How to avoid order reduction when Lawson methods integrate nonlinear initial boundary value problems

Abstract

It is well known that Lawson methods suffer from a severe order reduction when integrating initial boundary value problems where the solutions are not periodic in space or do not satisfy enough conditions of annihilation on the boundary. However, in a previous paper, a modification of Lawson quadrature rules has been suggested so that no order reduction turns up when integrating linear problems subject to time-dependent boundary conditions. In this paper, we describe and thoroughly analyse a technique to avoid also order reduction when integrating nonlinear problems. This is very useful because, given any Runge–Kutta method of any classical order, a Lawson method can be constructed associated to it for which the order is conserved.

Appendix

1.1 Proof of Theorem 3.1

Using Lemma 3.1 in [3],

$$\begin{aligned} {\bar{v}}_n(s)= & {} u(t_n)+s \varphi _1(s A_0) A u(t_n), \nonumber \\ \bar{{\tilde{v}}}(s)= & {} u(t_n)+s A u(t_n)+s^2 \varphi _2(s A_0) A^2 u(t_n), \nonumber \\ {\bar{w}}_{n,i}(s)= & {} e^{s A_0} f(t_n+c_i k, {\bar{K}}_{n,i}), \nonumber \\ \bar{{\tilde{w}}}_{n,i}(s)= & {} e^{s A_0}[f(t_n+c_i k, {\bar{K}}_{n,i})-f(t_n,u(t_n))]+f(t_n,u(t_n)) \nonumber \\&+\,s \varphi _1(s A_0) A f(t_n,u(t_n)). \end{aligned}$$

(7.1)

Then,

$$\begin{aligned} {\bar{K}}_{n,i}= & {} u(t_n)+ c_i k \varphi _1(c_i k A_0) A u(t_n)+k\sum _{j=1}^{i-1} a_{ij} e^{(c_i-c_j)kA_0} f(t_n+c_j k, {\bar{K}}_{n,j})\nonumber \\ \end{aligned}$$

(7.2)

$$\begin{aligned}= & {} u(t_n)+O(k), \qquad i=1,\dots ,s, \end{aligned}$$

(7.3)

$$\begin{aligned} {\bar{u}}_{n+1}= & {} u(t_n)+k A u(t_n)+ k^2 \varphi _2(k A_0) A^2 u(t_n) \nonumber \\&+\,k \sum _{i=1}^s b_i \bigg [e^{(1-c_i)k A_0}[f(t_n+c_i k,{\bar{K}}_{n,i})-f(t_n,u(t_n))]+f(t_n,u(t_n))\nonumber \\&+\,(1-c_i)k \varphi _1((1-c_i)k A_0) A f(t_n,u(t_n))\bigg ] \nonumber \\= & {} u(t_n)+k[Au(t_n)+f(t_n,u(t_n))]+O(k^2)= u(t_n)+k {\dot{u}}(t_n)+O(k^2),\nonumber \\ \end{aligned}$$

(7.4)

where, for the last line, (3.9), (A4) together with (7.3) and the first condition of (2.11) have been used. From this, the first result on the local error follows.

As for the second result, looking at the term in $k^2$ in ${\bar{u}}_{n+1}$ and using that $u\in C^3([0,T],X)$, we can notice that

$$\begin{aligned}&{A_0^{-1} \rho _{n+1}= k^3 (k A_0)^{-1}\left( \varphi _2(k A_0)-\frac{1}{2} I\right) A^2 u(t_n)+\frac{k^2}{2} A_0^{-1} A^2 u(t_n)} \nonumber \\&\quad +\,k^2 \sum _{i=1}^s b_i(1-c_i) ((1-c_i)k A_0)^{-1}[e^{(1-c_i)k A_0}-I][f(t_n+c_i k,{\bar{K}}_{n,i})-f(t_n,u(t_n))]\nonumber \\&\quad +\,k \sum _{i=1}^s b_i A_0^{-1} [f(t_n+c_i k,{\bar{K}}_{n,i})-f(t_n,u(t_n))] \nonumber \\&\quad +\,k^3 \sum _{i=1}^s b_i(1-c_i)^2 ((1-c_i)k A_0)^{-1}(\varphi _1((1-c_i)k A_0)- I)A f(t_n,u(t_n)) \nonumber \\&\quad +\,k^2 \sum _{i=1}^s b_i(1-c_i)A_0^{-1} A f(t_n,u(t_n))-\frac{k^2}{2} \ddot{u}(t_n)+O(k^3). \end{aligned}$$

