On the choice of mesh for a singularly
perturbed problem with a corner singularity
Sebastian Franz∗1 , R. Bruce Kellogg2 , and Martin Stynes3
1
2
3
Department of Mathematics and Statistics, University of Limerick, Limerick,
Ireland; [email protected]
Department of Mathematics, University of South Carolina, Columbia, SC 29208,
USA; [email protected]
Department of Mathematics, National University of Ireland, Cork, Ireland;
[email protected]
Summary. A singularly perturbed elliptic problem is considered on the unit square.
Its boundary data has a jump discontinuity at one corner of the square, so the solution of the problem exhibits a singularity there. To solve the problem numerically,
the Galerkin finite element method is tested on various tensor-product meshes. It is
demonstrated that the Shishkin mesh does not yield satisfactory results, but meshes
with a sufficient degree of mesh grading will yield convergence in certain norms,
uniformly in the singular perturbation parameter.
1 Introduction
Consider the singularly perturbed boundary value problem
−ε∆u − pux + qu = f in Q := (0, 1)2 ,
(1a)
u(x, 0) = gs (x), u(x, 1) = gn (x) for 0 < x < 1,
(1b)
u(0, y) = gw (y), u(1, y) = ge (y) for 0 < y < 1.
(1c)
Here the coefficients p and q are constants with p > 0 and q > 0, while the
parameter ε lies in (0, 1]. The functions f, gw , ge , gs , gn are assumed to satisfy
f ∈ C 2`,α (Q), gw , ge , gs , gn ∈ C 2`,α ([0, 1]) for some non-negative integer ` and
α ∈ (0, 1).
This problem was examined in [3, 4] while assuming an arbitrary but
known degree of compatibility between the boundary data and the solution
at the four corners of Q. Pointwise bounds on the solution u and its derivatives were derived in these papers. These bounds showed how the singularities induced by the degree of incompatibility of the data at the corners of Q
∗
Research supported by Science Foundation Ireland under the Research Frontiers
Programme 2008; Grant 08/RFP/MTH1536
2
Sebastian Franz, R. Bruce Kellogg, and Martin Stynes
interacted with the boundary layers in u that are caused by the convectiondominated nature of (1). These are characteristic layers along the sides y = 0
and y = 1 of Q and an exponential outflow layer along the side x = 0. It
is noteworthy that the corner singularities at (1, 0) and (1, 1) induce large
derivatives in the solution that have only a mild decay away from x = 1. In
contrast, the corner singularities at (0, 0) and (0, 1) induce large derivatives
in the solution that decay rapidly away from x = 0. These facts are shown in
the derivative bounds of [3, 4]. (Note that in [3, 4] the convective direction is
the reverse of that of the present paper.)
Bounds such as these are useful in the analysis of numerical methods
for (1). In [2] they were used in the particular case where the boundary data
is continuous at each corner of Q but no higher degree of compatibility was
assumed; a Galerkin finite element method using bilinears on a tensor product
Shishkin mesh was analysed and shown to converge, uniformly in the parameter ε, in the energy norm associated with this problem. Although the Shishkin
mesh was designed only for the boundary layers described above, [2] shows
that it is also able to handle the mild singularity associated with the corners
of the domain (the solution of this problem lies in H 2 (Ω)).
A natural question now arises: if one reduces the compatibility still further
by allowing the boundary data to have a jump discontinuity at one corner, will
the Shishkin mesh still be adequate? That is, for the same Galerkin method,
will one still obtain uniform convergence in some reasonable norm? In the
present paper we shall investigate and answer this question.
2 The test problem and its numerical computation
We construct a test problem of the form (1). The boundary data of this
problem will have a discontinuity at the inflow corner (1, 0) and be continuous
at the other three corners.
For simplicity take q ≡ 0. To begin the construction, consider the halfplane problem
Lv := −ε∆v − pvx = 0 for x < 1,
v(1, y) = g(y) for y ∈ (−∞, ∞).
Its solution v is given (see [3, Section 3.1]) by the formula
Z
1
1 − x p(1−x)/(2ε) ∞
e
(2)
g(t) K1 (pr/(2ε)) dt,
v(x, y) =
2πε
r
−∞
p
where r = (1 − x)2 + (y − t)2 and K1 is a modified Bessel function of the
second kind. Here the convective direction is into the half-plane and the function v has no boundary layer. Let us take the case: p = 1 and g(y) = 1 for
y > 0, g(y) = 0 for y < 0. Then
Z
1 − x (1−x)/(2ε) ∞ 1
v(x, y) =
e
K1 (r/(2ε)) dt.
