FEM-analysis on graded meshes for turning point problems exhibiting an interior layer

We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted graded meshes proposed by Liseikin. We prove $\epsilon$-uniform error estimates in the energy norm. Furthermore, for linear elements we are able to prove optimal order $\epsilon$-uniform convergence in the $L^2$-norm on these graded meshes.


Introduction
We consider singularly perturbed boundary value problems of the type where 0 < ε ≪ 1 is a small parameter and a, c, f are sufficiently smooth with for a point x 0 ∈ (−1, 1). Thus, the solution of (1.1) exhibits an interior layer of "cusp"-type at the simple interior turning point x 0 . In the literature (see i.e. [1], [3, p. 95], [6, Lemma 2.3]) the bounds for such interior layers are well known. We have where the parameter λ satisfies 0 < λ <λ := c(x 0 )/|a ′ (x 0 )|. The estimate also holds for λ =λ, if λ is not an integer. Otherwise there is an additional logarithmic factor, see references cited above. For convenience we assume x 0 = 0 in the following.
In the last decades a multitude of numerical methods has been developed to solve singularly perturbed problems with turning points and interior layers. For a general review we refer to [5]. Many authors have considered finite difference methods. A selection of possible schemes for problems of the form (1.1) may be found in [2] and the references therein. Also some layer-adapted meshes have been proposed to handle interior layers of "cusp"-type. As an example Liseikin [3] proved the ε-uniform first order convergence of an upwind finite difference method on special graded meshes. Moreover, Sun and Stynes [6] studied finite elements on a piecewise uniform mesh.
Under certain assumptions, we prove ε-uniform convergence in the energy norm of the form |||u − u N ||| ε ≤ CN −k for finite elements of order k, where C may depend on α and k, see Theorem 3.5. On the basis of a supercloseness result we also give an optimal error estimate in the L 2 -norm of the form for linear finite elements, see Theorem 3.10. Numerical experiments confirm our theoretical results. Notation: In this paper C denotes a generic constant independent of ε and the number of mesh points. Furthermore, for an interval I the usual Sobolev spaces H 1 (I), H 1 0 (I), and L 2 (I) are used. The spaces of continuous and k times continuously differentiable functions on I are written as C(I) and C k (I), respectively. Let (·, ·) I denote the usual L 2 (I) inner product and · I the L 2 (I)-norm. We will also use the supremum norm on I given by · ∞,I and the semi-norm in H 1 (I) given by |·| 1,I . If I = (−1, 1), the index I in inner products, norms, and semi-norms will be omitted. Additionally, for all v ∈ H 1 ((−1, 1)) we define a weighted energy norm by |||v||| ε := ε |v| 2 1 + v 2 1/2 .
Further notation will be introduced later at the beginning of the sections where it is needed.

The graded meshes proposed by Liseikin
The basic idea of Liseikin is to find a transformation ϕ(ξ, ε) that eliminates the singularities of the solution when it is studied with respect to ξ. In our case the approach can be condensed to the task to find ϕ : The outcome of this approach is the mesh generating function where 0 < α ≤ λ. By construction we have ϕ(0, ε) = 0 and ϕ(±1, ε) = ±1. Note that Liseikin derived the same transformation indirectly. Based on the principle of equidistribution, he used basic majorants of the solution derivatives to find basic layer-damping transformations. This procedure allows to handle also various other types of singularities, see e.g. [3,Chapter 6]. Now, the mesh points are generated by x i = ϕ( i N , ε), i = −N, . . . , N . We will denote the lengths of the mesh intervals by Motivated by Lemma A.2 and A.3, we define and estimate a special constant κ dependent on α ∈ (0, 1] and ε ∈ (0, 1] by It remains to check whether or not ϕ defined in (2.2) satisfies (2.1). An easy calculation shows which can be bounded independent of ε due to (2.3). Since the arguments are very similar thanks to the symmetry of the mesh, we will consider the case ξ ≥ 0 only. The next lemmas comprise some basic results concerning the mesh points and mesh intervals. Their proofs are deferred to Appendix B. The argumentation substantially uses the property (2.1). Here, the derivative of ϕ comes into play since the mean value theorem guarantees the estimate If 0 < α ≤ 1/k with k ∈ N, k ≥ 1 and ε ≤ h 2/α then we have In general, we have for 0 < α ≤ 1