(7.5)

Using (2.7), (7.3), (A4) and (3.9), the first, third and fifth terms are $O(k^3)$. As for the fourth one, considering (7.2), it can be written as

$$\begin{aligned}&k^2 \sum _{i=1}^s b_i A_0^{-1} f_u(t_n,u(t_n))\left[ c_i \varphi _1(c_i k A_0) A u(t_n)+\sum _{j=1}^{i-1} a_{ij} e^{(c_i-c_j)k A_0} f(t_n+c_j k, {\bar{K}}_{n,j})\right] \\&\quad +k^2 \sum _{i=1}^s b_i c_i A_0^{-1} f_t(t_n,u(t_n))+O(k^3) \\&=k^3 \sum _{i=1}^s b_i A_0^{-1} f_u(t_n,u(t_n))A_0 c_i (k A_0)^{-1}[\varphi _1(c_i k A_0) -I] A u(t_n) \\&\quad +k^2 \left( \sum _{i=1}^s b_i c_i\right) A_0^{-1} f_u(t_n,u(t_n))A u(t_n) \\&\quad +k^3 \sum _{i=1}^s b_i\sum _{j=1}^{i-1} a_{ij} A_0^{-1} f_u(t_n,u(t_n))A_0 (k A_0)^{-1}[e^{(c_i-c_j)k A_0}-I] f(t_n, u(t_n)) \\&\quad +k^2 \sum _{i=1}^s b_i \sum _{j=1}^{i-1} a_{ij} A_0^{-1} f_u(t_n,u(t_n))f(t_n,u(t_n)) \\&\quad +k^2 \sum _{i=1}^s b_i c_i A_0^{-1} f_t(t_n,u(t_n))+O(k^3) \\&=\frac{k^2}{2} A_0^{-1}[f_u(t_n,u(t_n)){\dot{u}}(t_n)+f_t(t_n,u(t_n))]+O(k^3), \end{aligned}$$

where, for the last equality, we have used (2.7) again, (3.10), the second condition in (2.11) and the fact that $\sum _{i=1}^s b_i c_i=1/2$ due to the second order of the Butcher tableau. Inserting this in (7.5) and simplifying notation,

$$\begin{aligned} A_0^{-1} \rho _{n+1}=\frac{k^2}{2}A_0^{-1} [A^2 u+f_u{\dot{u}}+f_t+A f-\ddot{u}]+O(k^3)=O(k^3), \end{aligned}$$

where the differentiation of (2.1) with respect to time shows that the term in bracket in the previous expression vanishes.

1.2 Proof of Theorem 3.3

Firstly notice that

$$\begin{aligned} e_{n+1,h}= & {} [U_{n+1,h}-{\bar{U}}_{n+1,h}]+\rho _{n+1,h}. \end{aligned}$$

(7.6)

Then, using (3.6), when considering Dirichlet boundary conditions, in which case $\partial Au(t_n)$ and $\partial f(t_n,u(t_n))$ are calculated exactly in terms of data, as ${\bar{U}}_{n+1,h}$ is the same as $U_{n+1,h}$ but starting from $P_h u(t_n)$ instead of $U_{n,h}$,

$$\begin{aligned}&{U_{n+1,h}-{\bar{U}}_{n+1,h}=e^{k A_{h,0}}[U_{n,h}-P_h u(t_n)]} \nonumber \\&\quad +\,k \sum _{i=1}^s b_i e^{(1-c_i)k A_{h,0}}[f(t_n+c_i k, K_{n,h,i})-f(t_n+c_i k, {\bar{K}}_{n,h,i})], \end{aligned}$$

