A MIXED DG METHOD FOR LINEARIZED INCOMPRESSIBLE
MAGNETOHYDRODYNAMICS
PAUL HOUSTON
∗,
DOMINIK SCHÖTZAU
† , AND
XIAOXI WEI
‡
Journal of Scientific Computing, vol. 40, pp. 281–314, 2009
Abstract. We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous Pk3 − Pk−1 elements whereas the magnetic
part of the equations is approximated by discontinuous Pk3 −Pk+1 elements. We carry out a complete
a-priori error analysis of the method and prove that the energy norm error is optimally convergent
in the mesh size. These results are verified in a series of numerical experiments.
Key words. Incompressible magnetohydrodynamics, mixed finite element methods, discontinuous Galerkin methods
1. Introduction. Incompressible magnetohydrodynamics (MHD) models the
interaction of viscous, electrically conducting incompressible fluids with electromagnetic fields. It has a number of industrial applications such as metallurgical engineering, electromagnetic pumping, stirring of liquid metals, and measuring flow quantities
based on magnetic induction; cf. [17].
The numerical simulation of incompressible MHD problems requires discretizing a
system of partial differential equations that couples the incompressible Navier-Stokes
equations with Maxwell’s equations. Various finite element methods (FEM) can be
found in the literature where the magnetic field is approximated by standard H 1 conforming finite elements, see, e.g., [3, 16, 19, 20] and the references therein. However,
in non-convex polyhedra of engineering interest, the magnetic field may have regularity
below H 1 and a nodal FEM discretization, albeit stable, can converge to a magnetic
field that misses certain singular solution components induced by reentrant vertices or
edges; see [13]. In the recent work [30], this drawback of nodal elements was overcome
by the use of Nédélec elements for the approximation of the magnetic field. Thereby,
a new variational setting for the formulation of incompressible MHD problems was
proposed. It is based on a mixed approach for the discretization of the Maxwell
operator and introduces a Lagrange multiplier related to the divergence constraint of
the magnetic field, cf. [27].
Over the last two decades, discontinuous Galerkin (DG) methods have become
an integral part of computational fluid mechanics and computational electromagnetics, see [11, 12, 22] and the references therein. DG methods are extremely versatile
and flexible; they can deal robustly with partial differential equations of almost any
kind, as well as with equations whose type changes within the computational domain.
Their intrinsic stability properties make them naturally suited for problems where
convection is dominant. Moreover, discontinuous Galerkin methods can easily handle
irregularly refined meshes and variable approximation degrees (hp-adaptivity). The
DG approximations of magnetic or electric fields can be based on standard polynomial
∗ School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7
2RD, UK, E-mail: Paul.Houston@nottingham.ac.uk.
† Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada.
E-mail: schoetzau@math.ubc.ca.
‡ Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada.
E-mail: weixiaoxi@math.ubc.ca.
1
shape functions, in contrast to curl-conforming or divergence-conforming elements
commonly used in computational electromagnetics. DG methods have already been
successfully applied to both ideal and viscous compressible MHD problems [26, 32].
In this paper, we propose and analyze an interior penalty discontinuous Galerkin
method for a linearized incompressible MHD model problem based on the mixed
formulation introduced in [30]. Our method combines the DG discretizations that
have been developed recently for incompressible flow problems and Maxwell’s equations. More specifically, the fluid unknowns are approximated using mixed discontinuous Pk3 − Pk−1 elements [8, 9, 31] while the magnetic variables are discretized with the
Pk3 − Pk+1 element pair proposed and analyzed in [24, 25]. We carry out a complete
a-priori error analysis for the proposed DG method, and show that the energy error
in all variables is convergent of order O(hk ) in the mesh size h. Our results further
show that the proposed DG method is able to correctly resolve the strongest magnetic singularities in non-convex polyhedral domains, in contrast to H 1 -conforming
elements.
The rest of the paper is organized as follows. In Section 2, we introduce an
interior penalty DG method for a linearized incompressible MHD model problem. In
Section 3, we state and discuss a-priori error estimates for the method. Section 4 is
devoted to the detailed proof of these estimates. In Section 5, we present a series
of numerical experiments validating our theoretical results. Finally, we present some
concluding remarks in Section 6.
2. Discretization of a model problem.
2.1. An MHD model problem. We consider the following linear and stationary MHD system based on the mixed formulation proposed in [30]: find the velocity
field u, the pressure p, the magnetic field b, and the scalar potential r such that
− ν ∆u + (w · ∇)u + γu + ∇p − κ (∇ × b) × d = f
κ νm ∇ × (∇ × b) + ∇r − κ ∇ × (u × d) = g
∇·u=0
∇·b=0
in
in
in
in
Ω,
Ω,
Ω,
Ω.
(2.1)
(2.2)
(2.3)
(2.4)
Here, we take Ω to be a Lipschitz polyhedron in R3 with unit outward normal n
on Γ = ∂Ω. We assume that Ω is simply-connected, and that its boundary Γ is
connected. The function w ∈ W 1,∞ (Ω)3 is a prescribed convective field, and d ∈
L∞ (Ω)3 is a given magnetic field. Typically these fields come from a linearization
process. The right-hand sides f and g are vector-valued source terms in L2 (Ω)3 .
The scalar function γ belongs to L∞ (Ω). We further assume that there is a positive
constant γ? such that
1
γ0 (x) := γ(x) − ∇ · w(x) ≥ γ? > 0,
2
x ∈ Ω.
(2.5)
Remark 2.1. The positivity of γ? in (2.5) is a purely technical (but standard)
assumption that facilitates dealing with the convection term in the error analysis.
However, in the absence of a reaction term (γ ≡ 0), the parameter γ? must be allowed
to be zero. While the proposed DG method is stable and well-defined in this case
as well, the error analysis becomes more involved and requires additional (duality)
arguments, see [8] for the Oseen operator.
Without loss of generality, we may assume that the length scale of Ω, and the
L∞ -norms of w and d are one. Then the equations contain three characteristic
2
parameters, namely the hydrodynamic Reynolds number Re = ν −1 , the magnetic
−1
Reynolds number Rm = νm
and the coupling number κ. The coupling number κ is
typically expressed as a function of the so-called Hartmann number Ha: κ = ννm Ha2 .
In many engineering applications (such as aluminum electrolysis), the magnitudes
of Rm and κ are of order one, whereas Re can be substantially larger; cf. [3]. We will
focus on this case and not make explicit our error estimates with respect to νm and κ.
We suppose that the boundary Γ of Ω can be partitioned into two disjoint parts.
That is, we have Γ = ΓDR∪ ΓN with ΓD ∩ ΓN = ∅. Throughout, we assume that ΓD is
non-empty and satisfies ΓD ds > 0. We then supplement the MHD system (2.1)–(2.4)
with the following boundary conditions:
u = uD
(pI − ν∇u)n = pN n
n × b = n × bD
r=0
on
on
on
on
ΓD ,
ΓN ,
Γ,
Γ.
(2.6)
(2.7)
(2.8)
(2.9)
Here, I is the identity matrix in R3×3 . We assume that pN ∈ L2 (ΓN ) and that
uD and bD are restrictions to the boundary of sufficiently smooth divergence-free
functions in Ω. Finally,Rwe notice that, if ΓN = ∅, the datum uD must satisfy the
compatibility condition Γ uD · n ds = 0.
Remark 2.2. The magnetic boundary conditions (2.8)–(2.9) are geared towards
Hartmann flow problems with insulating wall conditions in the setting of [17, Section 3.7.1]. The method can be readily extended to other types of electromagnetic
boundary conditions. For example, with only minimal changes, it is possible to specify
both b · n and (∇ × b) × n on Γ, corresponding to perfectly conducting wall conditions;
cf. [16, 17, 30].
Let Γ− = { x ∈ Γ : w(x) · n(x) < 0 } be the inflow boundary of Γ. We adopt the
(physically reasonable) hypothesis that
w(x) · n(x) ≥ 0
for all x ∈ ΓN .
(2.10)
Obviously, we then have Γ− ⊆ ΓD .
Remark 2.3. As in [30], the scalar potential r is the Lagrange multiplier associated with magnetic divergence constraint. By taking the divergence of (2.2), we see
that −∆r = ∇ · g in Ω, r = 0 on Γ. In particular, we have r = 0 provided that the
function g is divergence-free. In this case, the MHD problem (2.1)–(2.4) is the same
as the one studied in [16] or the linearized version of the one considered in [20].
To define the weak formulation of the MHD system, we introduce the Sobolev
spaces
V = u ∈ H 1 (Ω)3 : u = 0 on ΓD ,
C = { b ∈ H(curl; Ω) : n × b = 0 on Γ } ,
S = H01 (Ω) and Q = L2 (Ω). In the case where ΓN = ∅, we also need to enforce the
mean values of functions in Q to be zero. The weak formulation of the incompressible
MHD system (2.1)–(2.4) then consists in finding (u, b, p, r) ∈ H 1 (Ω)3 × H(curl; Ω) ×
Q × S, with u = uD on ΓD and n × b = n × bD on Γ, such that
A(u, v) + O(u, v) + C(v, b) + B(v, p) = (f , v)Ω − hpN n, viΓN ,
M (b, c) − C(u, c) + D(c, r) = (g, c)Ω ,
B(u, q) = 0,
D(b, s) = 0
3
for all (v, c, q, s) ∈ V × C × Q × S. Here, the bilinear forms are given by
Z
Z
A(u, v) =
ν ∇u : ∇v dx,
O(u, v) = ( (w · ∇)u + γ u) · v dx,
ZΩ
ZΩ
M (b, c) =
κ νm (∇ × b) · (∇ × c) dx,
C(v, b) =
κ (v × d) · (∇ × b) dx,
Ω
Z
ZΩ
B(u, q) = −
(∇ · u) q dx,
D(b, s) =
b · ∇s dx.
Ω
Ω
Under the above assumptions, the well-posedness of this problem follows from standard stability properties and the theory of mixed finite elements; see also [30] and the
references therein.
