C. R. Acad. Sci. Paris, t. 337, Série I, p. 71–74, 2003
Analyse numérique/Numerical Analysis
Mixed finite elements for incompressible
magneto-hydrodynamics
Anna Schneebeli a , Dominik Schötzau a
a
Department of Mathematics, University of Basel, Rheinsprung 21, CH-4051 Switzerland
E-mail: schneeba@math.unibas.ch, schotzau@math.unibas.ch
Abstract.
We present a new mixed finite element discretization for three-dimensional stationary incompressible magneto–hydrodynamics. The fluid variables are discretized by standard
inf-sup stable velocity-pressure pairs and the magnetic ones by a mixed approach using
Nédélec’s elements of the first kind. The resulting method is shown to be quasi-optimally
convergent. c 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier
SAS
Méthodes d’éléments finis mixtes pour la magnéto-hydrodynamique incompressible
Résumé.
Nous présentons une nouvelle méthode d’éléments finis mixtes pour les équations stationnaires tridimensionnelles de la magnéto-hydrodynamique incompressible. La partie fluide
est discrétisée par des couples d’espaces standards vitesse-pression, stables selon la condition inf-sup, et la partie magnétique par une approche mixte utilisant les éléments de
Nédélec de première espèce. Nous montrons que la méthode qui en résulte converge de
façon quasi-optimale. c 2003 Académie des sciences/Éditions scientifiques et médicales
Elsevier SAS
1.
Introduction
Incompressible magneto-hydrodynamics (MHD) describes the flow of a viscous, incompressible and electrically conducting fluid and arises in several engineering applications such as liquid metals in magnetic
pumps or aluminum electrolysis. Over the last few years, several finite element approximations have been
proposed for such problems that are based on nodal (i.e. H 1 -conforming) finite elements for the magnetic
field, combined with standard discretizations of the fluid variables. We mention here only [1, 4, 5, 6, 7] and
the references therein. However, it has been known for some time that in non-convex polyhedra of engi3 and that a nodal FEM discretization,
neering practice, the magnetic field may have regularity below H 1
albeit stable, can converge to a magnetic field that misses certain singular solution components induced by
reentrant vertices or edges; see [2]. In this note, we present an alternative mixed finite element approximation for incompressible MHD problems based on the Sobolev space H
. We use Nédélec’s first
family of elements for the discretization of the magnetic field, inf-sup stable velocity-pressure pairs for the
hydrodynamic unknowns and standard elements for an additional Lagrange multiplier related to the divergence constraint on the magnetic field. The resulting method is shown to lead to quasi-optimal error bounds
in general Lipschitz polyhedra.
(
)
(url; )
c 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
71
A. Schneebeli, D. Schötzau
(
)
For a Lipschitz domain R3 , we write k ks to denote the norm in the usual Sobolev space H s
,
s > . The L2 -based inner product is ; . We use the same notation for vector fields. We define
H01
as the subspace of functions in H 1
with zero trace on . The space H
is the space of
3 with
vector fields ~ 2 L2
~ 2 L2 3 , endowed with the graph norm k kurl . H0
is the
subspace of H
of functions with zero tangential trace.
0
(
)
2.
(
)
(
)
(url; )
( )
(
)
(
)
url
(url; )
(url; )
Mixed formulation of incompressible magneto-hydrodynamics
Let be a bounded Lipschitz polyhedron in R3 . For simplicity, we assume that is simply-connected, and
that its boundary is connected. The incompressible MHD problem we consider is to find the velocity
field ~u, the pressure p, the magnetic field ~b, and the scalar function r satisfying
Rs 1 ~u + (~u r)~u + rp S url ~b ~b = f~
R 1 S url(url ~b) S url(~u ~b) rr = ~g
m
in ;
in :
in ;
div ~u = div ~b = 0
(1)
Here, Rs is the hydrodynamic Reynolds number, Rm the magnetic Reynolds number, S the coupling
3 are given source terms. We complete the above system with the homogeneous
number, and f~, ~g 2 L2
boundary conditions ~u ~ , ~n ~b ~ , and r ~ on , with ~n denoting the outward normal unit vector
~b
to . Note that the scalar function r is the Lagrange multiplier associated to the constraint
. Its
purpose is to render the formulation and its discretization stable; cf. [3]. By taking the divergence of the
r
~g in , r
on . In particular, we have r for a
second equation in (1), we obtain
solenoidal source term ~g.