(7.7)

where, recursively, for $i=1,\dots ,s$,

$$\begin{aligned}&{K_{n+1,h,i}-{\bar{K}}_{n+1,h,i}=e^{c_i k A_{h,0}}[U_{n,h}-P_h u(t_n)]}\nonumber \\&\quad +\,k \sum _{j=1}^{i-1} a_{ij}e^{(c_i-c_j)k A_{h,0}}[f(t_n+c_j k, K_{n,h,j})-f(t_n+c_j k, {\bar{K}}_{n,h,j})]. \end{aligned}$$

(7.8)

In such a way, it is inductively proved that $K_{n+1,h,i}-{\bar{K}}_{n+1,h,i}=O(e_{n,h})$ and finally, using (7.7) and (7.6),

$$\begin{aligned} e_{n+1,h}=e^{k A_{h,0}} e_{n,h}+O(k e_{n,h})+\rho _{n+1,h}, \end{aligned}$$

(7.9)

from what the result follows from Theorem 3.2 by a summation-by-parts argument and a discrete Gronwall lemma in the same way than the proof of Theorem 22 in [4] for Strang method.

On the other hand, when considering Robin/Neumann boundary conditions, as, according to Remark 3.1, $\partial f(t_n,u(t_n))$ is just calculated approximately with an error which is $O(e_{n,h})$, using (3.6) again,

$$\begin{aligned}&{U_{n+1,h}-{\bar{U}}_{n+1,h}=e^{k A_{h,0}} e_{n,h} + k^2 \varphi _2( k A_{h,0})C_h O(e_{n,h})}\\&\quad +\,k \sum _{i=1}^{s} b_i \bigg [e^{(1-c_i)k A_{h,0}}[f(t_n+c_i k, K_{n,h,i})-f(t_n+c_i k, {\bar{K}}_{n,h,i})] \\&\quad + (1-c_i)k \varphi _1((1-c_i)k A_{h,0}) C_h O(e_{n,h})\bigg ], \end{aligned}$$

where $K_{n+1,h,i}-{\bar{K}}_{n+1,h,i}$ is the same as in (7.8) because $\partial u(t_n)$ is given exactly in terms of data with this type of boundary conditions. Then, using (2.7),

$$\begin{aligned}&{U_{n+1,h}-{\bar{U}}_{n+1,h}=e^{ k A_{h,0}} e_{n,h} + k [\varphi _1( k A_{h,0})-I] A_{h,0}^{-1} C_h O(e_{n,h})} \\&\quad +\,k \sum _{i=1}^{s} b_i \bigg [e^{(1-c_i)k A_{h,0}}O(e_{n,h}) + [e^{(1-c_i)k A_{h,0}}-I]A_{h,0}^{-1} C_h O(e_{n,h}) \bigg ]. \end{aligned}$$

Using now (H2c), it follows that $U_{n+1,h}-{\bar{U}}_{n+1,h}=e^{ k A_{h,0}} e_{n,h}+O(k e_{n,h})$, from what (7.9) applies again and the result follows in the same way as above.

1.3 Proof of Theorem 4.3

For the proof, as in Theorem 3.3, we must consider the decomposition (7.6) where ${\bar{U}}_{n+1,h}$ is calculated as $U_{n+1,h}$ but starting from $P_h u(t_n)$ and calculating the boundaries in (4.2) in an exact way. In contrast, according to Table 1, when considering $U_{n+1,h}$, the boundaries in (4.2) can just be calculated approximately.

More precisely, with Dirichlet boundary conditions, the terms on the boundary for the stages in (4.2) can be calculated exactly. However, when calculating $U_{n+1,h}$, $\partial A^2 u(t_n)$ and $\partial A f(t_n,u(t_n))$ can just be calculated except for $O(\nu _h+\frac{e_{n,h}}{h^\gamma })$. Because of this,

$$\begin{aligned} U_{n+1,h}-{\bar{U}}_{n+1,h}= & {} e^{k A_{h,0}} e_{n,h}+k^3 \varphi _3(k A_{h,0})C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }\right) \\&+\,k \sum _{i=1}^s b_i \bigg [ e^{(1-c_i) k A_{h,0}}[f(t_n+c_i k, K_{n,h,i})-f(t_n+c_i k, {\bar{K}}_{n,h,i})] \\&+\,(1-c_i)^2 k^2 \varphi _2((1-c_i)k A_{h,0})C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }\right) \bigg ], \end{aligned}$$