2.2. Meshes and trace operators. We consider a family of regular and shaperegular triangulations Th that partition the domain Ω into tetrahedra {K}. We denote
by FhI the set of all interior faces of Th , and by FhB the set of all boundary faces. We
always assume that FhB can be divided into two disjoint sets FhD and FhN of Dirichlet
and Neumann faces, respectively. That is, we assume that FhB = FhD ∪ FhN , where
ΓD = ∪F ∈FhD F and ΓN = ∪F ∈FhN F . As usual, hK denotes the diameter of the
element K, and hF is the diameter of the face F . The mesh size of Th is given by
h = maxK∈Th hK . Finally, we write nK to denote the outward unit normal vector on
the boundary ∂K of K.
Next, we introduce the average and jump operators. To do so, let F = ∂K ∩ ∂K 0
be an interior face shared by K and K 0 and let x ∈ F . Let ϕ be a generic piecewise
smooth function (scalar-, vector- or tensor-valued) and denote by ϕ and ϕ0 the traces
of ϕ on F taken from within the interior of K and K 0 , respectively. Then, we define
the mean value of ϕ at x ∈ F as {{ϕ}} = (ϕ + ϕ0 )/2. Furthermore, let u be a piecewise
smooth function and u a piecewise smooth vector-valued field. Analogously, we define
the following jumps at x ∈ F :
[[u]] = u nK + u0 nK 0 ,
[[u]]T = nK × u + nK 0 × u0 ,
[[u]] = u ⊗ nK + u0 ⊗ nK 0 ,
[[u]]N = u · nK + u0 · nK 0 .
On a boundary face F = ∂K ∩Γ, we set accordingly {{ϕ}} = ϕ, [[u]] = u n, [[u]] = u⊗n,
[[u]]T = n × u and [[u]]N = u · n.
2.3. Interior penalty formulation. For k ≥ 1, we define Pk (Th ) = { p ∈
L2 (Ω) : p|K ∈ Pk (K), K ∈ Th }, with Pk (K) denoting the polynomials of total
degree at most k on K, and introduce the finite element spaces
Vh = Pk (Th )3 ,
Ch = Pk (Th )3 ,
Qh = Pk−1 (Th ),
Sh = Pk+1 (Th ),
where we also impose zero mean value of the functions in Qh in the case ΓN = ∅.
We consider the following discontinuous Galerkin method: find (uh , bh , ph , rh ) ∈
Vh × Ch × Qh × Sh such that
Ah (uh , v) + Oh (uh , v) + Ch (v, bh ) + Bh (v, ph ) = Fh (v),
Mh (bh , c) − Ch (uh , c) + Dh (c, rh ) = Gh (c),
Bh (uh , q) = huD · n, qiΓD ,
Dh (bh , s) − Jh (rh , s) = 0
4
(2.11)
(2.12)
(2.13)
(2.14)
for all (v, c, q, s) ∈ Vh × Ch × Qh × Sh . Here, the forms Ah , Oh and Bh are related
to the discretization of the Oseen operator. We take the ones proposed and studied
in [8, 9, 21, 31]. The forms Mh , Dh and Jh are related to the discretization of the
Maxwell operator. We choose the ones corresponding to the non-stabilized Pk3 − Pk+1
interior penalty methods proposed and analyzed in [24, 25]. Finally, the form Ch
couples the Maxwell equations to the Oseen problem. These forms are defined next.
First, the form Ah is chosen as the standard interior penalty form
X Z
X Z
{{ν∇u}} : [[v]] ds
Ah (u, v) =
ν ∇u : ∇v dx −
K∈Th
K
X
−
F ∈FhI ∪FhD
Z
{{ν∇v}} : [[u]] ds +
F
F ∈FhI ∪FhD
F
X
F ∈FhI ∪FhD
νa0
hF
Z
[[u]] : [[v]] ds.
F
The parameter a0 > 0 is a sufficiently large stabilization parameter; see Proposition 2.4 below. For the convective form, we take the usual upwind form defined by
X Z
Oh (u, v) =
((w · ∇)u + γ u) · v dx
K∈Th
K
X Z
+
K∈Th
w · nK (ue − u) · v ds −
Z
∂K− \Γ−
w · nu · v ds.
Γ−
Here, ue is the value of the trace of u taken from the exterior of K and ∂K− = { x ∈
∂K : w(x) · nK (x) < 0 } is the inflow boundary of K. The form Bh related to the
divergence constraint on u is defined by
X Z
X Z
Bh (u, q) = −
(∇ · u) q dx +
{{q}}[[u]]N ds.
K∈Th
K
F ∈FhI ∪FhD
F
Next, we define the forms for the discretization of the Maxwell operator. The
form Mh for the curl-curl operator is given by
X Z
X Z
Mh (b, c) =
κ νm (∇ × b) · (∇ × c) dx −
{{κνm ∇ × b}} · [[c]]T ds
K
K∈Th
−
X Z
F ∈Fh
F
F ∈F
F
h
X κνm m0 Z
[[b]]T · [[c]]T ds.
{{κνm ∇ × c}} · [[b]]T ds +
hF
F
F ∈Fh
As for the diffusion form, to ensure stability, the stabilization parameter m0 > 0 must
be chosen large enough, see Proposition 2.4 below. The form Dh for the divergence
constraint on b is given by
X Z
X Z
Dh (b, s) =
b · ∇s dx −
{{b}} · [[s]] ds.
K∈Th
K
F ∈Fh
F
The form Jh is a stabilization term that ensures the H 1 -conformity of the multiplier rh .
It is given by
Z
X
s0
[[r]] · [[s]] ds,
Jh (r, s) =
κνm hF F
F ∈Fh
5
with s0 > 0 denoting a positive stabilization parameter. The dependence on νm and κ
is chosen so as to suitably balance the multiplier terms in our error analysis.
Finally, for the coupling form Ch in our DG formulation, we take a discontinuous
Galerkin version of the bilinear form C, namely:
Z
X Z
X
κ
Ch (v, b) =
κ
{{v × d}} · [[b]]T ds.
(v × d) · (∇ × b) dx −
K
K∈Th
F ∈FhI ∪FhN
F
With these forms, the source terms Fh (v) and Gh (c) must be chosen as
Z
X Z
Fh (v) =
f · v dx −
ν∇v : (uD ⊗ n) ds
Ω
F ∈F D
F
h
X νa0 Z
X Z
+
uD · v ds −
κ(v × d) · (n × bD ) ds
hF F
F
F ∈FhD
F ∈FhN
Z
X Z
−
w · nuD · v ds −
pN n · v ds,
Γ−
F ∈FhN
F
and
Z
g · c dx −
Gh (c) =
Ω
X Z
F ∈FhB
κνm (∇ × c) · (n × bD ) ds
F
X κνm m0 Z
(n × bD ) · (n × c) ds
+
hF
F
F ∈FhB
X Z
−
κ(uD × d) · (n × c) ds,
F ∈FhD
F
respectively.
2.4. Stability. The stability properties of the above DG forms have been well
established in the recent literature on DG methods. To review them, we introduce
the norms
X
X
2
kuk21,h =
k∇uk2L2 (K) +
h−1
F k[[u]]kL2 (F ) ,
K∈Th
F ∈FhI ∪FhD
1
2
kuk2V = νkuk21,h + kγ0 uk2L2 (Ω) +
1
1 X
k|w · n| 2 [[u]]k2L2 (F ) .
2
F ∈Fh
In the last term, n denotes any of the outward normals on F . For the magnetic field,
we define
X
X
2
|b|2C = κνm
k∇ × bk2L2 (K) + κνm
h−1
F k[[b]]T kL2 (F ) ,
F ∈Fh
K∈Th
kbk2C
=
κνm kbk2L2 (Ω)
+
|b|2C .
Finally, on the magnetic multiplier space, we introduce
X
X
−1
−1
2
krk2S = κ−1 νm
k∇rk2L2 (K) + κ−1 νm
h−1
F k[[r]]kL2 (F ) .
F ∈Fh
K∈Th
6
First, we recall the following coercivity properties of Ah , Oh and Mh , see [4, 8, 24]
and the references therein.
Proposition 2.4. Under the assumption (2.5), there is a threshold value a0 > 0,
independent of the mesh size, ν, νm and κ, such that for every a0 ≥ a0 there is a
constant C > 0 independent of the mesh size, ν, νm and κ such that
Ah (u, u) + Oh (u, u) ≥ Ckuk2V ,
u ∈ Vh .
Moreover, there is a threshold value m0 > 0, independent of the mesh size, ν, νm ,
and κ, such that for m0 ≥ m0 there is a constant C > 0 independent of the mesh size,
ν, νm and κ such that
Mh (b, b) ≥ C|b|2C ,
b ∈ Ch .
Next, we recall that the velocity/pressure pair Vh × Qh is inf-sup stable; cf. [7,
Remark II.2.10] and [21, Proposition 10]:
Proposition 2.5. There is a stability constant C > 0 independent of the mesh
size, ν, νm and κ such that
inf sup
p∈Qh u∈Vh
Bh (u, p)
≥ C > 0.
kuk1,h kpkL2 (Ω)
There is no inf-sup condition available for the pair Ch × Sh . However, the underlying conforming spaces are stable, see [27]. To discuss this, we introduce the
conforming spaces Cch = Ch ∩ C and Shc = Sh ∩ S. The space Cch is the Nédélec finite
element space of the second type of order k [27, 28], with zero tangential trace on Γ.
The space Shc is the space of continuous polynomials of degree k + 1, with zero trace
on Γ. Thus, we may decompose Ch and Sh into
Ch = Cch ⊕ C⊥
h,
Sh = Shc ⊕ Sh⊥ ,
(2.15)
respectively. Obviously, the norms of the jumps
X
X
2
−1
2
|b|2C ⊥ = κνm
h−1
|r|2S ⊥ = κ−1 νm
h−1
F k[[b]]T kL2 (F ) ,
F k[[r]]kL2 (F ) ,
F ∈Fh
F ∈Fh
⊥
define norms on C⊥
h and Sh , respectively. The following norm-equivalence results
from [24, Theorem 4.1] are essential to our error analysis.
Proposition 2.6. There is a constant C > 0, independent of the mesh size, ν,
νm and κ, such that
CkbkC ≤ |b|C ⊥ ≤ kbkC ,
CkrkS ≤ |r|S ⊥ ≤ krkS ,
⊥
for any b ∈ C⊥
h and r ∈ Sh .
For the conforming pair Cch × Shc , the following properties of the forms M and D
hold true, see [24, Lemma 5.3] for a proof.