~
H01 3 , Q L2 =R, C~
H0
, and S
H01 , the weak
By introducing the spaces V
~ Q C~ S such that
formulation of (1) reads: find ~u; p; ~b; r 2 V
(
)
=0
=0
=0
= div = 0 := (
) := (
)
:= (url; )
(
)
~ ~v );
as (~u; ~v ) + 0 (~u; ~u; ~v ) 1 (~b; ~v ; ~b) + bs (p; ~v ) = (f;
am (~b;~) 2 (~b; ~u;~) + bm (r;~) = (~g ;~);
bs (q; ~u) = bm (s; ~b) = 0;
) 2 V~ Q C~ S . Here, we use the forms
as (~u; ~v ) := Rs 1 (r~u; r~v );
bs (q; ~v ) := (q; div ~v );
~ ~v );
1 (d~; ~v ; ~b) := S (url ~b d;
1 ~ r)~u; ~v) 1 ((w~ r)~v; ~u):
0 (w
~ ; ~u; ~v ) := ((w
2
2
:=
div = 0
0
(
)
(2)
(
for any ~v ; q;~; s
am (~b;~) := Rm1 S (url ~b; url ~);
bm (s;~) := (rs;~);
~ url ~);
2 (d~; ~u;~) := S (~u d;
The recent results in [9] show that the weak formulation in (2) is well-posed and that the following existence
and uniqueness result holds.
~ ~g 2 L2 3 , there exists at least one solution ~u; p; ~b; r in V~ Q C~ S
T HEOREM 1. – For any f;
of the mixed formulation in (2). Moreover, there exists a constant C
solely depending on such that for
1
C maxf1;S g [kf~k2 +k~gk2 ℄ 2
small data with minfR 2 ;R 20S 2 g 0 < the solution is unique.
s
m (
)
(
1
3.
Finite element discretization
Let Th be a regular and quasi-uniform partition of
of the element K 2 Th , and set h
K 2Th hK .
= max
72
)
into tetrahedra fK g. We denote by hK the diameter
Mixed finite elements for incompressible magneto-hydrodynamics
~h V~ and Qh Q, which
To discretize the Navier-Stokes operator, we use standard finite element pairs V
are based on the mesh Th and are assumed to be inf-sup stable independently of the mesh-size. We further
assume the following standard approximation property to hold:
inf~ k~u
~v2Vh
( )
(
)
~v k1 +
inf kp
q2Q
h
q k0 6 Chminfs;kg k~uks+1 + kpks
(
)
(3)
1
3 H s , s > 1 , and an approximation order k > .
for ~u; p 2 H s+1
2
For the Maxwell operator, we use Nédélec’s first family of spaces [8] combined with a standard H 1 conforming space. To this end, let Pk K be the space of polynomials of total degree k > on K
and Pek K the space of homogeneous polynomials of degree k on K . The space Dk K denotes the
polynomials ~
p in Pek K 3 that satisfy ~p ~x ~x
on K . For k > , we then set
( )
( )
( )
( )
( ) =0
0
1
C~ h = f ~ 2 C~ j ~ujK 2 Pk 1 (K )3 Dk (K ); K 2 Th g;
Sh = f s 2 S j sjK 2 Pk (K ); K 2 Th g:
~h Qh C~ h Sh such that
The finite element approximation of (2) is: find (~uh ; ph ; ~bh ; rh ) 2 V
~ ~v );
as (~uh ; ~v ) + 0 (~uh ; ~uh ; ~v ) 1 (~bh ; ~v ; ~bh ) + bs (ph ; ~v ) = (f;
(4)
am (~bh ;~) 2 (~bh ; ~uh ;~) + bm (rh ;~) = (~g ;~);
bs (q; ~uh ) = bm (s; ~bh ) = 0;
~h Qh C~ h Sh .
for any (~v ; q;~; s) 2 V
Using discrete Helmholtz decompositions, it can be easily seen that we have rh 0 for a solenoidal source
term ~g . Furthermore, a discrete version of Theorem 1 has been established in [9].
