Robust Error Estimation for Singularly Perturbed Fourth Order Problems

. We consider two-dimensional singularly perturbed fourth order problems and estimate on properly constructed layer-adapted errors of a mixed method in the associated energy norms and balanced norms. This paper is a shortened version of [4]

The solution of this problem lies in H 2 0 which means, a conforming finite element discretisation requires C 1 -elements. They are not very popular in 2d or 3d, which leads to the widely usage of mixed or non-conforming methods. In this paper we want to study mixed finite element methods of order p. For non-singularly perturbed problems (ε = 1) and p-th order finite-element approximation for u and w = ∆u, the classical error estimate for the discrete solutions u h and w h on a standard shape-regular mesh with p ≥ 2, is known, see [3,11].
Here we want to consider the singularly perturbed case and estimation in energy and balanced norms. We use standard notation for Sobolev spaces, where · 0 is the L 2 -norm, | · | k the seminorm in H k and · k the full H k -norm. Furthermore, we denote by u, v D the L 2 -scalar product over a domain D ⊂ Ω, a subscript we drop if D = Ω.

Solution decomposition and meshes
For our numerical analysis to work we assume a decomposition of the solution u of problem (1a)+(1b) into a smooth part, boundary layers and corner layers: Here S stands for the smooth part, E k with k = 1, 2, 3, 4 for a boundary layer and E k with k = 12, 23, 34, 41 for a corner layer. More precisely, we assume and similarly for the other components of the decomposition. These assumptions are reasonable, see e.g in 1d in [12] or for smooth domains in [2, §12.4.3].
With above assumption on the solution decompositions we have |E 1 (λ, y)| ≤ CεN −σ . These layers are therefore weak layers as their influence vanishes with decreasing ε in a pointwise sense. Note also that the small and the large meshwidths satisfy

Numerical method and analysis
Using w = ε∆u ∈ H 2 (Ω), we rewrite the fourth-order problem as a system and obtain a weak formulation: where c = c + ∇b. The associated bilinear form is given by a : ( and a corresponding energy norm by By a direct calculation we have coercivity of the bilinear form w.r.t. the energy norm Therefore, above mixed formulation has a unique solution. Now let us define the discrete space on a rectangularly divided mesh T N . We use Here Q p (τ ) is the polynomial space on τ , with polynomial degrees at most p in each coordinate direction. The discrete problem now reads:

Estimation in the energy norm
The analysis of our method works, as usual, with the help of suitable interpolation operators and their error estimates. Let us define the interpolation operators I : C(Ω) → V 0 (Ω) and J : C(Ω) → V (Ω) in the sense of [5, p. 108] and [8]. For these interpolation operators hold the anisotropic interpolation-error estimates by [1,10]. Using these, standard techniques and the definition of the mesh, see i.e. [14, Section III.3.5], we obtain the following interpolation-error estimates.
Lemma 1. We have for σ ≥ p + 1 We can also use supercloseness estimates based on integral identities from [7,6,17,16]. They yield on each cell τ Let us now consider a rectangular domain T = {τ } with 1,T and 2,T being its left and right boundary. With inverse and Holder inequalities we conclude also the estimates If v x = 0 or χ = 0 on i,T for i = 1 or i = 2, then the sum containing i in (4b) can be omitted. The proof of the following theorem is based purely on interpolation-error estimates, supercloseness estimates and standard techniques for singularly perturbed problems.
Then it holds for the discrete error on a Shishkin mesh with σ ≥ p + 2 the supercloseness result Having these discrete-error and interpolation-error estimates, it is easy to conclude the following error estimates.
Theorem 3. On a Shishkin mesh with σ ≥ p + 2 holds for the exact solution (u, w) and the discrete solution

Estimation in a balanced norm
For a typical layer function E 1 and a non-layer function S the energy-norm yields 0 ≤ Cε and |||(S, ε∆S)||| 2 ≤ C and the the layer is not seen for ε → 0 in the energy norm. Introducing a balanced norm 2 0 ≤ C and |||(S, ε∆S)||| 2 b ≤ C, the layer is seen in this balanced norm. Unfortunately, our method is not coercive with respect to this norm. In order to prove error estimates we have to combine some more ideas. In [9] a special interpolation operator is constructed which uses different interpolants for a decomposition of w = ε∆u into smooth and layer components and is for layer components zero in the subdomain where the layer components are small enough. In [13] the idea of using suitable projections in estimating balanced norms is introduced. For this purpose we define the Ritz-projection πu ∈ V 0 by b∇(πu − u), ∇ψ + c · ∇(πu − u), ψ + d(πu − u), ψ = 0 for all ψ ∈ V 0 . Then we have for ψ = πu − u h ∈ V 0 and ϕ =Jw − w h ∈ V .
Using these ideas we can prove error estimates in the stronger balanced norm.
Remark 5. Comparing the results of the Theorems 3 and 4 we observe a reduction of the convergence order by one when measuring in the stronger norm. Numerically, there is no reduction visible. Moreover, combining Theorems 3 and 4 we obtain which is the corresponding uniform result on Shishkin meshes to the classical estimate (2) for ε = 1.
These boundary conditions introduce even weaker boundary layers, but above analysis can be adapted for this case too, and the same results in the energy norm hold. This norm is again not balanced, but can be made stronger by properly defining the weights in its definition. In this stronger norm an error estimate of the same order as in the energy norm holds in the case of constant b.
The extension to problems of type with either of the boundary conditions considered above, can also be done. Here the boundary layers are stronger than the ones considered so far. Still the analysis can be applied.
Further information, the full proofs and numerical results can be found in [4].