For error analysis and in experiments, we consider ${p}_{k}=p$ for all $k\in \mathcal{J}$. Thus we denote ${S}_{hp}={S}_{h\mathsf{p}}$. Now we would like to analyze the error estimates of the approximate solution ${u}_{h}^{l}$, $l=0,1,\dots $ , obtained by method (33). For simplicity, we consider a uniform partition ${t}_{l}=l\tau $, $l=0,1,\dots ,M$, of the time interval $[0,T]$ with time step $\tau =T/M$, where $M>1$ is an integer.

Let

${\mathrm{\Pi}}_{h}{u}^{l}$ be the standard

${S}_{hp}$-interpolation of

${u}^{l}=u({t}_{l})$, (

$l=0,\dots ,M$) satisfying (

*cf.* [

18]) for all

$v\in {H}^{p+1}({I}_{k})$,

${I}_{k}\in {\mathcal{T}}_{h}$,

${\parallel {\mathrm{\Pi}}_{h}v-v\parallel}_{{L}^{2}({I}_{k})}\le \tilde{c}{h}_{k}^{p+1}{|v|}_{{H}^{p+1}({I}_{k})},$

(38)

${|{\mathrm{\Pi}}_{h}v-v|}_{{H}^{1}({I}_{k})}\le \tilde{c}{h}_{k}^{p}{|v|}_{{H}^{p+1}({I}_{k})},$

(39)

for a generic constant

$\tilde{c}>0$ independent of

*h* and

*v*. We set

${\xi}_{h}^{l}={u}_{h}^{l}-{\mathrm{\Pi}}_{h}{u}^{l}\in {S}_{hp},\phantom{\rule{2em}{0ex}}{\eta}_{h}^{l}={\mathrm{\Pi}}_{h}{u}^{l}-{u}^{l}\in {H}^{p+1}(\mathrm{\Omega},{\mathcal{T}}_{h}).$

(40)

Then the error

${e}_{h}^{l}={u}_{h}^{l}-{u}^{l}$ can be expressed as

${e}_{h}^{l}={\xi}_{h}^{l}+{\eta}_{h}^{l},\phantom{\rule{1em}{0ex}}l=0,1,\dots ,M.$

(41)

Setting

${\xi}_{h}^{l+1}$ in (33), we get

${\mathcal{A}}_{h}^{\mu}({u}_{h}^{l+1}-{u}_{h}^{l},{\xi}_{h}^{l+1})+\tau ({b}_{hL}^{\epsilon}({u}_{h}^{l},{u}_{h}^{l+1},{\xi}_{h}^{l+1})-{b}_{hN}^{\epsilon}({u}_{h}^{l},{\xi}_{h}^{l+1}))=0.$

(42)

On the other hand, from (18) it follows

${\mathcal{A}}_{h}^{\mu}(\frac{\partial u}{\partial t}({t}_{l+1}),{\xi}_{h}^{l+1})+{b}_{h}^{\epsilon}(u({t}_{l+1}),{\xi}_{h}^{l+1})=0.$

(43)

Multiplying (43) by

*τ*, subtracting from (42) and using again the linearity of the form

${\mathcal{A}}_{h}^{\mu}$, we get

$\begin{array}{r}{\mathcal{A}}_{h}^{\mu}({u}_{h}^{l+1}-{u}_{h}^{l}-\tau \frac{\partial u}{\partial t}({t}_{l+1}),{\xi}_{h}^{l+1})\\ \phantom{\rule{1em}{0ex}}+\tau ({b}_{hL}^{\epsilon}({u}_{h}^{l},{u}_{h}^{l+1},{\xi}_{h}^{l+1})-{b}_{hN}^{\epsilon}({u}_{h}^{l},{\xi}_{h}^{l+1})-{b}_{h}^{\epsilon}(u({t}_{l+1}),{\xi}_{h}^{l+1}))=0.\end{array}$

(44)

Since

${u}_{h}^{l+1}-{u}_{h}^{l}={\xi}_{h}^{l+1}+{\eta}_{h}^{l+1}+{u}^{l+1}-{u}^{l}-{\xi}_{h}^{l}-{\eta}_{h}^{l}$, we can rewrite equation (

44) in the following way:

$\begin{array}{r}{\mathcal{A}}_{h}^{\mu}({\xi}_{h}^{l+1}-{\xi}_{h}^{l},{\xi}_{h}^{l+1})\\ \phantom{\rule{1em}{0ex}}=-{\mathcal{A}}_{h}^{\mu}({u}^{l+1}-{u}^{l}-\tau \frac{\partial u}{\partial t}({t}_{l+1}),{\xi}_{h}^{l+1})-{\mathcal{A}}_{h}^{\mu}({\eta}_{h}^{l+1}-{\eta}_{h}^{l},{\xi}_{h}^{l+1})\\ \phantom{\rule{2em}{0ex}}+\tau ({b}_{hL}^{\epsilon}({u}_{h}^{l},{u}_{h}^{l+1},{\xi}_{h}^{l+1})-{b}_{hN}^{\epsilon}({u}_{h}^{l},{\xi}_{h}^{l+1})-{b}_{h}^{\epsilon}(u({t}_{l+1}),{\xi}_{h}^{l+1})).\end{array}$

(45)

For the term on the right-hand side of equation (

45), we use decomposition

$(m-n)m=\frac{1}{2}({m}^{2}-{n}^{2}+{(m-n)}^{2})$ and linearity of the form

${\mathcal{A}}_{h}^{\mu}$. Together with (30), we finally get

$\begin{array}{r}\frac{1}{2}\{{\mathcal{A}}_{h}^{\mu}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})-{\mathcal{A}}_{h}^{\mu}({\xi}_{h}^{l},{\xi}_{h}^{l})+{\mathcal{A}}_{h}^{\mu}({\xi}_{h}^{l+1}-{\xi}_{h}^{l},{\xi}_{h}^{l+1}-{\xi}_{h}^{l})\}\\ \phantom{\rule{1em}{0ex}}=-{\mathcal{A}}_{h}^{\mu}({u}^{l+1}-{u}^{l}-\tau \frac{\partial u}{\partial t}({t}_{l+1}),{\xi}_{h}^{l+1})-{\mathcal{A}}_{h}^{\mu}({\eta}_{h}^{l+1}-{\eta}_{h}^{l},{\xi}_{h}^{l+1})\\ \phantom{\rule{2em}{0ex}}+\tau \{{b}_{hL}^{\epsilon}({u}_{h}^{l},{u}_{h}^{l+1},{\xi}_{h}^{l+1})-{b}_{hL}^{\epsilon}({u}_{h}^{l},u({t}_{l+1}),{\xi}_{h}^{l+1})\\ \phantom{\rule{2em}{0ex}}+{b}_{hL}^{\epsilon}({u}_{h}^{l},u({t}_{l+1}),{\xi}_{h}^{l+1})-{b}_{hL}^{\epsilon}(u({t}_{l+1}),u({t}_{l+1}),{\xi}_{h}^{l+1})\\ \phantom{\rule{2em}{0ex}}+{b}_{hN}^{\epsilon}(u({t}_{l+1}),{\xi}_{h}^{l+1})-{b}_{hN}^{\epsilon}({u}_{h}^{l},{\xi}_{h}^{l+1})\}.\end{array}$

(46)

For next estimates, we use the following lemmas.

**Lemma 2** *Under assumptions* (R)

*for* ${t}_{l},{t}_{l+1}\in [0,T]$,

*the following hold*:

$\left|({u}^{l+1}-{u}^{l}-\tau \frac{\partial u}{\partial t}({t}_{l+1}),{\xi}_{h}^{l+1})\right|\le c{\tau}^{2}{\parallel {\xi}_{h}^{l+1}\parallel}_{2},$

(47)

$\left|({\eta}_{h}^{l+1}-{\eta}_{h}^{l},{\xi}_{h}^{l+1})\right|\le c\tau {h}^{2(p+1)}{\parallel {\xi}_{h}^{l+1}\parallel}_{2},$

(48)

$\left|{a}_{h}({u}^{l+1}-{u}^{l}-\tau \frac{\partial u}{\partial t}({t}_{l+1}),{\xi}_{h}^{l+1})\right|\le c{\tau}^{2}\left(\right|{\xi}_{h}^{l+1}{|}_{1,2}+{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2}),$

(49)