where $K_{n,h,i}-{\bar{K}}_{n,h,i}=O(e_{n,h})$ as in the proof of Theorem 3.3. Therefore, using (2.7) and (H2c),

$$\begin{aligned} U_{n+1,h}-{\bar{U}}_{n+1,h}= & {} e^{k A_{h,0}} e_{n,h}+k^2 \left( \varphi _2(k A_{h,0})-\frac{1}{2}I\right) O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }\right) +O(k e_{n,h}) \\&+\,k^2 \sum _{i=1}^s b_i(1-c_i) (\varphi _1(k A_{h,0})-I)O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }\right) , \end{aligned}$$

from what, using condition (4.4),

$$\begin{aligned} e_{n+1,h}=e^{k A_{h,0}}e_{n,h}+O(k e_{n,h})+O(k^2 \nu _h)+\rho _{n+1,h}. \end{aligned}$$

The classical argument of convergence and the first part of Theorem 4.2 leads then to the first result of this theorem for Dirichlet boundary conditions. For the second part, the second part of Theorem 4.2 must be used, apart from (3.13) and the additional regularity (4.5).

On the other hand, with Robin/Neumann boundary conditions, there is some error when approximating the boundaries for both the stages and the numerical solution. More precisely, using Table 1 and (4.2),

$$\begin{aligned} K_{n,h,i}-{\bar{K}}_{n,h,i}= & {} e^{c_i k A_{h,0}} e_{n,h} +c_i^2 k^2 \varphi _2(c_i k A_{h,0}) C_h O(e_{n,h}) \\&+\,k \sum _{j=1}^{i-1} a_{ij}[O(e_{n,h})+(c_i-c_j)k \varphi _1((c_i-c_j)k A_{h,0}) C_h O(e_{n,h})] \\= & {} e^{c_i k A_{h,0}} e_{n,h}+c_i k [\varphi _1(c_i k A_{h,0})-I] O(e_{n,h})\\&+\,k \sum _{j=1}^{i-1} a_{ij}\bigg [O(e_{n,h})+ [e^{(c_i-c_j)k A_{h,0}}-I] O(e_{n,h})\bigg ] \\= & {} e^{c_i k A_{h,0}} e_{n,h}+O(k e_{n,h})=O(e_{n,h}), \end{aligned}$$

and then

$$\begin{aligned} U_{n+1,h}-{\bar{U}}_{n+1,h}= & {} e^{k A_{h,0}}e_{n,h}+k^3 \varphi _3(k A_{h,0})C_h O\left( \mu _{k,1}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \\&+\,k \sum _{i=1}^s b_i \bigg [ e^{(1-c_i)k A_{h,0}}[f(t_n+c_i k,K_{n,h,i})-f(t_n+c_i k,{\bar{K}}_{n,h,i})] \\&+\,(1-c_i)k \varphi _1((1-c_i)k A_{h,0})C_h O(k \mu _{k,1}+e_{n,h}) \\&+\,(1-c_i)^2 k^2 \varphi _2((1-c_i)k A_{h,0})C_h O\left( \mu _{k,1}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \bigg ] \\= & {} e^{k A_{h,0}}e_{n,h}+k^2 \left[ \varphi _2(k A_{h,0})-\frac{1}{2}I\right] O\left( \mu _{k,1}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \\&+\,k \sum _{i=1}^s b_i \bigg [ O(e_{n,h})+[e^{(1-c_i)k A_{h,0}}-I]O(k \mu _{k,1}+e_{n,h}) \\&+\,(1-c_i)k [\varphi _1(k A_{h,0})-I]O\left( \mu _{k,1}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \bigg ]. \end{aligned}$$

From this, under condition (4.4),

$$\begin{aligned} e_{n+1,h}=e^{k A_{h,0}}e_{n,h}+O(k e_{n,h}+k^2 \mu _{k,1}+k^2 \nu _h)+\rho _{n+1,h}, \end{aligned}$$

so that, using the first part of Theorem 4.2 and the classical argument of convergence, $e_{n,h}=O(k^2+\varepsilon _h+k \nu _h+k \mu _{k,1})$. Again, under the second set of hypotheses in Theorem 4.2 and using (3.13) and the regularity (4.5), the finer result $e_{n,h}=O(k^3+k \varepsilon _h+\eta _h+k \mu _{k,1}+k \nu _h)$ can be achieved.