Proposition 2.7. There exists a constant C > 0 independent of the mesh size,
ν, νm and κ such that
M (b, b) ≥ Ckbk2C ,
7
for any b in Xch , where
Xch = { b ∈ Cch : D(b, s) = 0
∀ s ∈ Shc }.
(2.16)
Furthermore, there exists a second constant C > 0 independent of the mesh size, ν,
νm and κ such that
infc sup
r∈Sh b∈Cc
h
D(b, s)
≥ C > 0.
kbkC krkS
Employing the above properties and applying arguments as in [24, Proposition
3.3], existence and uniqueness of discrete approximations can be readily shown provided that a0 ≥ a0 , m0 ≥ m0 and s0 > 0.
3. A-priori error estimates. In this section, we present and discuss the main
result of this paper: a-priori error estimates for the proposed DG method.
We shall suppose that the solution (u, b, p, r) of the MHD problem satisfies the
regularity properties
(u, p) ∈ H σ+1 (Ω)3 × H σ (Ω),
τ
3
τ
3
(b, ∇ × b, r) ∈ H (Ω) × H (Ω) × H
τ +1
(Ω),
for σ > 21 ,
(3.1)
1
2.
(3.2)
for τ >
These regularity assumptions are realistic. This can be seen from the regularity properties of the Maxwell operator and the linearized Navier-Stokes operator in
polyhedral domains, respectively, see [2, 14]. In particular, the strongest magnetic
singularities satisfy assumption (3.2).
The following theorem represents the main result of this article.
Theorem 3.1. Let the solution (u, b, p, r) of the MHD problem satisfy the regularity assumptions stated in (3.1) and (3.2). Further, let (uh , bh , ph , rh ) denote the
DG approximation defined in (2.11)–(2.14). Assuming (2.5) holds, the errors can be
bounded by
ku − uh kV + kb − bh kC + kp − ph kL2 (Ω) + kr − rh kS
1
≤ Chmin{σ,k} kukH σ+1 (Ω) + ν − 2 kpkH σ (Ω)
+ Chmin{τ,k} kbkH τ (Ω) + k∇ × bkH τ (Ω) + krkH τ +1 (Ω) ,
where C is a positive constant, independent of the mesh size and ν.
Remark 3.2. For smooth solutions, the estimate in Theorem 3.1 ensures convergence rates of order O(hk ) in the mesh size h. This rate is optimal in the approximation of the velocity, the pressure and the magnetic field in the respective norms,
but suboptimal by one order in the approximation of the multiplier r with respect to
the norm k · kS . This is due to the fact that we are using polynomials of degree k + 1
to approximate r. The same suboptimal result is observed for the conforming Nédélec
family of the second type [28, 27]. On the other hand, the use of polynomials of
degree k + 1 for the magnetic multiplier leads to optimal convergence rates in the L2 error in the magnetic field b, in contrast to the use polynomials of degree k, cf. the
discussion in [23, 24].
Remark 3.3. Our error estimates also hold in the case where the conforming
Nédélec pair Cch × Shc is used for the approximation of b and r. While these spaces
have less degrees of freedom than their discontinuous counterparts Ch and Sh , the use
8
of discontinuous approximations for the magnetic field has several advantages. For example, DG approximations can be based on standard polynomial shape functions which
greatly facilitates the implementation of higher-order elements and magnetic boundary
conditions. Moreover, they are naturally suited to deal with irregularly refined meshes
and variable approximation degrees (hp-adaptivity).
4. Proof of the error estimates.
4.1. Preliminaries. For the purpose of our analysis, we set V(h) = V + Vh ,
C(h) = C + Ch and S(h) = S + Sh . Using the lifting operators constructed in [4, 31]
and [23, 24], it is then possible to extend the discrete bilinear forms Ah , Bh , Mh , Dh
eh : V(h) × V(h) → R, B
eh : V(h) × Q → R, M
fh : C(h) × C(h) → R
to bilinear forms A
e h : C(h) × S(h) → R, respectively. The extended forms are continuous:
and D
eh (u, v)| ≤ C νkuk1,h kvk1,h
|A
∀ u, v ∈ V(h),
(4.1)
fh (b, c)| ≤ C |b|C |c|C
|M
eh (u, q)| ≤ Ckuk1,h kqkL2 (Ω)
|B
∀ b, c ∈ C(h),
(4.2)
∀ u ∈ V(h), q ∈ Q,
(4.3)
∀ c ∈ C(h), r ∈ S(h),
(4.4)
e h (c, r)| ≤ C kckC krkS
|D
with constants C > 0 that are independent of the mesh size, ν, νm and κ. Moreover,
the extended forms are constructed in such a way that
eh (u, v) = Ah (u, v),
A
eh (u, p) = Bh (u, p),
B
fh (b, c) = Mh (b, c),
M
e h (b, r) = Dh (b, r),
D
(4.5)
for all discrete functions u, v ∈ Vh , b, c ∈ Ch , p ∈ Qh and r ∈ Sh , as well as
eh (u, v) = A(u, v),
A
f
Mh (b, c) = M (b, c),
eh (u, p) = B(u, p),
B
e
Dh (b, r) = D(b, s),
(4.6)
for all u, v ∈ V, b, c ∈ C, p ∈ Q and r ∈ S.
Suppose now that (u, b, p, r) is the solution of the MHD equations. We define,
for any v ∈ Vh , c ∈ Ch and s ∈ Sh , the following functionals
eh (u, v) + Oh (u, v) + Ch (v, b) + B
eh (v, p) − Fh (v),
RA (v) = A
fh (b, c) − Ch (u, c) + D
e h (c, r) − Gh (c),
RM (c) = M
e h (b, s) − Jh (r, s).
RD (s) = D
The terms RA , RM and RD measure how well the analytical solution satisfies the
DG formulation when it is rewritten in terms of the extended bilinear forms. Indeed,
if now (uh , bh , ph , rh ) is the DG approximation and eu = u − uh , eb = b − bh ,
ep = p − ph and er = r − rh are the errors, then the following error equations can be
shown to hold:
eh (eu , v) + Oh (eu , v) + Ch (v, eb ) + B
eh (v, ep ),
RA (v) = A
fh (eb , c) − Ch (eu , c) + D
e h (c, er ),
RM (c) = M
e h (eb , s) − Jh (er , s),
RD (s) = D
for any v ∈ Vh , c ∈ Ch , and s ∈ Sh . We remark that the third equation of our DG
eh . That is, from the
method is consistent when it is rewritten in terms of the form B
9
eh in [31] we see that
definition of B
eh (u − uh , q) = 0,
B
q ∈ Qh .
(4.7)
Proceeding as in [23, 24, 31], we readily obtain the following bound.
Proposition 4.1. Let the solution (u, b, p, r) of the MHD problem satisfy the
smoothness assumptions in (3.1) and (3.2). Then, we have
1
|RM (c)| ≤ |c|C ⊥ E(u, b, p),
|RA (v)| ≤ 2ν 2 kvk1,h E(u, b, p),
|RD (s)| ≤ |s|S ⊥ E(u, b, p)
for all v ∈ Vh , c ∈ Ch and s ∈ Sh , where E(u, b, p) can be bounded by
1
1
E(u, b, p) ≤ Chmin{σ,k} ν 2 kukH σ+1 (Ω) + ν − 2 kpkH σ (Ω)
+ Chmin{τ,k} kbkH τ (Ω) + k∇ × bkH τ (Ω) ,
with constants C > 0 that are independent of the mesh size and ν.
Let us also establish some continuity properties of the coupling and convection
forms Ch and Oh . To that end, we need to introduce the following trace semi-norms:
X
X
2
|u|2−1/2,Fh =
hK kuk2L2 (∂K) ,
|u|21/2,Fh =
h−1
K kukL2 (∂K) .
K∈Th
K∈Th
Proposition 4.2. The coupling form Ch satisfies
|Ch (u, b)| ≤ C kukL2 (Ω) + |u|−1/2,Fh |b|C
∀ u ∈ V(h), b ∈ C(h),
|Ch (u, b)| ≤ CkukL2 (Ω) |b|C
∀ u ∈ Vh , b ∈ C(h).
Moreover, the convection form Oh can be bounded by
|Oh (u, v)| ≤ C kuk1,h + kukL2 (Ω) + |u|1/2,Fh kvkL2 (Ω)
for all u ∈ V(h), v ∈ Vh . The constants C > 0 are independent of the mesh size
and ν.
Proof: Applying the Cauchy-Schwarz inequality and taking into account the shaperegularity of the meshes, we immediately obtain the first estimate
|Ch (u, b)| ≤ CkukL2 (Ω) k∇ × bkL2 (Ω) + C
X
hK kuk2L2 (∂K)
21
|b|C .
K∈Th
To prove the second continuity estimate, we use the following discrete trace inequality: for any polynomial u ∈ Pk (K), K ∈ Th , we have
−1
kukL2 (∂K) ≤ ChK 2 kukL2 (K) .
(4.8)
The constant C > 0 only depends on the polynomial degree k and the shape-regularity
constants of the meshes. We then readily obtain
|u|−1/2,Fh ≤ CkukL2 (Ω)
from which the second estimate follows.
10
∀ u ∈ Vh ,
To prove the estimate for Oh , we apply the Cauchy-Schwarz inequality to the
form Oh , take into account the shape-regularity of the meshes and use the bound (4.8).
This results in
|Oh (u, v)| ≤ kwkL∞ (Ω) kuk1,h + kγkL∞ (Ω) kukL2 (Ω) kvkL2 (Ω)
X
12 X
21
−1
2
2
+ CkwkL∞ (Ω)
hK kukL2 (∂K)
hK kvkL2 (∂K)
K∈Th
≤ C kuk1,h + kuk
L2 (Ω)
K∈Th
+ |u|1/2,Fh kvk
L2 (Ω)
.
2
This proves the estimate for Oh .
4.2. Error bounds. Let us now denote by (u, b, p, r) the solution of the MHD
problem and by (uh , bh , ph , rh ) its DG approximation. We split the velocity error as
follows:
eu = u − uh = (u − ΠB u) + (ΠB u − uh ) = ηu + ξu ,
(4.9)
where we use the Brezzi-Douglas-Marini (BDM) projection ΠB onto Vh ∩H(div; Ω) of
degree k for the approximation of the velocity, see [7, Proposition III.3.6]. This allows
us to use an exactly divergence-free approximation of the velocity and to decouple
the velocity error from the pressure error which is crucial for bounding the convection
and coupling terms. For the other fields, specific approximations will be chosen at a
later point.
