A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem

Abstract

We propose a continuous interior penalty finite element method designed for a third-order singularly perturbed problem. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Moreover, we show numerical experiments which support our theoretical findings.

Appendices

Appendices

1.1 Appendix 1: Proof of existence of weak solutions

First we verify the inf-sup condition

$$\begin{aligned} \mathcal{S}:=\sup _{0\ne w\in Y}\frac{B_{G}(v,w)}{|w|_{H^1(\Omega )}}\ge C\Vert v\Vert _X. \end{aligned}$$

(48)

To do that we estimate $|v|_{H^1(\Omega )}$ and $\varepsilon |v|_{H^2(\Omega )}$ separately. We get easily

$$\begin{aligned} |v|_{H^1(\Omega )}^2\le C B_{G}(v,v)=C \frac{B_{G}(v,v)}{|v|_{H^1(\Omega )}}|v|_{H^1(\Omega )}, \end{aligned}$$

and therefore

$$\begin{aligned} |v|_{H^1(\Omega )}\le C\,\mathcal{S}. \end{aligned}$$

For estimating $\varepsilon |v|_{H^2(\Omega )}$ we use the possibility to estimate the $L^2$-norm by the sum of the $H^{-1}$-norm and the $H^{-1}$-norm of the derivative, see Steinbach (2008, Theorem 2.17). It follows

$$\begin{aligned} |v|_{H^2(\Omega )}\le C(|v''|_{H^{-1}(\Omega )}+|v'''|_{H^{-1}(\Omega )}). \end{aligned}$$

Now

$$\begin{aligned} |v''|_{H^{-1}(\Omega )}=\sup _{0\ne w\in Y} \frac{\langle v'',w\rangle }{|w|_{H^1(\Omega )}} \le |v|_{H^1(\Omega )}\le C\mathcal{S}. \end{aligned}$$

Moreover,

$$\begin{aligned} \varepsilon |v'''|_{H^{-1}(\Omega )}=\varepsilon \sup _{0\ne w\in Y} \frac{\langle v''',w\rangle }{|w|_{H^1(\Omega )}} =\varepsilon \sup _{0\ne w\in Y} \frac{(-v'',w')}{|w|_{H^1(\Omega )}}. \end{aligned}$$

Taking into account

$$\begin{aligned} \varepsilon (v'',w')=-B_{G}(v,w)+(av',w')+(bv'+cv,w), \end{aligned}$$

we get

$$\begin{aligned} \varepsilon |v'''|_{H^{-1}(\Omega )}\le \sup _{0\ne w\in Y}\frac{B_{G}(v,w)}{|w|_{H^1(\Omega )}} +\sup _{0\ne w\in Y}\frac{|A(v,w)|}{|w|_{H^1(\Omega )}}\le C\,\mathcal{S}. \end{aligned}$$

Consequently, we have proved (48).

Finally we consider the equation

$$\begin{aligned} -\varepsilon (v'',w')+A(v,w)=0. \end{aligned}$$

Taking $v\in C_0^\infty (\Omega )$, from

$$\begin{aligned} \varepsilon \langle w'',v'\rangle =-A(v,w) \end{aligned}$$

we conclude that w belongs to the space $H^2(\Omega )$. Moreover, it satisfies the homogenous adjoint problem

$$\begin{aligned} \varepsilon (w'',v')+A(w,v)=0 \end{aligned}$$

(49)

with the additional boundary condition $w'(0)=0$. Setting $w=v$ in (49), we get

$$\begin{aligned} \frac{\varepsilon }{2}(w')^2(1)+\min \{\alpha , \gamma \}\Vert w\Vert ^2_{H^1(\Omega )}\le 0, \end{aligned}$$

thus $w=0$.

Now, the unique solvability of the Galerkin formulation (5)–(6) follows and moreover, stability with respect to the norm in X.

1.2 Appendix 2: Proof of Lemma 2

On an interval $I_i\subset \Omega _f$, $i=N/2+1,\ldots ,N$, starting from the standard interpolation estimate (23) and (4), we first have

$$\begin{aligned} |E-E^I|^2_{H^q(I_i)}&\le Ch_i^{2(k+1-q)}\Vert E^{(k+1)}\Vert ^2_{L^2(I_i)} \le C\varepsilon ^{-2k}h_i^{2(k+1-q)}\int _{I_i}\mathrm{e}^{-2(1-x)/\varepsilon }\,\mathrm{d}x \nonumber \\&= C\varepsilon ^{-2k+1}h_i^{2(k+1-q)}\left( \mathrm{e}^{-2(1-x_i)/\varepsilon }-\mathrm{e}^{-2(1-x_{i-1})/\varepsilon }\right) . \end{aligned}$$