Remark 2.3
Note that an estimate similar to the first one of Lemma 2.2 can also be found in [7]. ♣
would be convenient. In fact, this is ensured for 0 < α ≤ λ/k which is already used in the proof of Lemma 2.1. ♣

FEM-analysis on graded meshes
This section follows the paper of Sun and Stynes [6], but while they studied linear finite elements on a layer-adapted piecewise uniform mesh, we shall use the graded mesh proposed by Liseikin instead. Besides our more general approach enables to analyse finite elements of higher order as well. We will only consider homogeneous Dirichlet boundary condition ν −1 = ν 1 = 0. This is no restriction at all since it can be easily ensured by modifying the right hand side f . Furthermore, due to [6, Lemma 2.1] we may assume without loss of generality that For v, w ∈ H 1 0 ((−1, 1)) we set B ε (v, w) := (εv ′ , w ′ ) + (av ′ , w) + (cv, w).
The weak formulation of (1.1) with ν −1 = ν 1 = 0 reads as follows: Find u ∈ H 1 0 ((−1, 1)) such that Let k ≥ 1 and let P k ((x a , x b )) denote the space of polynomial functions of maximal order k over (x a , x b ). We define the trial and test space V N by Then the discrete problem is given by: Letφ 0 , . . . ,φ k denote the Lagrange basis functions on the reference interval [0, 1] with respect to the points 0 =x 0 <x 1 < . . . <x k = 1. We shall denote by u I ∈ V N the interpolant of u which is defined on each mesh interval (x i−1 , x i ) by , for all j = 0, . . . , k + 1 the standard interpolation theory leads to the error estimates: holds.

Finite elements of higher order
In the following we shall present the analysis for finite elements of order k ≥ 1 for problems of the form (1.1). We assume that λ ∈ (0, k + 1) which is the most difficult case. Otherwise all crucial derivatives of the solution could be bounded by a generic constant independent of ε and consequently optimal order ε-uniform estimates could be proven with standard methods on uniform meshes.

Lemma 3.1
Let u be the solution of (1.1) and u N the solution of (3.2) on an arbitrary mesh. Then we have Proof: By the coercivity of B ε (·, ·) and due to orthogonality, we have Integrating by parts, we obtain Hence, triangle inequality and Cauchy-Schwarz inequality yield Now, for −N + 1 ≤ i ≤ N an inverse inequality and the fact that a is smooth with a(0) = 0 imply Using this bound to estimate (3.6), we get by Cauchy-Schwarz' inequality Combining this and (3.5) completes the proof.
see also Lemma 3.7. Aside from the fact that their argumentation works for linear elements only, such an estimate would not enable optimal estimates for finite elements of higher order. ♣ The next two lemmas give bounds for the interpolation error on the layer-adapted mesh proposed by Liseikin.
Combining the above estimates for j = 0 and using symmetry on [−1, 0] we get (3.7). This estimate together with the above estimates for j = 1 immediately gives (3.8).
It remains to estimate the second term in Lemma 3.1. Lemma 3.4 Let u be the solution of problem (1.1). Let u I ∈ V N interpolate to u on the mesh generated by (2.2) with 0 < α ≤ min{λ/(k + 1), 1/(2(k + 1))}. Then (3.9) Proof: The proof is similar to the proof of Lemma 3.3 but advanced in some way.
where we used (3.3), (1.2), and Lemma 2.1. Hence, Now, let i = 1 and x ∈ (x 0 , x 1 ). We consider two different cases. First, if ε ≥ h 2/α then as above Lemma 2.1 yields and therefore If ε ≤ h 2/α we estimate the integral directly. We have by ( Since, thanks to symmetry the sum for i = −N + 1, . . . , 0 can be bounded analogously, the proof is completed. Now, we are able to prove the ε-uniform error estimate of P k -FEM in the energy norm.

