Stokes Problem with Slip Boundary Conditions of Friction Type: Error Analysis of a Four-Field Mixed Variational Formulation

Abstract

In this work, a finite element approximation of the Stokes problem under a slip boundary condition of friction type, known as the Tresca boundary condition, is considered. We treat the approximate problem of a four field mixed formulation using the ${\mathbb {P}}^{1}$-bubble element for the velocity field, ${\mathbb {P}}^{1}$ element for the pressure field and the ${\mathbb {P}}^{1}$ element for the Lagrange multipliers $\lambda _{n}$ and $\lambda _{t}$ defined on the slip boundary. The multiplier $\lambda _{t}$ is introduced to regularize the non-differentiable problem, whereas $\lambda _{n}$ treats the impermeability condition. Existence and uniqueness results for both continuous and discrete problems are proven and an a priori error estimate is established. Numerical realization of such problem is discussed and some numerical tests are provided.

Appendix. Estimate of \(N^0\)

For simplicity we suppose that $\varGamma =[0,1]$, that $\varGamma _{ns} = [{\hat{x}},1]$ and we study the orthogonal projection of $L^2(\varGamma )$ on $\tilde{W_h}$ assuming that ${{{\mathscr {T}}}}_h|_\varGamma$ is a uniform mesh. Let $N=\dim ({\tilde{W}}_h)$. Since $\pi _h u_t = \sum _{i=1}^N \alpha _i \varphi _i$ with vector $\alpha = (\alpha _1,\ldots , \alpha _N)\in {\mathbb {R}}^N$ solution of the linear system $A \alpha = \ell ,$ where $A = (a_{ij})$ is the $N\times N$ mass matrix with $a_{ij} = \int _\varGamma \varphi _i \varphi _j \mathrm {d}\varGamma$, $1\le i,j,\le N$ and $\ell = (\ell _i)$ with $\ell _i = \int _\varGamma u_t \varphi _i \mathrm {d}\varGamma$, $1\le i \le N$.

We consider the elements $e_k$ of ${{\mathscr {T}}}_h|_\varGamma$ such that $e_k\cap \varGamma _{ns} \ne \emptyset$, that is $\varGamma _{ns,h}=\cup _{k=k_0}^{N-1} e_k (\supset \varGamma _{ns})$ with ${\hat{x}} \in e_{k_0}$ (we have supposed here that the elements $e_k$ are numbered from left to right). Then the vector ${\tilde{\alpha }}=(\alpha _{k_0+1},\ldots \alpha _N)$ is solution of ${\tilde{A}} {\tilde{\alpha }} = {\tilde{\ell }},$ where ${\tilde{A}}$ and ${\tilde{\ell }}$ are, respectively, the tridiagonal matrix and the right-hand side vector defined by

$$\begin{aligned} \frac{h}{6} \left( \begin{array}{cccccc} 4 &{}\quad 1 &{}\quad 0 &{}\quad \ldots &{}\quad \ldots &{}\quad 0 \\ 1 &{}\quad 4 &{}\quad 1 &{}\quad 0 &{}\quad \ldots &{}\quad 0 \\ &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \\ &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \\ 0 &{}\quad \ldots &{}\quad 0 &{}\quad 1 &{}\quad 4 &{}\quad 1 \\ 0 &{}\quad \ldots &{}\quad \ldots &{}\quad 0 &{}\quad 1 &{}\quad 2 \end{array} \right) , \quad \text{ and }\quad \left( \begin{array}{c} \ell _{k_0+1} - \frac{h}{6}\alpha _{k_0} \\ 0 \\ \vdots \\ \vdots \\ 0\\ 0 \end{array} \right) . \end{aligned}$$

The solution of this system can be calculated explicitly using the properties of Chebyshev’s polynomials of the first and second kind $T_k$ and $U_k$, respectively (see [19]). In fact, renumbering from $k=1$ to $n=N-k_0$ and using the recurrence relations $T_{k+1}(\lambda ) = 2\lambda T_k(\lambda ) - T_{k-1}(\lambda )$ and $U_{k+1}(\lambda ) = 2\lambda U_k(\lambda ) - U_{k-1}(\lambda )$, we have that $\tilde{\alpha _k} = CT_k(-2) + DU_{k-1}(-2)$, with $C,D\in {\mathbb {R}}$, satisfies the $k-$th homogeneous equation of the linear system ${\tilde{A}} {\tilde{\alpha }} = {\tilde{\ell }}$, for $2 \le k \le n$. With the first, second and n-th equations of the linear system we can write a $3\times 3$ linear system for $\tilde{\alpha _1}$, C and D. The resolution of this system leads to

$$\begin{aligned} \tilde{\alpha _k} = \frac{6}{h}{\tilde{\ell }}_1 (a T_k(\lambda ) + b U_{k-1}(\lambda )),\quad k=2,\ldots , n, \end{aligned}$$

where $\lambda = -2$, $a=-1$ and

$$\begin{aligned} b = \frac{T_{n-1}(\lambda ) + 2 T_n(\lambda )}{U_{n-2}(\lambda ) + 2 U_{n-1}(\lambda )}. \end{aligned}$$

Using the explicit expression of the Chebyshev’s polynomials and denoting $r_{1,2} = -2 \pm \sqrt{3}$ (the roots of the characteristic equation $r^2 -2\lambda r + 1 = 0$) we have that

$$\begin{aligned} a T_k(\lambda ) + b U_{k-1}(\lambda ) = r_1 ^k \frac{1}{\frac{r_1^{n-1}(1 + 2r_1)}{r_2^{n-1}(1 + 2r_2)} -1} + r_2^k \frac{1}{\frac{r_2^{n-1}(1 + 2r_2)}{r_1^{n-1}(1 + 2r_1)} -1}. \end{aligned}$$

The oscillating behaviour of the solution is due to the fact that $r_1$ and $r_2$ are negative numbers. The decreasing behaviour is explained by the fact that $|r_1| < 1$, $|r_2| >1$ and $n>>1$. In fact, we have that

$$\begin{aligned} \frac{1}{\frac{r_1^{n-1}(1 + 2r_1)}{r_2^{n-1}(1 + 2r_2)} -1} \rightarrow -1,\quad n\rightarrow \infty , \end{aligned}$$

and

$$\begin{aligned} \frac{1}{\frac{r_2^{n-1}(1 + 2r_2)}{r_1^{n-1}(1 + 2r_1)} -1} \rightarrow 0, \quad n\rightarrow \infty ,\quad \text{ like }\quad \frac{r_1^n}{r_2^{n}}. \end{aligned}$$

That is, for n large enough (i.e. for h small enough), $a T_k(\lambda ) + b U_{k-1}(\lambda )$ behaves like $r_1^k$.

Next, the coefficient ${\tilde{\ell }}_1= \ell _{k_0+1} - \frac{h}{6}\alpha _{k_0}$ can be approximated with a first order quadrature rule by $-\frac{3}{24}h u_t(x_{k_0}) + \frac{h}{6}\delta$, where $\delta = u_t(x_{k_0}) - \alpha _{k_0}$. Hence, $\tilde{\alpha _k}$ behaves also like ${r_1^k}$. Therefore we have that $|\tilde{\alpha _k}| \le h^2$ for

$$\begin{aligned} k \ge 2 \frac{\log (h)}{\log (|r_1|)}. \end{aligned}$$

So that $N^0 \approx 2 \frac{|\log (h)|}{|\log (|r_1|)|}$.