(50)

From (21) and the properties of the mesh-generating and mesh-characterizing functions, we get

$$\begin{aligned} \sinh (h_i/\varepsilon )&\le Ch_i/\varepsilon =C(\phi (t_{i-1})-\phi (t_i))=C\int _{t_i}^{t_{i-1}}\phi '(t)\,\mathrm{d}t =C\int _{t_{i-1}}^{t_i}\frac{\psi '(t)}{\psi (t)}\,\mathrm{d}t \\&\le C\psi (t_{i-1})^{-1}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t = C\mathrm{e}^{(1-x_{i-1})/(\tau \varepsilon )}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t, \end{aligned}$$

in order to derive

$$\begin{aligned} \mathrm{e}^{-2(1-x_i)/\varepsilon }-\mathrm{e}^{-2(1-x_{i-1})/\varepsilon }&= \mathrm{e}^{-2(1-x_{i-1})/\varepsilon } (\mathrm{e}^{2h_i/\varepsilon }-1) \\&= 2\mathrm{e}^{-2(1-x_{i-1})/\varepsilon } \mathrm{e}^{h_i/\varepsilon }\sinh (h_i/\varepsilon ) \\&\le C\mathrm{e}^{-2(1-x_{i-1})/\varepsilon }\mathrm{e}^{(1-x_{i-1})/(\tau \varepsilon )}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t. \end{aligned}$$

Applying the last inequality into (50) gives us

$$\begin{aligned} |E-E^I|^2_{H^q(I_i)}&\le C\varepsilon ^{-2k+1}h_i^{2(k+1-q)} \mathrm{e}^{-2(1-x_{i-1})/\varepsilon }\mathrm{e}^{(1-x_{i-1})/(\tau \varepsilon )}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)} \mathrm{e}^{\left( \frac{2(k+1-q)}{\tau \varepsilon }-\frac{2}{\varepsilon }+\frac{1}{\tau \varepsilon }\right) (1-x_{i-1})} \int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t. \end{aligned}$$

Here we have used (21) as well as

$$\begin{aligned} \frac{2(k+1-q)}{\tau \varepsilon }-\frac{2}{\varepsilon }+\frac{1}{\tau \varepsilon } =\frac{2k+3-2\tau -2q}{\tau \varepsilon }\le 0 \qquad \text{ for }\quad \tau \ge k+3/2. \end{aligned}$$

Now taking sum over all intervals from the fine part of the mesh we get

$$\begin{aligned} |E-E^I|^2_{H^q(\Omega _f)}&= \sum _{i=N/2+1}^N|E-E^I|^2_{H^q(I_i)} \nonumber \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)}\int _{1/2}^{1}\psi '(t)\,\mathrm{d}t \nonumber \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)} \end{aligned}$$

due to

$$\begin{aligned} \int _{1/2}^{1}\psi '(t)\,\mathrm{d}t=\psi (1)-\psi (1/2)=1-N^{-1}. \end{aligned}$$

Thus, we have proved (28).

Using similar arguments we derive

$$\begin{aligned} \Vert (E-E^I)^{(q)}\Vert _{L^\infty (\Omega _f)}&\le C\max _{i=N/2+1,\ldots ,N}h_i^{k+1-q}\Vert E^{(k+1)}\Vert _{L^\infty (I_i)} \nonumber \\&\le C\varepsilon ^{-q+1}(N^{-1}\max |\psi '|)^{k+1-q}\max _{i=N/2+1,\ldots ,N} \mathrm{e}^{\frac{(k+1-q)(1-x_{i-1})}{\tau \varepsilon }}\mathrm{e}^{-\frac{1-x_i}{\varepsilon }} \nonumber \\&\le C\varepsilon ^{-q+1}(N^{-1}\max |\psi '|)^{k+1-q}, \end{aligned}$$

since

$$\begin{aligned}&{\exp \left( \frac{(k+1-q)(1-x_{i-1})}{\tau \varepsilon }-\frac{1-x_i}{\varepsilon }\right) } \\&\qquad = \exp \left( \left( \frac{k+1-q}{\tau \varepsilon }-\frac{1}{\varepsilon }\right) (1-x_{i-1})+\frac{h_i}{\varepsilon }\right) \\&\qquad \le C\exp \left( \frac{k+1-\tau -q}{\tau \varepsilon }(1-x_{i-1})\right) \le C. \end{aligned}$$

Hence, the inequality (29) is also proven.