Special features of linear finite elements
In this section we present some special features of linear finite elements. So, we shall assume k = 1.
The following two lemmas hold for arbitrary meshes and are borrowed from [6]. In particular, they show that for linear finite elements the L 2 -norm interpolation error estimate (3.7) suffices to prove the ε-uniform convergence in the energy norm.  Let u be the solution of (1.1) and u N ∈ V N (k = 1) the solution of (3.2) on an arbitrary mesh. Then we have Proof: As in the proof of Lemma 3.1 the coercivity of B ε (·, ·) and orthogonality yield Integrating by parts and applying the Cauchy-Schwarz inequality, we have where we used (3.11) and u I ∞ ≤ u ∞ ≤ C.
The next lemma provides an auxiliary inequality that will be needed later. We defer its proof to Appendix C.

Lemma 3.8
Let e ∈ V N (k = 1) on an arbitrary mesh and −N ≤ L < R ≤ N . Then where e i = e(x i ).
Proof: See Appendix C.
We now prove the ε-uniform error estimate in the energy and L 2 -norm.

Theorem 3.10
Let u be the solution of (1.1) and u N ∈ V N (k = 1) the solution of (3.2) on a mesh generated by (2.2) with 0 < α ≤ min{λ/2, 1/4}. Then we have Proof: The bound in the energy norm is already given in Theorem 3.5, but also follows easily from the triangle inequality, Lemma 3.7, (3.10), and (3.7). To prove the bound in the L 2 -norm only the supercloseness result of Lemma 3.9 and (3.7) have to be used.

Remark 3.11
For linear elements in [6, Theorem 5.1] the presumably non-optimal L 2 -norm estimate is proven for a discrete solution calculated on a piecewise-equidistant mesh. The argumentation there is similar to the proof of Lemma 3.9. But, in one of the occurring terms there are problems when h i = h i+1 since the difference |h i − h i+1 | is not sufficiently small on the piecewise-equidistant mesh. For the graded meshes generated by (2.2) we have the estimates of Lemma 2.2. Thus, these problems can be circumvented. ♣

Numerical experiments
Now, we shall present some numerical results to verify the theoretical findings of this paper. Therefore, we study a test problem taken from [6] whose solution exhibits typical interior layer behaviour of "cusp"-type. All computations where performed using a FEM-code based on SOFE by Lars Ludwig [4]. In general, the parameter α needed to generate the graded mesh was chosen as α = α 0 min{λ/(k + 1), 1/(2(k + 1))} with α 0 = 1. Exceptions are explicitly stated. For given errors E ε,N we calculate the rates of convergence by (ln E ε,N − ln E ε,2N ) / ln 2.
Example 4.1 (see [6]) We consider the singularly perturbed turning point problem where the right hand side f (x) is chosen such that the solution u(x) is given by Note that the parameter λ in the problem coincide with the quantityλ = c(0)/|a ′ (0)|.
In Figure 1 the energy norm error is plotted for finite elements of order k = 1, . . . , 4 applied to Example 4.1 with ε = 10 −8 and λ = 0.005. The expected convergence behaviour, cf. Theorem 3.5, can be clearly seen. The numerical results suggest that the energy norm error is almost independent of ε. Anyway it stays stable for small ε, see Table 1.
Furthermore, we study the influence of varying λ and α 0 . Therefore, we consider Example 4.1 with fixed ε = 10 −8 on correspondent layer-adapted meshes with N = 1024. In Figure 2 the energy norm error is plotted against λ for α 0 = 1 (left) and against α 0 for λ = 0.005 (right), respectively. In both cases the error is almost constant in the studied ranges. Thus, it seems to be plausible to presume the method to be robust in α.
Finally, in Table 2 we compare the energy norm and the L 2 -norm error for linear finite elements. As predicted by theory, cf. Theorem 3.10, the L 2 -error is uniformly convergent of second order whereas the error in the |||·||| ε -norm converges with order one only.

Acknowledgement
The author would like to thank Hans-Görg Roos for helpful comments and discussions.

A Auxiliary lemmas
In this section we provide some auxiliary lemmas and prove some basic inequalities that are needed in the paper.

Proof:
We study the function The first derivative of f is calculated to be .
The upper bound is verified using the convexity of x → 2 x . We have The bound of the last lemma is exact for c = 1 and asymptotically exact for α ց 0. ♣