For notational convenience, we introduce the product norm
X
1
|||(u, b, p, r)|||2 = kuk21,h + kuk2L2 (Ω) + |u|21/2,Fh +
k|w · n| 2 [[u]]k2L2 (F )
F ∈Fh
+
kbk2C
+ν
−1
kpk2L2 (Ω)
+
krk2S .
Finally, we decompose bh and rh into
bh = bch + b⊥
h,
rh = rhc + rh⊥ ,
(4.10)
⊥
c
c
⊥
⊥
with bch ∈ Cch , b⊥
h ∈ Ch , rh ∈ Sh and rh ∈ Sh , in accordance to (2.15).
4.2.1. Error in u and b. We first prove two technical lemmas.
Lemma 4.3. There are constants C > 0 and Cε > 0 independent of the mesh size
and ν such that
2
⊥ 2
kξu k2V + |b⊥
h |C ⊥ + |rh |S ⊥
≤ εkb − bh k2C + CE(u, b, p)2 + Cε |||(ηu , b − c, p − q, r − s)|||2 ,
for any ε > 0, c ∈ Cch , q ∈ Qh , and s ∈ Shc . The constant Cε depends on ε.
Proof: Fix c ∈ Cch , q ∈ Qh , s ∈ Shc and ε > 0. As in (4.9), we write
eb = b − bh = (b − c) + (c − bh )= ηb + ξb ,
ep = p − ph = (p − q) + (q − ph ) = ηp + ξp ,
er = r − rh = (r − s) + (s − rh ) = ηr + ξr .
We now proceed in the following steps.
11
(4.11)
Step 1: We first observe that, since the functions bch and c are conforming in Cch ,
we have
c
⊥
|b⊥
h |C ⊥ = |c − bh − bh |C ⊥ = |c − bh |C ⊥ ≤ |ξb |C .
Similarly, from the conformity of rhc and s in Shc ,
|rh⊥ |S ⊥ = |s − rhc − rh⊥ |S ⊥ = |s − rh |S ⊥ = |ξr |S ⊥ .
(4.12)
Taking into account these two bounds, we have
2
⊥ 2
2
2
2
kξu k2V + |b⊥
h |C ⊥ + |rh |S ⊥ ≤ kξu kV + |ξb |C + |ξr |S ⊥ .
(4.13)
To bound the right-hand side above, we observe (4.5), use the stability results for
eh + Oh , M
fh in Proposition 2.4, the fact that Jh (ξr , ξr ) = s0 |ξr |2 ⊥ , and add and
A
S
subtract the coupling and multiplier terms. Thereby, we obtain
C1 kξu k2V + |ξb |2C + |ξr |2S ⊥
eh (ξu , ξu ) + Oh (ξu , ξu ) + Ch (ξu , ξb ) + B
eh (ξu , ξp )
≤A
fh (ξb , ξb ) − Ch (ξu , ξb ) + D
e h (ξb , ξr )
+M
eh (ξu , ξp ) − D
e h (ξb , ξr ) + Jh (ξr , ξr ).
−B
From this estimate and the error equations in Section 4.1, we now readily conclude
that
C1 kξu k2V + |ξb |2C + |ξr |2S ⊥ ≤ T1 + T2 + T3 + T4 ,
(4.14)
where
e u , ξu ) − Oh (ηu , ξu ) − Ch (ξu , ηb ) − B
eh (ξu , ηp ),
T1 = RA (ξu ) − A(η
fh (ηb , ξb ) + Ch (ηu , ξb ) − D
e h (ξb , ηr ),
T2 = RM (ξb ) − M
eh (ηu , ξp ),
T3 = B
e h (ηb , ξr ) − Jh (ηr , ξr ).
T4 = −RD (ξr ) + D
Step 2: We now bound the terms T1 − T4 under the additional assumption that
c belongs to the kernel Xch defined in (2.16).
eh
To bound T1 , we use the estimate of RA in Proposition 4.1, the continuity of A
eh in (4.1) and (4.3), respectively, and the bounds for Ch and Oh in Proposiand B
tion 4.2. Upon application of the arithmetic-geometric mean inequality, we readily
obtain that
|T1 | ≤ Ckξu kV E(u, b, p) + |||(ηu , ηb , ηp , 0)|||
C1
kξu k2V + CE(u, b, p)2 + C|||(ηu , ηb , ηp , 0)|||2 .
≤
2
Similarly, from Proposition 4.1, (4.2), (4.4) and Proposition 4.2, we have
|T2 | ≤ C|ξb |C E(u, b, p) + |||(ηu , ηb , 0, 0)||| + Ckξb kC kηr kS
≤ C|ξb |C E(u, b, p) + |||(ηu , ηb , 0, 0)||| + Ckeb kC kηr kS + Ckηb kC kηr kS .
12
Using the arithmetic-geometric mean inequality again, we have that, for all ε > 0,
|T2 | ≤
C1
C1
|ξb |2C +
εkb − bh k2C + CE(u, b, p)2 + Cε |||(ηu , ηb , 0, ηr )|||2 .
2
2
Next, we claim that T3 = 0. To see this, we note that ηu = u − ΠB u belongs to
H(div; Ω). It follows that [[ηu ]]N = 0 on interior faces. In addition, by virtue of [7,
Proposition III.3.7] and since ∇ · u = 0, we have that ∇ · ηu = 0 in Ω. Then, using
eh in [31] and the defining properties of the BDM projection (cf. [7,
the definition of B
Proposition III.3.6]), we conclude that
X Z
e
T3 = Bh (ηu , ξp ) =
ηu · n ξp ds = 0;
F ∈FhD
F
thereby, proving that T3 = 0.
For the term T4 , we first note, since s ∈ Shc , we have Jh (ηr , ξr ) = 0. Furthermore,
e h (ηb , ξr ) = D
e h (ηb , s − rh ) = D
e h (ηb , s − rc ) − D
e h (ηb , rh⊥ ).
D
h
From property (4.6), we conclude that
e h (ηb , s − rhc ) = D(b, s − rhc ) − D(c, s − rhc ).
D
Both terms on the right-hand side are zero: the first one due to the fourth equation in
the weak formulation of the MHD problem and the second one due to the assumption
that c ∈ Xch . As a consequence, we obtain
e h (ηb , r⊥ ).
T4 = −RD (ξr ) − D
h
e h in (4.4),
From Proposition 4.1 and the continuity of D
|T4 | ≤ C|ξr |S ⊥ E(u, b, p) + Ckηb kC krh⊥ kS .
The norm-equivalence in Proposition 2.6 and the identity (4.12) yield
krh⊥ kS ≤ C|rh⊥ |S ⊥ = C|ξr |S ⊥ .
These results and the arithmetic-geometric mean inequality readily show that
|T4 | ≤
C1
|ξr |2S ⊥ + C E(u, b, p)2 + kηb k2C .
2
Combining (4.14) and the bounds for T1 through T4 implies that
C1
kξu k2V + |ξb |2C + |ξr |2S ⊥
2
C1
≤
εkb − bh k2C + CE(u, b, p)2 + Cε |||(ηu , ηb , ηp , ηr )|||2 .
2
Dividing the previous estimate by
C1
2
and using (4.13) yield
2
⊥ 2
kξu k2V + |b⊥
h |C ⊥ + |rh |S ⊥
≤ εkb − bh k2C + CE(u, b, p)2 + Cε |||(ηu , b − c, p − q, r − s)|||2 ,
13
(4.15)
provided that c ∈ Xch .
Step 3: We show that, in estimate (4.15), the approximation c ∈ Xch can be
replaced by any c ∈ Cch . To that end, take c ∈ Cch and look for a ∈ Cch such that
e h (a, s) = D
e h (b − c, s)
D
∀ s ∈ Shc .
By (4.4), the right-hand side is a continuous functional on Shc . Since Xch is non-empty
e h = D on Cc × S c , cf. (4.6), the inf-sup condition for D in Proposition 2.7
and D
h
h
implies that there exists at least one non-trivial solution a ∈ Cch satisfying
kakC ≤ C kb − ckC ,
e h and the
with a constant C > 0 only depending on the continuity constant of D
discrete inf-sup constant of D on Cch × Shc in Proposition 2.7, see [7, Equation II.2.20].
By construction, we have a + c ∈ Xch , since, due to (4.6) and the fourth equation of
the weak formulation of the MHD system, there holds
e h (b, s) = D(b, s) = 0
D
∀ s ∈ Shc .
Consequently, (a + c) can be used as an approximation in (4.15). In addition,
kb − (a + c)kC ≤ kb − ckC + kakC ≤ C kb − ckC ,
and inequality (4.15) holds for any c ∈ Cch , which completes the proof.
2
Lemma 4.4. There exists a constant C > 0 independent of the mesh size and ν
such that
2
⊥ 2
2
kb − bh k2C ≤ C kξu k2V + |b⊥
h |C ⊥ + |rh |S ⊥ + |||(ηu , b − c, 0, r − s)||| ,
for any c ∈ Cch and s ∈ Shc .
Proof: Let c ∈ Cc and s ∈ Shc . Again, we split the errors in b and r into two parts
and adopt the same notation as in (4.11). We now proceed in two steps.
Step 1: We first consider the case where the approximation c ∈ Cch to the magnetic
field b is such that
c − bch ∈ Xch .
(4.16)
It can be readily shown that non-trivial approximations of this type exist. To show
this, consider the problem: find c ∈ Cch such that
e h (c, s) = D
e h (bc , s)
D
h
∀ s ∈ Shc .
(4.17)
As before, the right-hand side is a continuous functional on Shc , cf. (4.4), and the discrete inf-sup condition for D in Proposition 2.7, cf. (4.6), ensures that problem (4.17)
admits at least one non-trivial solution c ∈ Cch which then satisfies property (4.16).