4.
Error analysis
The mixed method (4) leads to quasi-optimal error bounds.
1
C maxf1;S g [kf~k2 +k~gk2 ℄ 2
T HEOREM 2. – Assume that minfR 2 ;R 20S 2 g 0 < 21 . Then we have the error bounds
m s
k~u
~uh k1 + kp
0
ph k0 + k~ ~h kurl + kr rh k1
6 C inf~ k~u ~v k1 + ~q2infQ kp qk0 +
~v 2Vh
h
inf~ k~b ~kurl + sinf
kr
2S
~2Ch
h
sk1 ;
with a constant C > that is independent of the mesh-size.
~ h and Sh yields the
Using Theorem 2, assumption (3) and standard approximation results for the spaces C
3 H s , r 2 H 1+s , ~b 2 H s 3 ,
following convergence rates, see [9]. Let ~u; p 2 H 1+s
~b 2 H s 3 , for a regularity exponent s > 1 . Then we have
2
url
(
)
( )
(
)
(
)
(
)
(
)
k~u
~uh k1 + kp phk0 + k~ ~h kurl + kr rh k1 6 Chminfs;kg k~uks+1 + kpks + k~bks + k url~bks + krks+1 :
5.
Numerical results
We present numerical results for the following linearized and two-dimensional variant of (1):
~u + rp url~b d~ = f;~ urlurl~b url(~u d~) rr = ~g; div ~u = div ~b = 0; in ;
where is the L-shaped polygon = ( 1; 1)2 n [0; 1) ( 1; 0℄ and d~ the prescribed magnetic field
d~ = ( 1; 1). We further choose f~, ~g , and the boundary conditions so that the solution to the above
73
A. Schneebeli, D. Schötzau
problem is given by the strongest corner singularity for the underlying elliptic operator. The corresponding
hydrodynamic variables ~u and p are then
~u(~x) =
((1 + ) sin() () + os() 0 ())
;
( (1 + ) os() () + sin() 0 ())
p(~x) = 1 ((1+ )2 0 ()+ 000 ())=(1 );
() = sin((1+ )) os(w)=(1+ ) os((1+ )) sin((1 )) os(w)=(1 )+os((1
)), and with 0:5445 and (; ) denoting the polar coordinates of ~x = (x1 ; x2 ). The pair (~b; r) is
given by ~b(~x) = r(2=3 sin (2=3)) and r(~x) 0. We point out that the magnetic field ~b does not belong
to H 1 (
)2 and thus cannot be correctly captured by nodal elements; see [2].
with
The finite element approximations to this MHD solution are computed on a sequence of successively refined
square meshes fTi gi>1 , employing the general purpose finite element library deal.II; see [10]. The
mesh-size in the mesh Ti is proportional to i . We use lowest order two-dimensional Nédélec’s elements
for the field ~b, corresponding to rotated Raviart-Thomas elements, bilinear elements for r and inf-sup stable
Q22 Q0 elements for ~u; p . In Table 1, we show the errors and numerical convergence rates that are
obtained for each of the solution components. The numbers clearly show convergence in accordance with
8 that
the theoretical results in Section 4. Note that the H 1 -error for r vanishes within the accuracy of
was used to iteratively solve the resulting linear systems. This test demonstrates the ability of our mixed
.
method to resolve highly singular solutions whose magnetic components have regularity below H 1
2
( )
10
(
)
mesh
1
2
3
4
5
total dofs
195
675
2499
9603
37635
H 1 -error in ~u
1.5510 0
1.1310 0 0.46
8.0310 1 0.50
5.5910 1 0.52
3.8610 1 0.53
L2 -error in p
2.1110 0
1.3810 0 0.61
8.9710 1 0.63
5.9510 1 0.59
4.0210 1 0.57
curl-error in ~b
1
8.74
1 0.57
5.91
1 0.56
4.00
1 0.55
2.72
1 0.55
1.86
10
10
10
10
10
H 1 -error in r
1.1710 9
1.9510 9
3.3610 9
2.0110 9
1.9810 9
Table 1: Energy errors and convergence rates for singular MHD solution.
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