Convergence and Optimality of the Adaptive Nonconforming Linear Element Method for the Stokes Problem

Abstract

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi-orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates.

Appendix: A Counter Example

We present an example in this appendix to show that if the prolongation operator $I_{h}^{\prime}$ defined by (5.2) is directly used to analyze the discrete reliability of the estimator, the constant for the established discrete reliability could depend on some key mesh refinement ratio

$$\gamma:=\max _{K\in \mathcal {T}_H\backslash \mathcal {T}_h}\max _{\mathcal {T}_h\ni T\subset K} \frac{h_K}{h_T}, $$

where $\mathcal {T}_{H}$ is some regular triangulation of Ω into triangles and $\mathcal {T}_{h}$ is some refinement of $\mathcal {T}_{H}$. To this end, we first give an example to demonstrate that there are generally no positive constants C independent of γ such that the following estimate holds true:

$$ \sum _{E\in \mathcal {E}_h\backslash \mathcal {E}_H}\int _E[u_H]\biggl\{\frac{\partial v_h}{\partial \nu_E}\biggr\}ds \leq C\biggl(\sum _{E\in \mathcal {E}_H\backslash \mathcal {E}_h}h_E^{-1} \bigl\|[u_H]\bigr\|_{L^2(E)}^2\biggr)^{1/2}\| \nabla _hv_h\|_{L^2(\varOmega )}, $$

(A.1)

where u _H ∈V _H is the finite element solution of the velocity on the mesh $\mathcal {T}_{H}$ and v _h is some element of V _h over the nested fine mesh $\mathcal {T}_{h}$. As usual, $\mathcal {E}_{H}$ (resp. $\mathcal {E}_{h}$) is the set of the edges of $\mathcal {T}_{H}$ (resp. $\mathcal {T}_{h}$). Denote V _H (resp. V _h ) as the nonconforming linear element space with respect to $\mathcal {T}_{H}$ (resp. $\mathcal {T}_{h}$). Denote [⋅] as the jump of some function across the edge E and {⋅} as the average of some function across the edge E. In addition, denote ν _E as the unit normal vector to E with the length h _E .

In the following, an example is given to show that u _H ∈V _H and v _h ∈V _h exist such that the above constant C depends on the ratio γ. For simplicity, let $\mathcal {T}_{H}$ consist of two triangles △ABC and △ACD as in Fig. 1. Let $\mathcal {T}_{h}$ be a uniform triangulation of Ω into 2×N ² triangles, cf. Fig. 1 for the case N=5. We stress that the idea and result can be easily extended to the mesh with the newest vertex bisection. For the sake of simplicity, let N=2k+1 with some nonnegative integer k. Let Z _i , i=−k,…,k, be the nodes of $\mathcal {T}_{h}$ whose coordinates are $(\frac {1}{N},\frac {2i}{N})$. Let $\phi_{Z_{i}}$ be the nodal basis function of the conforming linear element space defined over $\mathcal {T}_{h}$ such that $\phi_{Z_{i}}(Z_{i})=1$ and $\phi_{Z_{i}}(Z)=0$ for any node Z other than Z _i . We choose u _H ∈V _H such that the jump is [u _H ]=y over the edge AC. We choose v _h as follows:

$$ v_h:=\sum _{i=-k}^{k}\operatorname {sign}(i) \phi_{Z_i}\quad \text{with } \operatorname {sign}(i):= \begin{cases} 1&\text{if }i>0, \\ 0 &\text{if }i=0, \\ -1&\text{if }i<0. \end{cases} $$

(A.2)

Note that $\{\frac{\partial \phi_{Z_{i}}}{\partial \nu_{E}}\}=N/2$ over the edge AC for i=−k,…,k. A direct calculation gives

$$ \int_{AC}[u_H]\biggl\{\frac{\partial v_h}{\partial \nu_E}\biggr \}ds=N/2-\frac{1}{2N}. $$

(A.3)

On the other hand, a direct calculation leads to

$$ \|\nabla _hv_h\|^2_{L^2(\varOmega )}\leq4N. $$

(A.4)

This indicates that the constant C in (A.1) should be $\mathcal{O}(\sqrt{N})$, which depends on the ratio $\gamma =\mathcal {O}(N)$ for this example.

Fig. 1

The meshes $\mathcal {T}_{H}$ and $\mathcal {T}_{h}$

For the analysis of the discrete reliability, a direct application of the prolongation operator $I_{h}^{\prime}$ as defined in (5.2) will lead to a similar estimate like (A.1), and, as a result, the constant for the established discrete reliability based on such an estimate will depend on the ratio γ. Note that in the analysis of optimality of the adaptive method it is possible to know that $\mathcal {T}_{h}$ is some refinement of $\mathcal {T}_{H}$ only by the newest vertex bisection [8, 19, 40]. Note, too, that there is no guarantee that γ is bounded. Therefore, the proof of the discrete reliability based on the prolongation operator $I_{h}^{\prime}$ as presented in [5, 30] may not lead to a uniform estimate as claimed.