Now let c ∈ Cch be such that (4.16) holds. We decompose the function ξb = c−bh
into
ξb = ξbc + ξb⊥ ,
ξbc ∈ Cch , ξb⊥ ∈ C⊥
h,
(4.18)
according to (2.15). Since the approximation c belongs to the conforming space Cch ,
we have
ξbc = (c − bch ),
14
ξb⊥ = −b⊥
h.
(4.19)
fh on Ch , see Proposition 2.4
Next, we bound kξbc kC . Due to the coercivity of M
c
⊥
and (4.5), and the fact that ξb = ξb − bh , we have
c
fh (ξbc , ξbc ) = M
fh (ηb + ξb , ξbc ) − M
fh (ηb , ξbc ) + M
fh (b⊥
C1 kξbc k2C ≤ M
h , ξb ).
Using the error equation from Section 4.1, we obtain that
fh (ηb + ξb , ξbc ) = RM (ξbc ) + Ch (ηu + ξu , ξbc ) − D
e h (ξbc , ηr + ξr ).
M
e h (ξ c , ηr + ξr ) can be
The term RM (ξbc ) is zero because ξbc ∈ Cch . The term −D
b
c
c
c
simplified as follows: since ξb = c − bh and s ∈ Sh , we deduce from (4.16) and (4.5)
that
e h (ξ c , ηr + ξr ) = −D
e h (ξ c , ηr ) − D
e h (ξ c , s − rc − r⊥ )
−D
b
b
b
h
h
c
c
⊥
e
e
= −Dh (ξb , ηr ) + Dh (ξb , rh ).
From the previous discussion, we conclude that
C1 kξbc k2C ≤ S1 + S2 ,
(4.20)
where
fh (ηb , ξbc ) + Ch (ηu , ξbc ) − D
e h (ξbc , ηr ),
S1 = −M
c
c
fh (b⊥
e c ⊥
S2 = M
h , ξb ) + Ch (ξu , ξb ) + Dh (ξb , rh ).
fh , D
e h and Ch in (4.2), (4.4) and Proposition 4.2,
The continuity properties of M
respectively, and the arithmetic-geometric mean inequality yield
|S1 | ≤
C1 c 2
kξb kC + C |ηb |2C + kηu k2L2 (Ω) + |ηu |2−1/2,Fh + kηr k2S .
4
Similarly,
C1 c 2
2
2
⊥ 2
kξb kC + C kb⊥
h kC + kξu kL2 (Ω) + krh kS
4
C1 c 2
2
2
⊥ 2
≤
kξb kC + C |b⊥
h |C ⊥ + kξu kV + |rh |S ⊥ ,
4
|S2 | ≤
where we have also used the norm-equivalence results in Proposition 2.6.
Combining (4.20) with the estimates for S1 and S3 , we conclude that
2
⊥ 2
2
kξbc k2C ≤ C kξu k2V + |b⊥
h |C ⊥ + |rh |S ⊥ + |||(ηu , ηb , 0, ηr )||| .
Therefore, the previous estimate, the decomposition (4.18)–(4.19), the triangle inequality and the norm equivalence in Proposition 2.6 yield that
kb − bh k2C ≤ C(kηb k2C + kξb k2C )
2
≤ C(kηb k2C + kξbc k2C + |b⊥
h |C ⊥ )
2
⊥ 2
2
≤ C(kξu k2V + |b⊥
h |C ⊥ + |rh |S ⊥ + |||(ηu , b − c, 0, r − s)||| ,
for any s ∈ Shc and c ∈ Cch satisfying (4.16).
15
Step 2: We now show that the last bound of Step 1 holds for c ∈ Cch arbitrary.
Proceeding as before, we can find a non-trivial function a ∈ Cch such that
c
e h (a, s) = D
e h (b − c − b⊥ , s)
D
h
∀ s ∈ Sh ,
(4.21)
⊥
kakC ≤ C kb − ckC + kbh kC .
Then, due to the properties in (4.5), (4.6) and the weak formulation of the equations
and the DG discretization, we have
e h ((a + c) − bch , s)
D((a + c) − bch , s) = D
e h (b − bh , s) = D(b, s) − Dh (bh , s) = 0,
=D
for any s ∈ Shc . Hence, (a + c) − bch ∈ Xch and a + c satisfies (4.16). It can then be
used as an approximation in the last inequality of Step 1. In view of (4.21) and the
norm-equivalence in Proposition 2.6, we obtain
kb − (a + c)kC ≤ kb − ckC + kakC ≤ Ckb − ckC + |b⊥
h |C ⊥ .
It follows that the last inequality of Step 1 holds for any approximation c ∈ Cch , which
completes the proof of the lemma.
2
We are now ready to bound the errors in u and b.
Theorem 4.5. There exists a constant C > 0, independent of the mesh size
and ν such that
⊥
ku − uh kV + kb − bh kC + |b⊥
h |C ⊥ + |rh |S ⊥
≤ CE(u, b, p) + C|||(ηu , b − c, p − q, r − s)|||,
for any c ∈ Cch , q ∈ Qh and s ∈ Shc .
Proof: Fix c ∈ Cch , q ∈ Qh and s ∈ Shc . Decomposing the errors as in (4.9) and (4.11),
we obtain from the triangle inequality, Lemma 4.4 and Lemma 4.3:
2
⊥ 2
ku − uh k2V + kb − bh k2C + |b⊥
h |C ⊥ + |rh |S ⊥
2
⊥ 2
≤ C kηu k2V + kξu k2V + kb − bh k2C + |b⊥
h |C ⊥ + |rh |S ⊥
2
⊥ 2
≤ C kξu k2V + |||(ηu , ηb , 0, ηr )|||2 + |b⊥
h |C ⊥ + |rh |S ⊥
≤ Cεkb − bh k2C + C E(u, p, b)2 + Cε |||(ηu , b − c, p − q, r − s)|||2 .
1
and bringing the term 21 kb − bh k2C to the left-hand side now readily
Choosing ε = 2C
implies the assertion.
2
4.2.2. Error in p and r. Next, we bound the errors in the pressure p and the
multiplier r.
Proposition 4.6. There is a constant C > 0 independent of the mesh size and ν
such that
kp − ph kL2 (Ω) ≤ C (E(u, b, p) + |||(ηu , b − c, p − q, r − s)|||) ,
for any c ∈ Cch , q ∈ Qh and s ∈ Shc .
Proof: We begin by recalling the Poincaré inequality for piecewise smooth functions
cf. [6, Remark 1.1]:
kvkL2 (Ω) ≤ Ckvk1,h ,
16
v ∈ V(h).
(4.22)
Let now c ∈ Cch , q ∈ Qh and s ∈ Shc . As before, we split the errors into two parts
and adopt the same notation as in (4.11). Obviously, by the triangle inequality
kp − ph kL2 (Ω) ≤ kηp kL2 (Ω) + kξp kL2 (Ω) .
(4.23)
We must then further estimate kξp kL2 (Ω) . To this end, we make use of the continuous
inf-sup condition over V ×Q; see, e.g., [7]. Therefore, we conclude that there is v ∈ V
such that
eh (v, ξp )
Ckξp kL2 (Ω) ≤ B
and
kvkH 1 (Ω) ≤ 1,
(4.24)
where we have also used (4.6). We now set vh = ΠB v with ΠB denoting the BDM
eh
projection into Vh ∩ H(div; Ω). By using the definition of the extended form B
in [31], (4.5), (4.6) and the properties of the BDM projection, we readily obtain
eh (vh , ξp ) = B
eh (v, ξp ).
B
(4.25)
In addition, the approximation property of the BDM projection and (4.24) guarantee
that
kvh k1,h ≤ kv − vh k1,h + kvk1,h ≤ CkvkH 1 (Ω) ≤ C.
(4.26)
We use the error equations from Section 4.1 to obtain
eh (vh , ξp ) = T1 + T2 + T3 + T4 ,
B
(4.27)
where
eh (ηu , vh ) − Oh (ηu , vh ) − B
eh (vh , ηp ),
T1 = RA (vh ) − A
T2 = −Ch (vh , b − bh ),
eh (ξu , vh ),
T3 = −A
T4 = −Oh (ξu , vh ).
Let us bound T1 , T2 , T3 and T4 .
eh , B
eh and Oh in (4.1),
For T1 , we use Proposition 4.1, the continuity results for A
(4.3) and Proposition 4.2, respectively, combined with the Poincaré inequality (4.22).
We obtain
1
|T1 | ≤ Ckvh k1,h ν 2 E(u, b, p) + |||(ηu , 0, ηp , 0)|||
1
≤ C ν 2 E(u, b, p) + |||(ηu , 0, ηp , 0)||| ,
where we have also used (4.26).
To estimate T2 , we use the continuity of Ch in Proposition 4.2 and the Poincaré
inequality (4.22):
|T2 | ≤ Ckvh kL2 (Ω) |b − bh |C ≤ Ckvh k1,h kb − bh kC .
From the bound for b − bh in Theorem 4.5 and (4.26), we thus conclude
|T2 | ≤ C E(u, b, p) + |||(ηu , ηb , ηp , ηr )||| .
17
eh , the bound for
To bound T3 , we proceed similarly: we use the continuity of A
ξu in Lemma 4.3 (with ε = 1), the error bound b − bh in Theorem 4.5 and (4.26).
We readily conclude that
1
|T3 | ≤ Cν 2 (E(u, b, p) + |||(ηu , ηb , ηp , ηr )|||) .
The term T4 is the reason for introducing the continuous velocity field v in (4.23).
To bound this term, we proceed as follows. We integrate by parts the form Oh and
write
Oh (ξu , vh ) = T4,1 + T4,2 + T4,3 ,
with
X Z
T4,1 = −
K∈Th
T4,2 =
K
K∈Th
X Z
K∈Th
X Z
(w · ∇)vh · ξu dx +
(γ − ∇ · w)ξu · vh dx,
K
w · nK ξu · (vh − vhe ) ds,
∂K+ \Γ+
Z
w · nξu · vh ds,
T4,3 =
Γ+
where Γ+ = {x ∈ Γ : w(x) · n(x) ≥ 0} and ∂K+ = {x ∈ ∂K : w(x) · nK ≥ 0}.
Employing the Poincaré inequality (4.22), the term T4,1 can be readily bounded by
|T4,1 | ≤ Ckξu kL2 (Ω) kvh k1,h + kvh kL2 (Ω) ≤ Ckξu kV kvh k1,h .