1.4 Proof of Theorem 5.3

As in the proof of Theorem 4.3, we must consider the decomposition (7.6) and then study the difference $U_{h,n+1}-{\bar{U}}_{n+1}$ taking into account that the boundaries for $U_{h,n+1}$ in (5.1) are just calculated approximately with an error which is given through Table 1.

More precisely, with Dirichlet boundary conditions,

$$\begin{aligned} U_{n+1,h}-{\bar{U}}_{n+1,h}= & {} e^{k A_{h,0}} e_{n,h}+k^3 \varphi _3(k A_{h,0})C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }\right) \nonumber \\&+\,k^4 \varphi _4(k A_{h,0})C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma k}+\frac{\mu _{k,1}}{h^\gamma }\right) \nonumber \\&+\,k \sum _{i=1}^s b_i \left[ e^{(1-c_i) k A_{h,0}}[f(t_n+c_i k, K_{n,h,i})-f(t_n+c_i k, {\bar{K}}_{n,h,i})] \right. \nonumber \\&+\,(1-c_i) k \varphi _1((1-c_i)k A_{h,0})C_h O\left( k^2 \nu _h+k^2 \frac{e_{n,h}}{h^\gamma }\right) \nonumber \\&+\,(1-c_i)^2 k^2 \varphi _2((1-c_i)k A_{h,0})C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }+ \frac{k\mu _{k,1}}{h^\gamma }\right) \nonumber \\&\left. +(1-c_i)^3 k^3 \varphi _3((1-c_i)k A_{h,0})C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma k}+\frac{\mu _{k,1}}{h^\gamma }\right) \right] , \end{aligned}$$

(7.10)

where

$$\begin{aligned} K_{n,h,i}-{\bar{K}}_{n,h,i}= & {} e^{c_i k A_{h,0}}e_{n,h}+c_i^3 k^3 \varphi _3(k A_{h,0}) C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }\right) \\&+\,k \sum _{j=1}^{i-1} a_{ij}\bigg [ e^{(c_i-c_j)k A_{h,0}}[f(t_n+c_j k,K_{n,h,j})-f(t_n+c_j k,{\bar{K}}_{n,h,j})] \\&+\,(c_i-c_j)^2 k^2 \varphi _2((c_i-c_j)k A_{h,0}) C_h O\left( \nu _h+\frac{e_{n,h}}{h^\gamma }\right) \bigg ] \nonumber \\= & {} O(e_{n,h}+k^2 \nu _h), \end{aligned}$$

and, for the last equality, (2.7), (H2c) and (4.4) have been used. Inserting this in (7.10) and using again (2.7), (H2c) and (4.4), it follows that

$$\begin{aligned} U_{n+1,h}-{\bar{U}}_{n+1,h}= e^{k A_{h,0}} e_{n,h}+ O(k^2 \nu _h+k e_{n,h}+k^2 \mu _{k,1}). \end{aligned}$$

From here,

$$\begin{aligned} e_{n+1,h}=e^{k A_{h,0}}e_{n,h}+O(k e_{n,h})+O(k^2 \nu _h+k^2 \mu _{k,1})+\rho _{n+1,h}, \end{aligned}$$

and using a discrete Gronwall Lemma and the first part of Theorem 5.2, the first part of the theorem follows for Dirichlet boundary conditions. For the second part, the second part of Theorem 5.2 must be used, apart from (3.13) and the additional regularity (5.7).