(3)
2πε
r
0
Singularly perturbed problem with corner singularity
3
This function has an interior layer along the line y = 0; see [3, 4].
Now set w(x, y) = v(x, y) − v(x, −y). Then Lw = 0 in the quarter-plane
x < 1, y > 0 and w(x, 0) = 0 for x < 1, w(1, y) = 1 for y > 0. For the test
problem of this paper we take u = wQ ; see Figure 1. It is easily seen that
u ∈ L∞ (Ω). This function has a characteristic boundary layer along the side
y = 0 of Q but no other layers; the absence of an outflow layer at x = 0 is
unimportant since here we are concerned only with the effect of the corner
singularity at (1, 0). This characteristic layer will require some form of mesh
refinement along y = 0. Furthermore, the singularity at (1, 0) may require
mesh refinement along the side x = 1.
Fig. 1. Plot of test problem u(x, y)
u(x,y)=v(x,y)−v(x,−y)
1.5
1
0.5
0
1
0.8
1
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0
0
A weak formulation of (1) is ε(∇u, ∇w) − (ux , w) = 0 for all w ∈ H01 (Ω).
To solve this numerically, a standard Galerkin finite element method will be
used; see (4) below.
The mesh on the unit square is yet to be chosen. We confine our attention
to tensor product meshes with possible refinement along the two sides y = 0
and x = 1. A standard Shishkin mesh would be refined only along y = 0. We
shall examine further geometric mesh refinement near y = 0 and x = 1, while
maintaining the tensor product structure of the mesh. The precise definition
of the mesh is given in (6).
On this mesh, our Galerkin method uses the space V N of continuous,
piecewise bilinears that vanish at boundary mesh points. The approximate
solution uN is defined to be the continuous piecewise bilinear function, with
4
Sebastian Franz, R. Bruce Kellogg, and Martin Stynes
values at the boundary mesh points given by the values of u, and with values
at the interior mesh points given by the solution to the linear system
N
ε(∇uN , ∇w) − (uN
x , w) = 0 ∀w ∈ V .
(4)
In (4), the parentheses are inner products in L2 (Q). The value of uN at the
point (1, 0) is, somewhat arbitrarily, taken to be 1.
Computational aspects
Matlab 7.5.0 is used in our numerical experiments. The exact solution u(x, y) =
v(x, y) − v(x, −y) is obtained by setting ξ = (1 − x)/(2ε), η = y/(2ε) and
τ = t/(2ε) to reformulate (3) as
Z
p
1
ξ ξ ∞
p
K1
ξ 2 + (η − τ )2 dτ
v(x, y) = v̂(ξ, η) = e
2
2
π
ξ + (η − τ )
0
= I1 + I2 ,
where
I1 =
=
ξ ξ
e
π
Z
ξ ξ
e
π
Z
∞
1
p
η
∞
0
K1
p
ξ 2 + (η − τ )2 dτ
ξ 2 + (η − τ )2
p
1
p
K1
ξ 2 + s2 ds
2
2
ξ +s
= v(ξ, 0),
ξ ξ
e
π
Z
ξ
= eξ
π
Z
I2 =
η
0
0
η
p
1
p
ξ 2 + (η − τ )2 dτ
K1
ξ 2 + (η − τ )2
p
1
p
ξ 2 + s2 ds.
K1
2
2
ξ +s
The changes of variable s = τ − η and s = η − τ were used in I1 and I2
respectively.
It follows that v(x, −y) = v̂(ξ, −η) = v(ξ, 0)
p− I2 , on making the change
of variable s 7→ −s in I2 . Thus, setting r̃(s) = ξ 2 + s2 , we have
Z
2ξ ξ η 1
e
K1 (r̃(s)) ds
u(x, y) = v(x, y) − v(x, −y) =
π
0 r̃(s)
Z
i
2 η
s2
ξ h r̃(s)
=
exp −
e K1 (r̃(s)) ds.
(5)
π 0
ξ + r̃(s) r̃(s)
Note that the improper integral of (3) has now been replaced by an integral
over a bounded interval; the above transformations have the effect of reducing roundoff error in the calculation. The term in square brackets is evaluated using the Matlab function besselk(1,z,1) with z = r̃. The integral is
Singularly perturbed problem with corner singularity
5
then computed using the built-in routine quadgk, an adaptive Gauss-Kronrod
quadrature.
To evaluate energy norm errors, we shall need the first-order derivatives
of u. For these we differentiate (5) directly, obtaining
uy (x, y) =
h
i
η2
ξ
e− ξ+r̃(η) er̃(η) K1 (r̃(η)) ,
πεr̃(η)
ux (x, y) =
Z η
s2
1+ξ
1
ξ 2 r̃
2ξ 2 r̃
− ξ+r̃(s)
−
e
e K1 (r̃(s)) −
−
e K0 (r̃(s)) ds.