For the term T4,2 , we use arguments as in the proof of Proposition 4.2 and the discrete
trace inequality (4.8) to obtain
|T4,2 | ≤ CkwkL∞ (Ω)
X
hK kξu k2L2 (∂K)
K∈Th
21
X
2
h−1
F k[[vh ]]kL2 (F )
21
F ∈FhI
≤ CkwkL∞ (Ω) kξu kL2 (Ω) kvh k1,h ≤ Ckξu kV kvh k1,h .
Finally, the term T4,3 can be written as
Z
Z
T4,3 =
w · nξu · (vh − v) ds +
Γ+
w · nξu · v ds ≡ T4,3,1 + T4,3,2 ,
Γ+
with the continuous velocity field v from (4.24). For the first integral above, we use
the approximation properties of the BDM projection in [7, Proposition III.3.6] and
obtain
21
Z
21 Z
|w · n||v − vh |2 ds
T4,3,1 ≤
|w · n||ξu |2 ds
Γ
Γ
1
≤ Ch 2 kξu kV kvkH 1 (Ω) .
To estimate the second integral, we use the trace theorem for functions in H 1 (Ω).
This yields
Z
T4,3,2 ≤
Γ
12 Z
21
2
|w · n||ξu | ds
|w · n||v| ds
≤ Ckξu kV kvkH 1 (Ω) .
2
Γ
18
As a consequence, we see that
T4,3 ≤ Ckξu kV kvkH 1 (Ω) .
Hence, from the above estimates, (4.24) and (4.26), we conclude that
|T4 | ≤ Ckξu kV .
From Lemma 4.3 (with ε = 1) and Theorem 4.5, we then obtain:
|T4 | ≤ C (E(u, b, p) + |||(b − c, p − q, r − s)|||) .
The desired estimate for the pressure now follows from (4.23), (4.27) and the
above estimates for T1 through T4 .
2
Finally, we bound the error in r.
Proposition 4.7. There is a constant C > 0 independent of the mesh size and ν
such that
kr − rh kS ≤ C E(u, b, p) + |||(ηu , b − c, p − q, r − s)||| ,
for any c ∈ Cch , q ∈ Qh and s ∈ Shc .
Proof: Let c ∈ Cch , q ∈ Qh and s ∈ Shc . As before, we adopt the notation from (4.11).
By the triangle inequality, we have
kr − rh kS ≤ kηr kS + kξr kS .
To bound the term kξr kS , we decompose ξr into
ξr = ξrc + ξr⊥ ,
ξrc ∈ Shc , ξr⊥ ∈ Sh⊥ ,
(4.28)
according to (2.15). Since s belongs to the conforming space Shc , we have
ξr⊥ = −rh⊥ .
ξrc = (s − rhc ),
(4.29)
By the triangle inequality and the norm-equivalence in Proposition 2.6, we have
kξr kS ≤ kξrc kS + krh⊥ kS ≤ kξrc kS + C|rh⊥ |S ⊥ .
(4.30)
The latter term can be bounded by Theorem 4.5:
|rh⊥ |S ⊥ ≤ C E(u, b, p) + |||(ηu , b − c, p − q, r − s)||| .
(4.31)
To bound the former term, we use (4.6) and the inf-sup condition for D in Proposition 2.7. Thereby, we obtain
Ckξrc kS ≤ sup
c∈Cch
e h (c, ξrc )
D
.
kckC
Using (4.28) and (4.29), we write
e h (c, ξrc ) = D
e h (c, ηr + ξr ) − D
e h (c, ηr ) + D
e h (c, rh⊥ ),
D
19
(4.32)
and use the error equation in Section 4.1 to conclude that
e h (c, ξrc ) = RM (c) − M
fh (ηb , c) + Ch (ηu , c)
D
fh (ξb , c) + Ch (ξu , c) − D
e h (c, ηr ) + D
e h (c, rh⊥ )
−M
for all c ∈ Cch . We note that RM (c) = 0 for c ∈ Cch . The continuity properties and
the norm-equivalence in Proposition 2.6 then yield
e h (c, ξrc )| ≤ CkckC |||(ηu , ηb , 0, ηr )||| + kξb kC + kξu kV + krh⊥ kS
|D
≤ CkckC |||(ηu , ηb , 0, ηr )||| + kb − bh kC + kξu kV + |rh⊥ |S ⊥ .
Combining the above estimates with Lemma 4.3 (with ε = 1) and Theorem 4.5 readily
gives the assertion.
2
4.2.3. Proof of Theorem 3.1. We are now ready to complete the proof of
Theorem 3.1. In (4.9), the approximation for the velocity u has already be chosen to
be the BDM projection ΠB u of degree k. We now approximate the remaining fields
as follows:
c = ΠN b,
q = Πk−1 p,
s = ΠS r,
(4.33)
where ΠN is the H(curl; Ω)-conforming Nédélec projection of the second kind of
degree k onto Cch ; see [28, 27], Πk−1 the L2 -projection of degree k − 1 onto Qh
and ΠS the H 1 -conforming nodal interpolation operator of degree k + 1 into Sh . The
approximation properties of these operators immediately yield the following result.
Proposition 4.8. Choosing the interpolants as in (4.33), there holds
|||(ηu , b − c, p − q, r − s)|||
1
≤ Chmin{σ,k} kukH σ+1 (Ω) + ν − 2 kpkH σ (Ω)
+Chmin{τ,k} kbkH τ (Ω) + k∇ × bkH τ (Ω) + krkH τ +1 (Ω) ,
with a constant C > 0 that is independent of the mesh size and ν.
The error estimate in Theorem 3.1 now follows directly from the error estimates
in Theorem 4.5, Proposition 4.6 and Proposition 4.7, in conjunction with the approximation results in Proposition 4.1 and Proposition 4.8.
5. Numerical Results. In this section we present a series of numerical experiments to highlight the practical performance of the mixed DG method introduced
in this article for the numerical approximation of incompressible MHD problems.
Throughout this section, we select the stabilization parameters as follows: a0 = α k 2 ,
m0 = µ k 2 and s0 = 1, α, µ > 0, cf. [24], for example. To ensure stability of the
underlying DG method we set α = µ = 10 in 2D; for 3D simulations, it is necessary
to increase α and µ to α = µ = 20.
All computations have been performed using the AptoFEM finite element software
package; see [18] for details. In order to solve the resulting system of linear equations,
we have employed a variety of open-source software: for relatively small numbers of
degrees of freedom, we exploit the MUltifrontal Massively Parallel Solver (MUMPS),
see [1] for details; for larger problems, we have used both the out of core version
of PARDISO [29], as well as the (parallel) additive Schwarz preconditioned GMRES
solver available in PETSc [5].
20
y
1
−1
0
x
1
−1
(a)
(b)
Fig. 5.1. Example 1. (a) Problem domain; (b) Initial unstructured triangular mesh.
5.1. Smooth solutions. First, we verify the theoretical error bound stated in
Theorem 3.1 for problems with smooth analytical solutions.
5.1.1. Example 1: 2D problem in an L–shaped domain. The first example
we consider is a two-dimensional version of the MHD problem (2.1)–(2.4). While the
Navier-Stokes operator has the same form in two dimensions, some care is required
for the curl-curl operator and the coupling terms in the equations; see [27, Page 51]
and [23] for details.
We consider the L–shaped domain Ω = (−1, 1)2 \ ([0, 1) × (−1, 0]) with ΓN =
{(1, y) : y ∈ (0, 1)} and ΓD = ∂Ω \ ΓN , cf. Figure 5.1(a). We set ν = νm = κ = 1,
w = (2, 1), γ = 0, d = (x, −y), and choose the forcing functions f and g and
boundary conditions so that the analytical solution of the two-dimensional variant of
(2.1)–(2.4) is of the form
u(x, y) = (−(y cos y + sin y)ex , y sin y ex ),
b(x, y) = (−(y cos y + sin y)ex , y sin y ex ),
p(x, y) = 2ex sin y,
r(x, y) = − sin πx sin πy.
Here, we investigate the asymptotic convergence of the interior penalty DG method
on a sequence of successively finer quasi-uniform unstructured triangular meshes for
k = 1, 2, 3, 4. In each case the meshes are constructed by uniformly refining the initial
mesh depicted in Figure 5.1(b).
In Figure 5.2 we plot the norms k · kV , k · kC , and k · kS of the errors eu = u − uh ,
eb = b − bh , and er = r − rh , respectively, against the square root of the number of
degrees of freedom in the finite element space Vh × Ch × Qh × Sh . Here, we observe
that both keu kV and keb kC converge to zero, for each fixed k, at the optimal rate
O(hk ), as the mesh is refined, in accordance with Theorem 3.1. On the other hand,
for this mixed-order method, ker kS converges at the rate O(hk+1 ), for each k, as h
tends to zero; this rate is indeed optimal, though this is not reflected by Theorem 3.1,
cf. also [24]. Additionally, in Figure 5.2(d), we plot the sum of the three error
contributions with respect to the square root of the number of degrees of freedom in
the finite element space. Clearly, as above, this converges to zero at the optimal rate
predicted by Theorem 3.1.
Secondly, we highlight the optimality of the proposed mixed method when the
components of the error are measured in terms of the L2 -norm. From Figure 5.3 we
observe that the L2 -norm of the error in both the approximation to the velocity field
u and the magnetic field b tend to zero at the expected optimal rate O(hk+1 ), for
21
0
10
0
10
1
1
1
1
−2
1
verroru
10
−6
1
2
1
−4
10
10
10
cerrorb
−2
10
1
3
3
−6
10
1
k=1
k=2
k=3
k=4
4
−8
10
2
10
sqdof
k=1
k=2
k=3
k=4
1
4
2
10
sqdof
(a)
(b)
0
10
0
10
1
1
1
2
−2
10
combinederror
serrorr
−4
10
3
1
4
−6
−8
10
1
−2
1
10
2
−4
10
1
−4
10
3
1
5
k=1
k=2
k=3
k=4
−6
10
2
2
1
k=1
k=2
k=3
k=4
4
2
10
sqdof
10
sqdof
(c)
(d)
Fig. 5.2. Example 1. Convergence with h-refinement: (a) keu kV ; (b) keb kC ; (c) ker kS ; (d)
keu kV + keb kC + ker kS .
each k, as h tends to zero. In agreement with Theorem 3.1, for each fixed k, the
L2 -norm of the error in the pressure p, denoted by ep = p − ph , tends to zero at the
optimal rate O(hk ) as the mesh is enriched, while ker kL2 (Ω) is of order O(hk+2 ) as h
tends to zero.