As for Robin/Neumann boundary conditions, with similar arguments,

$$\begin{aligned}&{K_{n,h,i}-{\bar{K}}_{n,h,i}} \\&\quad =e^{c_i k A_{h,0}} e_{n,h}+c_i^2 k^2 \varphi _2(k A_{h,0}) \\&\qquad C_h O(e_{n,h})+c_i^3 k^3 \varphi _3(k A_{h,0}) C_h O\left( \mu _{k,1}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \\&\qquad +\,k \sum _{j=1}^{i-1} a_{ij}\bigg [e^{(c_i-c_j) k A_{h,0}} [f(t_n+c_j k, K_{n,h,j})-f(t_n+c_j k, {\bar{K}}_{n,h,j})] \\&\qquad +\,(c_i-c_j) k \varphi _1((c_i-c_j)k A_{h,0}) C_h O\left( e_{n,h}+k\left( \mu _{k,1}+\frac{e_{n,h}}{k}\right) \right) \\&\qquad +\,(c_i-c_j)^2 k^2 \varphi _2((c_i-c_j)k A_{h,0}) C_h O\left( \mu _{k,1}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \bigg ] \\&\quad = e^{c_i k A_{h,0}} e_{n,h}+O(k e_{n,h}+k^2 \mu _{k,1}+k^2 \nu _h)=O(e_{n,h}+k^2 \mu _{k,1}+k^2 \nu _h), \end{aligned}$$

from what

$$\begin{aligned}&{U_{n+1,h}-{\bar{U}}_{n+1,h}} \\&\quad = e^{k A_{h,0}}e_{n,h}+ k^2 \varphi _2(k A_{h,0})C_h O(e_{n,h})+ k^3 \varphi _3(k A_{h,0})C_h O\left( \mu _{k,1}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \\&\qquad +\,k^4 \varphi _4(k A_{h,0})C_h O\left( \mu _{k,1}+\mu _{k,2}+\frac{e_{n,h}}{k^2}+\nu _h+\frac{e_{n,h}}{k h^\gamma }\right) \\&\qquad +\,k \sum _{i=1}^s b_i \bigg [ e^{(1-c_i)k A_{h,0}}[f(t_n+c_i k,K_{n,h,i})-f(t_n+c_i k,{\bar{K}}_{n,h,i})] \\&\qquad +\,(1-c_i)k \varphi _1((1-c_i)k A_{h,0})C_h O\left( k e_{n,h}+k^2 \mu _{k,1}+k^2 \mu _{k,2}+k^2 \nu _h+\frac{k^2}{h^\gamma }e_{n,h}\right) \\&\qquad +\,(1-c_i)^2 k^2 \varphi _2((1-c_i)k A_{h,0})C_h O\left( k \mu _{k,1}+k \mu _{k,2}+\frac{e_{n,h}}{k}+\nu _h+\frac{e_{n,h}}{h^\gamma }\right) \\&\qquad +(1-c_i)^3 k^3 \varphi _3((1-c_i)k A_{h,0})C_h O\left( \mu _{k,1}+ \mu _{k,2}+\frac{e_{n,h}}{k^2}+\nu _h+\frac{e_{n,h}}{k h^\gamma }\right) \bigg ] \\&\quad = e^{k A_{h,0}}e_{n,h}+O(k e_{n,h}+k^2 \mu _{k,1}+k^3 \mu _{k,2}+k^2 \nu _h). \end{aligned}$$

From this,

$$\begin{aligned} e_{n+1,h}=e^{k A_{h,0}}e_{n,h}+O(k e_{n,h}+k^2 \mu _{k,1}+k^3 \mu _{k,2}+k^2 \nu _h)+\rho _{n+1,h}, \end{aligned}$$

so that, using the first part of Theorem 5.2 and the classical argument of convergence, $e_{n,h}=O(k^3+\varepsilon _h+k \nu _h+k \mu _{k,1}+k^2 \mu _{k,2})$. Again, under the second set of hypotheses in Theorem 5.2 and using (3.13) and the regularity (5.7), the finer result $e_{n,h}=O(k^4+k \varepsilon _h+\eta _h+k \mu _{k,1}+k^2 \mu _{k,2}+k \nu _h)$ is achieved.