πε 0
r̃(s)
r̃(s)3
r̃(s)2
3 Numerical results
Following the discussion in Section 2, we define the nodes of our tensor product
N × N mesh as follows:
xi = 1 − [1 − (i/N )]1+δx ,
1+δy
yj = λy (4j/N )
,
i = 0, . . . , N,
j = 0, . . . , N/4,
4j/N − 1
,
yj = λy + (1 − λy )
3
(6)
j = N/4, . . . , N,
where we assumed that
√
1
λy := min{1/4, 2 ε ln N } ≤ .
4
Here the positive integer N is the number of mesh intervals in each coordinate
direction. The two parameters δx and δy are non-negative and user-chosen.
For δx = 0 we have an equidistant mesh in the x-direction while δx > 0 gives a
mesh that is graded approaching x = 1. The choice δy = 0 produces a Shishkin
mesh (i.e., piecewise equidistant) in the y-direction while δy > 0 would provide
additional grading in the layer region near y = 0. Away from the layer, i.e., for
y > λy , we always have an equidistant mesh in the y-direction. See Figure 2.
We refer to the members of this family as geometric meshes. For other
work on the use of mesh refinement in a problem with both boundary layers
and corner singularities, we mention [1], which deals with a reaction-diffusion
problem in an L-shaped domain, and with zero boundary conditions. The
corner singularity is not as severe as in our problem, but the issue of approximating a function which has both types of singular behaviour is the same.
Our mesh in the y direction is an S-type mesh; these meshes were introduced (and analysed for problems without singularities) by Roos and Linß
in [5].
When the Galerkin method (4) is applied on this mesh, the integrals are
evaluated exactly. To measure the errors in the computed solution, we use
the exact solution u as described in (5). The discontinuity in the boundary
6
Sebastian Franz, R. Bruce Kellogg, and Martin Stynes
Fig. 2. 8 × 8 meshes with (i) δx = δy = 0 (Shishkin mesh)
1
1
(ii) δx = δy = 1
λy r
λy r
0
1
0
1
Table 1. Geometric mesh with δx = δy = 0 (i.e., Shishkin mesh)
N
16
32
64
128
256
ku − uN k0,Ω
2.692e-02 2.11
6.229e-03 1.96
1.604e-03 1.73
4.850e-04 1.38
1.859e-04
ku − uN k1,m
3.161e-02 1.10 1.62
1.475e-02 0.47 0.63
1.068e-02 0.38 0.49
8.204e-03 0.38 0.47
6.298e-03
condition implies p
that u ∈
/ H 1 (Q), the usual Sobolev space. Thus, setting
m(x, y) = min{ε, (x − 1)2 + y 2 }, we weaken the standard energy norm to
Z
1/2
kvk1,m =
m|∇v|2 + v 2
.
(7)
Q
In [3, 4] it is shown that the solution u satisfies |∇u(x, y)| ≤ C (1 − x)2 +
−1/2
y2
. Therefore kuk1,m < ∞.
When computing errors in the L2 norm and the norm k · k1,m , we approximate the error integrals by Gaussian quadrature with 5 × 5 point evaluations
in each mesh cell. Computations show that Gaussian quadrature with fewer
points gives inaccurate results.
In Tables 1–4 we fix ε = 10−6 (a small value that ensures the problem is
singularly perturbed) and results are presented for various values of δx and δy .
The rates of convergence are computed from the hypothesis error = N −rate
in columns 3 and 5, while the rates of column 6 are computed assuming that
error = (N −1 ln N )rate .
Table 1 is for the case δx = δy = 0, i.e., the standard Shishkin mesh. It is
clear from the table that the L2 convergence rate is less than 2, and rates for
the energy norm convergence rates in column 5 and 6 are both less than 1.