5.1.2. Example 2: 3D problem in the unit cube. The second example is a
3D problem with a smooth analytical solution. Here, we set Ω = (0, 1)3 ⊂ R3 with
ΓD = ∂Ω and ΓN = ∅, ν = νm = κ = 1, w = (2, 1, 1), γ = 0, and d = (x, −y, 1),
and select f and g, together with appropriate inhomogeneous boundary conditions,
so that the solution of the incompressible MHD system (2.1)–(2.4) is given by
u = (−(y cos y + sin y + z cos z)ex , y sin y ex , z sin z ex ),
b = (−(y cos y + sin y + z cos(z))ex , y sin y ex , z sin z ex ),
p = 2ex (sin y + sin z) − p0 ,
r = sin πx sin πy sin πz,
where p0 = 4(−1 + e + cos 1 − e cos 1).
In Table 5.1 we investigate the asymptotic rate of convergence of the error in the
approximation of the hydrodynamic variables; here, l denotes the computed rate of
convergence. To this end, we show keu kL2 (Ω) , keu kV , and kep kL2 (Ω) computed on
a sequence of uniformly refined tetrahedral meshes for k = 1, 2. As in the previous
example, we again observe optimal rates of convergence for all three measures of the
error. Indeed, in accordance with Theorem 3.1, both keu kV and kep kL2 (Ω) tend to
zero at the optimal rate O(hk ), for each fixed k, as the mesh is refined. Additionally,
22
0
10
0
10
−2
1
10
1
1
−2
1
1
−6
10
−8
10
−10
10
10
3
l2errorp
l2erroru
−4
10
1
−4
1
k=1
k=2
k=3
k=4
5
−6
10
3
k=1
k=2
k=3
k=4
2
10
sqdof
(b)
0
10
1
l2errorb
−4
−2
1
10
−6
10
2
1
3
1
10
3
−4
l2errorr
−2
10
−10
4
10
sqdof
(a)
10
1
2
0
−8
2
10
4
10
10
1
2
k=1
k=2
k=3
k=4
10
1
−6
10
4
−8
10
1
5
−10
10
2
10
sqdof
4
1
(c)
5
1
k=1
k=2
k=3
k=4
6
2
10
sqdof
(d)
Fig. 5.3. Example 1. Convergence with h-refinement: (a) keu kL2 (Ω) ; (b) kep kL2 (Ω) ; (c)
keb kL2 (Ω) ; (d) ker kL2 (Ω) .
k
1
2
DOFs uh /ph keu kL2 (Ω)
576/48
6.400e-2
4608/384
1.647e-2
36864/3072
4.213e-3
294912/24576 1.072e-3
1440/192
5.082e-3
11520/1536
6.802e-4
92160/12288 8.417e-5
l
–
1.96
1.97
1.97
–
2.90
3.01
keu kV
1.168
5.823e-1
2.896e-1
1.442e-1
1.262e-1
3.162e-2
7.822e-3
l
kep kL2 (Ω)
–
7.321e-1
1.00 4.564e-1
1.01 2.606e-1
1.01 1.400e-1
–
3.424e-1
2.00 8.488e-2
2.02 2.127e-2
l
–
0.68
0.81
0.90
–
2.01
2.00
Table 5.1
Example 2: Convergence of keu kL2 (Ω) , keu kV , and kep kL2 (Ω) with h-refinement.
we observe that keu kL2 (Ω) is of optimal order O(hk+1 ) as h tends to zero.
The corresponding errors for the magnetic variables are shown in Tables 5.2 &
5.3. Here, we clearly observe the optimality of the approximation to the magnetic
field b. Indeed, from Table 5.2 we observe that keb kL2 (Ω) and keb kC converge to zero
at the optimal rates O(hk+1 ) and O(hk ), respectively, for each fixed k, as the mesh is
refined. As in the previous example, we again observe that ker kL2 (Ω) and ker kS are
of order O(hk+2 ) and O(hk+1 ), respectively, as the mesh is uniformly refined.
5.2. Example 3: 2D problem with a singular solution. To verify the ability
of the proposed interior penalty DG method to capture the strongest magnetic (and
23
k
1
2
DOFs bh
576
4608
36864
294912
1440
11520
92160
keb kL2 (Ω)
7.289e-2
2.076e-2
5.486e-3
1.399e-3
6.082e-3
7.953e-4
1.006e-4
l
–
1.81
1.92
1.97
–
2.94
2.98
keb kC
7.324e-1
3.445e-1
1.668e-1
8.184e-2
4.920e-2
1.146e-2
2.767e-3
l
–
1.09
1.05
1.03
–
2.10
2.05
Table 5.2
Example 2: Convergence of keb kL2 (Ω) and keb kC with h-refinement.
k
1
2
DOFs rh
480
3840
30720
245760
960
7680
61440
ker kL2 (Ω)
1.038e-1
1.766e-2
2.350e-3
2.924e-4
1.327e-2
8.567e-4
5.135e-5
l
–
2.56
2.91
3.01
–
3.95
4.06
ker kS
1.546
5.098e-1
1.363e-1
3.405e-2
2.559e-1
3.430e-2
4.210e-3
l
–
1.60
1.90
2.00
–
2.90
3.03
Table 5.3
Example 2: Convergence of ker kL2 (Ω) and ker kS with h-refinement.
hydrodynamic) singularities, we consider a problem in which the precise regularity of
the analytical solution is known. To this end, we again let Ω be the L-shaped domain
employed in Example 1 above with ΓN = {(1, y) : y ∈ (0, 1)} and ΓD = ∂Ω \ ΓN .
We choose ν = νm = κ = 1, and set w = 0, γ = 0 and d = (−1, 1). Hence, the
Navier-Stokes operator coincides with the Stokes equations. We further choose f
and g, and appropriate inhomogeneous boundary conditions so that the solution to
this problem is given by the strongest corner singularities for the underlying elliptic
operators. That is, in polar coordinates (ρ, φ) around the origin, the hydrodynamic
solution components u and p are taken to be
λ
ρ ((1 + λ) sin(φ)ψ(φ) + cos(φ)ψ 0 (φ))
u(ρ, φ) = λ
,
ρ (−(1 + λ) cos(φ)ψ(φ) + sin(φ)ψ 0 (φ))
p(ρ, φ) = −ρλ−1 ((1 + λ)2 ψ 0 (φ) + ψ 000 (φ))/(1 − λ),
where
ψ(φ) = sin((1 + λ)φ) cos(λw)/(1 + λ) − cos((1 + λ)φ)
− sin((1 − λ)φ) cos(λw)/(1 − λ) + cos((1 − λ)φ),
and λ ≈ 0.54448373678246. The magnetic pair (b, r) is taken as
b(x) = ∇(ρ2/3 sin (2/3φ)),
r(x) ≡ 0.
We point out that the magnetic field b does not belong to H 1 (Ω)2 and thus cannot be
correctly captured by nodal elements. In fact, for this example, we have that (u, p) ∈
H 1+λ (Ω)2 × H λ (Ω) and b ∈ H 2/3 (Ω)2 . Thus, the limiting regularity exponent, cf.
24
k
1
2
3
DOFs uh /ph keu kL2 (Ω)
144/24
1.311e-1
576/96
4.638e-2
1.632e-2
2304/384
9216/1536
5.837e-3
36864/6144
2.120e-3
288/72
6.089e-2
1152/288
2.405e-2
4608/1152
9.434e-3
18432/4608
3.837e-3
73728/18432 1.630e-3
480/144
3.094e-2
1920/576
1.198e-2
7680/2304
4.844e-3
30720/9216
2.054e-3
122880/36864 9.046e-4
l
–
1.50
1.51
1.48
1.46
–
1.34
1.35
1.30
1.24
–
1.37
1.31
1.24
1.18
keu kV
1.910
1.352
9.419e-1
6.510e-1
4.482e-1
1.075
7.382e-1
5.065e-1
3.474e-1
2.383e-1
7.498e-1
5.151e-1
3.532e-1
2.422e-1
1.661e-1
l
kep kL2 (Ω)
–
1.443
0.50
1.064
0.52 7.690e-1
0.53 5.436e-1
0.54 3.789e-1
–
1.520
0.54 9.010e-1
0.54 5.852e-1
0.54 3.910e-1
0.54 2.649e-1
–
9.219e-1
0.54 5.809e-1
0.54 3.779e-1
0.54 2.527e-1
0.54 1.713e-1
l
–
0.44
0.47
0.50
0.52
–
0.75
0.62
0.58
0.56
–
0.67
0.62
0.58
0.56
Table 5.4
Example 3: Convergence of keu kL2 (Ω) , keu kV , and kep kL2 (Ω) with h-refinement.
(3.1) and (3.2), appearing in Theorem 3.1 is λ which stems from the regularity of the
hydrodynamic variables.
In this example we study the asymptotic convergence of the interior penalty DG
method on the sequence of successively finer quasi-uniform unstructured triangular
meshes employed in Example 1, cf. Figure 5.1(b), with k = 1, 2, 3. Table 5.4 presents
the L2 -norm of the error in both the computed velocity uh and pressure ph , as well
as the k · kV -norm of the error in uh . In agreement with Theorem 3.1 we see that
both keu kV and kep kL2 (Ω) tend to zero at the optimal rate O(hλ ) as h tends to zero.
The rate of convergence of keu kL2 (Ω) is observed to be between O(h1.2 ) and O(h1.5 )
approximately as the mesh is uniformly refined.
From Table 5.5 we observe that both keb kL2 (Ω) and keb kC are approximately
O(h) as h tends to zero. For this latter error, this rate is higher than what we would
expect from Theorem 3.1. However, this same behavior of the error has also been
observed in the case of simply approximating the time-harmonic Maxwell operator in
isolation, cf. [24]. In contrast, from Table 5.6, we observe that ker kS converges to zero
at the rate O(h2/3 ) as the mesh is refined. In terms of the numerical approximation
of the time-harmonic Maxwell operator in isolation, this rate is indeed optimal, cf.
