtained by successively solving the equations
n
(5.1)
(5.2)
Me
(n+l
n)At +Ma (n+12t-’n)-Mc’n+l/2--,rn+l/22
M,
(’n+3/2--’n+l/2)
At
+ MTcn+l
0.
1632
PETER B. MONK
,.-n+l/2
is the right-hand side representing the current J at tn+l/2. These
obtained from (1.11)-(1.12) using centered differences in time. The
are
equations
matrix M is a symmetric, positive-definite block diagonal matrix of size dim Uh
dim Uh. M is positive semidefinite with the same block structure as M. Mc is
is also symmetric and positive-definite and of size dim Vh
dim Uh dim Vh.
to be well conditioned.
dim Vh. Furthermore, if Th is uniformly regular we expect
To timestep (5.1)-(5.2), given t ’ and T/n+1/2, we first compute ’+1 from (4.55)
by solving the block-diagonal problem with matrix Mc/At + Ma/2. Here the diagonal
blocks correspond to the unknowns on each element. Once +1 is known, we can
use 7+/2 and t +1 in (5.2) to compute 7,+3/2. This involves solving a system
with matrix
The stability of the above leapfrog timestepping method applied to Maxwell’s
equations has not yet been analyzed. It has, however, been used with great success for timestepping finite difference methods for Maxwell’s equations (cf. [28]), so
there is reason to believe it will be successful for the finite element scheme. Indeed,
computations in -2 using the two-dimensional analogue of Ndlec’s elements, and
leapfrog timestepping, suggest that the method analyzed in this paper will possess
good accuracy and dispersion properties when applied in 7 3 [19].
If a fully implicit scheme is used (although this appears a difficult choice in practice) we can show that the method is stable and hence convergent (cf. [1] where an
implicit scheme for Maxwell’s equations in two dimensions is used and analyzed).
Specifically, if we apply the Crank-Nicolson scheme to (1.11)-(1.12) we obtain the
fully discrete problem of computing a sequence of vector fields (n, 7"/n) 0 <_ n
Here
M
M
M.
(5.4)
M,
At
+ MT
2
0.
This fully discrete scheme is stable. To prove stability, take ,+1/2 0, 0 _< n < oc
and multiply (5.3) by (n+ + t)T on the left. Then multiply (5.4) on the left by
(7 +1 + Tin) T and adding the resulting equations we obtain
_
(n+ + n)TMe (+l_n)) + nn+ nn) TM, .( n+l--nAt
_(n+l + n)TMa (’+1+)
2
)
The above equality is the discrete analogue of the continuous conservation of energy
for Maxwell’s equations. Since M is positive semidefinite, we conclude that
--n+l
(n+l)TMcn+i + (7 n+l) M’]’tT-
(n)TMn + (n)TM, nn.
And iterating this equality implies that
(g+)TM+ + (tn+)TM, n +1 <_ (g)TM + (n)VM, n
M
.
are symmetric positive definite, we conclude that
Since the matrices M and
the fully discrete problem is stable. Furthermore, if the choice of Ndlec space and
discontinuous space analyzed in this paper are used, the method converges with error
O((/kt) 2 + h k) provided the continuous solution (E(t), H(t)) is smooth enough.
A MIXED METHOD FOR MAXWELL’S EQUATIONS
1633
Although the Crank-Nicolson method has a large work estimate since it is fully
implicit, the above analysis shows that it is possible to obtain an accurate and stable
fully discrete method for Maxwell’s equations using the finite element spaces analyzed
in this paper. Clearly, an important problem is to investigate other timestepping
methods to find a fast and stable method for implementing the three-dimensional
problem (cf. [1]). Three-dimensional numerical tests are also needed.
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