An analysis that we shall publish elsewhere shows that for δx ≥ 1 and
δy ≥ δx /2, the L2 error between u and its piecewise bilinear nodal interpolant
Singularly perturbed problem with corner singularity
7
Table 2. Geometric mesh with δx = 1 and δy = δx /2 = 1/2
N
16
32
64
128
256
ku − uN k0,Ω
2.323e-02 2.59
3.852e-03 2.57
6.496e-04 2.43
1.202e-04 2.06
2.874e-05
ku − uN k1,m
2.639e-02 1.46 2.16
9.570e-03 0.67 0.90
6.032e-03 0.58 0.75
4.022e-03 0.58 0.71
2.697e-03
Table 3. Geometric mesh with δx = 3/2 and δy = 5/4
N
16
32
64
128
256
ku − uN k0,Ω
2.592e-02 3.13
2.963e-03 2.96
3.808e-04 2.69
5.889e-05 1.92
1.559e-05
ku − uN k1,m
2.823e-02 1.92 2.84
7.441e-03 0.84 1.14
4.155e-03 0.78 1.00
2.421e-03 0.81 1.01
1.378e-03
Table 4. Geometric mesh with δx = δy = 2
N
16
32
64
128
256
ku − uN k0,Ω
2.168e-02 3.53
1.881e-03 3.02
2.317e-04 1.95
5.993e-05 1.88
1.632e-05
ku − uN k1,m
2.463e-02 1.85 2.73
6.813e-03 0.90 1.22
3.648e-03 0.87 1.12
1.995e-03 0.92 1.14
1.055e-03
is bounded by CN −2 (ln N )5/2 , where the constant C is independent of ε and
N . Thus we take δx = 1 and δy = 1/2 in Table 2, whose results are clearly
superior to those of Table 1. In particular we now have O(N −2 ) convergence
in the L2 -norm; but the energy norm rates still fail to achieve first-order
convergence.
Thus we try a further refinement of the mesh. Table 3 shows the results
for δx = 3/2 and δy = 5/4. Here, in addition to O(N −2 ) convergence in L2 ,
we have O(N −1 ln N ) convergence in the energy norm.
Further increases of δx and δy give limited improvements of these results;
see Table 4 for δx = δy = 2.
Outflow corner discontinuity
The situation is somewhat different if the boundary data discontinuity is
moved from (1, 0) to the outflow corner (0, 0). One can again start from [3,
Section 3.1] and, similarly to Section 2, construct a function that we still call u
which satisfies Lu = 0 on Q with a jump discontinuity in its boundary condition at the corner (0, 0). This function has an outflow boundary layer along
8
Sebastian Franz, R. Bruce Kellogg, and Martin Stynes
the side x = 0 and no other layer. Nevertheless we shall consider mesh refinement near y = 0 since the solutions of almost all boundary value problems
associated with the operator L will have characteristic boundary layers here.
To solve the problem numerically we examine a mesh that is refined geometrically near the side y = 0 of Q exactly as in (6), and has a standard
Shishkin mesh structure in the x variable: piecewise equidistant, and fine near
x = 0 with mesh transition point λx = min{1/2, 2ε ln N }. In the case δy = 0
one gets a Shishkin mesh that is fine near x = 0 and y = 0.
When measuring the error in our numerical solutions,
p the energy norm (7)
is of course modified by redefining m(x, y) = min{ε, x2 + y 2 }. When our
problem is solved numerically on the above Shishkin mesh, the results obtained
are broadly similar to those of Table 1. Thus some mesh refinement is needed
to obtain a more satisfactory rate of convergence in the energy norm.
Table 5. Outflow discontinuity, geometric mesh with δy = 0.5
N
16
32
64
128
256
ku − uN k0,Ω
3.517e-03 1.81
1.002e-03 1.91
2.671e-04 1.97
6.810e-05 2.04
1.659e-05
ku − uN k1,m
1.401e-01 0.66 0.98
8.861e-02 0.72 0.97
5.391e-02 0.74 0.95
3.227e-02 0.73 0.91
1.944e-02
Table 5 displays numerical results for the geometric mesh with δy = 0.5; it
shows that we have nearly O(N −1 ln N ) convergence in the energy norm. This
is in contrast to Tables 1–3 for the inflow corner discontinuity where much
more mesh refinement was needed to attain this order of convergence.
References
1. Vladimir B. Andreev and Natalia Kopteva. Pointwise approximation of corner
singularities for a singularly perturbed reaction-diffusion equation in an L-shaped
domain. Math. Comp., 77(264):2125–2139, 2008.
2. Sebastian Franz, R. Bruce Kellogg, and Martin Stynes. Galerkin and streamline
diffusion finite element methods on a Shishkin mesh for a convection-diffusion
problem with corner singularities. (Submitted for publication).
3. R. Bruce Kellogg and Martin Stynes. Corner singularities and boundary layers in
a simple convection-diffusion problem. J. Differential Equations, 213(1):81–120,
2005.
4. R. Bruce Kellogg and Martin Stynes. Sharpened bounds for corner singularities
and boundary layers in a simple convection-diffusion problem. Appl. Math. Lett.,
20(5):539–544, 2007.
5. H.-G. Roos and T. Linß. Sufficient conditions for uniform convergence on layeradapted grids. Computing, 63(1):27–45, 1999.