[24], though this is not reflected in Theorem 3.1. Finally, we note that the L2 -norm
of the error in the approximation to the variable r tends to zero at the rate O(h4/3 )
as h tends to zero.
5.3. Hartmann channel flow. Finally, we consider 2D and 3D Hartmann channel flow problems; cf. [17]. Note that assumption (2.5) is not satisfied in these examples. However, the particular structure of the solutions implies that (w · ∇)u = 0 and
assumption (2.5) is not relevant in this context.
5.3.1. Example 4: 2D Hartmann flow. In the domain Ω = (0, L) × (−1, 1),
L 1, we consider the steady 2D unidirectional flow under a constant pressure
gradient −G in the x–direction, where theoretically G can be any real number, and
25
k
1
2
3
DOFs bh
144
576
2304
9216
36864
288
1152
4608
18432
73728
480
1920
7680
30720
122880
keb kL2 (Ω)
2.601e-1
1.492e-1
7.699e-2
4.038e-2
2.265e-2
2.244e-1
1.124e-1
5.371e-2
2.679e-2
1.452e-2
1.888e-1
9.013e-2
4.156e-2
2.004e-2
1.055e-2
l
–
0.80
0.96
0.93
0.84
–
1.00
1.07
1.00
0.88
–
1.07
1.12
1.05
0.93
keb kC
4.091e-1
2.291e-1
1.112e-1
5.280e-2
2.657e-2
3.649e-1
1.785e-1
8.065e-2
3.654e-2
1.766e-2
3.090e-1
1.448e-2
6.363e-2
2.812e-2
1.320e-2
l
–
0.84
1.04
1.07
0.99
–
1.03
1.15
1.14
1.05
–
1.09
1.19
1.18
1.09
Table 5.5
Example 3: Convergence of keb kL2 (Ω) and keb kC with h-refinement.
k
1
2
3
DOFs rh
144
576
2304
9216
36864
240
960
3840
15360
61440
360
1440
5760
23040
92160
ker kL2 (Ω)
2.397e-1
1.150e-1
4.860e-2
1.946e-2
7.664e-3
1.944e-1
8.588e-2
3.498e-2
1.382e-2
5.419e-3
1.621e-1
6.932e-2
2.784e-2
1.095e-2
4.290e-3
l
–
1.06
1.24
1.32
1.34
–
1.18
1.30
1.34
1.35
–
1.23
1.32
1.35
1.35
ker kS
2.107
1.768
1.265
8.384e-1
5.387e-1
2.728
2.066
1.412
9.193e-1
5.868e-1
3.188
2.323
1.559
1.008
6.415e-1
l
–
0.25
0.48
0.59
0.64
–
0.40
0.55
0.62
0.65
–
0.46
0.58
0.63
0.65
Table 5.6
Example 3: Convergence of ker kL2 (Ω) and ker kS with h-refinement.
practically any achievable pressure change produced by external forces. We set
G
cosh(yHa)
G sinh(yHa)
w=
1−
, 0 , d=
−y , 1 ,
νHa tanh(Ha)
cosh(Ha)
κ
sinh(Ha)
γ = 0, f = 0, and g = 0. Additionally, we impose the boundary conditions
u=0
pn = pN n
n × b = n × bD
on y = ±1,
on x = 0 and x = L,
on Γ,
26
Fig. 5.4. Example 4. Initial unstructured triangular mesh.
0
10
0
10
1
1
1
−1
cerrorb
verroru
10
1
1
−2
2
10
−3
10
1
−1
10
−2
10
1
3
−3
10
2
10
k=1
k=2
k=3
3
3
2
10
sqdof
10
(a)
3
10
sqdof
(b)
0
0
10
10
−1
10
−1
10
1
1
1
l2errorp
2
serrorr
2
1
k=1
k=2
k=3
−2
10
−2
10
1
1
1
−3
10
−4
10
1
3
2
10
k=1
k=2
k=3
3
3
2
10
sqdof
2
−3
10
4
k=1
k=2
k=3
10
(c)
3
10
sqdof
(d)
Fig. 5.5. Example 4. Convergence with h-refinement: (a) keu kV ; (b) keb kC ; (c) ker kS ; (d)
kep kL2 (Ω) .
where
bD = (0, 1),
pN
G2
= −Gx −
2κ
sinh(yHa)
−y
sinh(Ha)
2
+ p0 ,
and p0 is any constant. The analytical solution to the incompressible MHD equations
is given by u = w, b = d, p = pN , r ≡ 0, where κ = ννm Ha2 . We note that the
fluid always moves in the direction in which the pressure decreases. We set L = 10,
ν = νm = 0.1, Ha = 10, G = 0.5, and p0 = 10.
Firstly, in Figure 5.5 we investigate the asymptotic convergence of the interior
penalty DG method on a sequence of successively finer quasi-uniform unstructured
triangular meshes for k = 1, 2, 3. In each case the meshes are constructed by uniformly
refining the initial unstructured mesh depicted in Figure 5.4. Here, we plot the norms
k · kV , k · kC , k · kS , and k · kL2 (Ω) of the errors eu , eb , er , and ep , respectively, with
respect to the square root of the number of degrees of freedom in the finite element
27
(a)
(b)
Fig. 5.6. Example 4. DG solution computed on the finest mesh with k = 1: (a) Velocity field;
(b) Magnetic field.
0.4
DG Solution
Analytical Solution
0.6
DG Solution
Analytical Solution
0.3
0.2
0.5
0.1
0.4
0
0.3
−0.1
0.2
−0.2
0.1
0
−1
−0.3
−0.5
0
0.5
−0.4
−1
1
(a)
−0.5
0
0.5
1
(b)
Fig. 5.7. Example 4. DG solution computed on the finest mesh with k = 1. Slices along x = 5,
−1 ≤ y ≤ 1 of the solution: (a) First component of the velocity field; (b) First component of the
magnetic field.
space Vh × Ch × Qh × Sh . As in the previous examples presented in Section 5.1, we
observe that keu kV , keb kC and kep kL2 (Ω) converge to zero, for each fixed k, at the
optimal rate O(hk ) as the mesh is refined, in accordance with Theorem 3.1, while
ker kS converges at the rate O(hk+1 ), for each k, as h tends to zero. Moreover, we
note that the L2 -norms of the error in the approximation to u, b and r tend to zero
optimally, cf. Section 5.1; for brevity, these results have been omitted.
Finally, in Figures 5.6 & 5.7 we show the DG solution computed on the finest mesh
with 41216 elements, employing k = 1; thereby, the total number of degrees of freedom
employed in the finite element space Vh × Ch × Qh × Sh is 783104. In particular,
from Figure 5.7, we observe extremely good agreement between the computed and
analytical solutions of the first components in the velocity and magnetic fields.
5.3.2. Example 5: 3D Hartmann flow. In this final example, we consider the
steady 3D unidirectional flow in the rectangular duct given by Ω = [0, L] × [−y0 , y0 ] ×
[−z0 , z0 ] with y0 , z0 L. We take w = u (cf. below), f = g = 0, γ = 0, d = (0, 1, 0),
28
(a)
(b)
Fig. 5.8. Example 5. DG solution computed on a uniform tetrahedral mesh with k = 1: (a)
Velocity field; (b) Magnetic field.
and consider solutions of the form
u = (u(y, z), 0, 0),
b = (b(y, z), 1, 0),
p = −Gx + p0 ,
r ≡ 0.
We enforce the boundary conditions
u=0
pn = pN n
n × b = n × bD
for y = ±y0 and z = ±z0 ,
for x = 0 and x = L,
on Γ,
with pN = −Gx + p0 and bD = (0, 1, 0). As before, G and p0 are arbitrary constants.
For this channel problem, the analytical solution can be expressed by Fourier series;
for details, we refer to [15]. Here, we set L = 10, y0 = 2, z0 = 1, ν = νm = 0.1, Ha =
5, G = 0.5, and p0 = 10.
In Figures 5.8 & 5.9 we show the DG solution computed on a uniform tetrahedral
mesh comprising of 30720 elements with the polynomial degree k = 1; this results in
a total of 1075200 degrees of freedom in the finite element space Vh × Ch × Qh × Sh .
In particular, from Figure 5.9, we observe that there is reasonably good agreement
between the computed and analytical solutions of the first components in the velocity
and magnetic fields on this relatively coarse mesh. However, here we do observe
some over-shoots in the computed solution, which are particularly evident in the
approximation to the magnetic field.
29
2
2
DG Solution
Analytical Solution
1.8
1.6
1
1.4
0.5
1.2
1
0
0.8
−0.5
0.6
−1
0.4
−1.5
0.2
0
−2
DG Solution
Analytical Solution
1.5
−1.5
−1
−0.5
0
0.5
1
1.5
−2
−2
2
−1.5
(a)
−1
−0.5
0
0.5
1
1.5
2
(b)
Fig. 5.9. Example 5. DG solution computed on a uniform tetrahedral mesh with k = 1. Slices
along x = 5, −1 ≤ y ≤ 1, z = 0, of the solution: (a) First component of the velocity field; (b) First
component of the magnetic field.
6. Conclusions. In this paper, we have proposed and analyzed a mixed DG
method for a linear incompressible magnetohydrodynamics problem. We have derived
a-priori error estimates and computationally verified them with a set of numerical
examples. We have further tested the methods for channel flow problems in both twoand three-dimensions. As mentioned in Remark 3.3, our analysis can be easily adapted
to the case where conforming Nédélec elements are used for the approximation of the
magnetic unknowns. It can also be extended to the divergence-conforming Pk3 − Pk−1
elements proposed in [10] for the approximation of the fluid variables.
The systematic development of efficient solution methods and preconditioners for
the proposed discretization is the subject of ongoing research. Indeed, a wide variety
of efficient and robust saddle point solvers are available in the literature for both the
Oseen operator and the Maxwell operator. However, these approaches need to be
extended to MHD problems of the form (2.1)–(2.4). We also refer the reader to [20]
and [30, Section 3.4] for discussions of iterative strategies that amount to the solution
of a sequence of decoupled Navier-Stokes and Maxwell problems. Ongoing work also
includes extensions of the method to fully nonlinear incompressible MHD systems.
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31