$\left|{a}_{h}({\eta}_{h}^{l+1}-{\eta}_{h}^{l},{\xi}_{h}^{l+1})\right|\le c\tau {h}^{2(p+1)}\left(\right|{\xi}_{h}^{l+1}{|}_{1,2}+{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2}),$

(50)

$\left|{J}_{h}^{\sigma}({u}^{l+1}-{u}^{l}-\tau \frac{\partial u}{\partial t}({t}_{l+1}),{\xi}_{h}^{l+1})\right|\le c{\tau}^{2}{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2},$

(51)

$\left|{J}_{h}^{\sigma}({\eta}_{h}^{l+1}-{\eta}_{h}^{l},{\xi}_{h}^{l+1})\right|\le c\tau {h}^{2(p+1)}{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2},$

(52)

*where* *c* *is a generic constant independent of* *h* *and* *τ*.

*Proof* The proof of these standard estimates can be found, for instance, in [15]. □

**Lemma 3** *Under assumptions* (R), (H)

*and for* ${t}_{l},{t}_{l+1}\in [0,T]$,

*the following hold*:

$\begin{array}{r}\left|{b}_{hL}^{\epsilon}({u}_{h}^{l},{u}_{h}^{l+1}-u({t}_{l+1}),{\xi}_{h}^{l+1})\right|\\ \phantom{\rule{1em}{0ex}}\le c\left(\right|{\xi}_{h}^{l+1}{|}_{1,2}+{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2})({h}^{p+1}+{\parallel {\xi}_{h}^{l+1}\parallel}_{2}),\end{array}$

(53)

$\begin{array}{r}|{b}_{hL}^{\epsilon}({u}_{h}^{l},u({t}_{l+1}),{\xi}_{h}^{l+1})-{b}_{hL}^{\epsilon}(u({t}_{l+1}),u({t}_{l+1}),{\xi}_{h}^{l+1})|\\ \phantom{\rule{1em}{0ex}}\le c\left(\right|{\xi}_{h}^{l+1}{|}_{1,2}+{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2})({h}^{p+1}+{\parallel {\xi}_{h}^{l}\parallel}_{2}+\tau ),\end{array}$

(54)

$\begin{array}{r}|{b}_{hN}^{\epsilon}(u({t}_{l+1}),{\xi}_{h}^{l+1})-{b}_{hN}^{\epsilon}({u}_{h}^{l},{\xi}_{h}^{l+1})|\\ \phantom{\rule{1em}{0ex}}\le c\left(\right|{\xi}_{h}^{l+1}{|}_{1,2}+{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2})({h}^{p+1}+{\parallel {\xi}_{h}^{l}\parallel}_{2}+\tau ),\end{array}$

(55)

*where* *c* *is a generic constant independent of* *h* *and* *τ*.

*Proof* Again, one can find the proof of these estimates in [15]. □

Since

${\mathcal{A}}_{h}^{\mu}({\xi}_{h}^{l+1}-{\xi}_{h}^{l},{\xi}_{h}^{l+1}-{\xi}_{h}^{l})$ is always nonnegative, applying previous lemmas gives us

$\begin{array}{r}\frac{1}{2}{\mathcal{A}}_{h}^{\mu}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})-\frac{1}{2}{\mathcal{A}}_{h}^{\mu}({\xi}_{h}^{l},{\xi}_{h}^{l})\\ \phantom{\rule{1em}{0ex}}\le c\{\tau (\tau +{h}^{p+1}){\parallel {\xi}_{h}^{l+1}\parallel}_{2}+\mu \tau (\tau +{h}^{p})\left(\right|{\xi}_{h}^{l+1}{|}_{1,2}+{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2})\\ \phantom{\rule{2em}{0ex}}+\mu \tau (\tau +{h}^{p}){J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2}\\ \phantom{\rule{2em}{0ex}}+\tau \left(\right|{\xi}_{h}^{l+1}{|}_{1,2}+{J}_{h}^{\sigma}{({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})}^{1/2})({\parallel {\xi}_{h}^{l+1}\parallel}_{2}+{h}^{p+1}+{\parallel {\xi}_{h}^{l}\parallel}_{2}+\tau )\}.\end{array}$

(56)

Multiplying by 2, applying the Young inequality and using the definition of the form

${\mathcal{A}}_{h}^{\mu}$, we obtain

$\begin{array}{r}{\parallel {\xi}_{h}^{l+1}\parallel}_{2}^{2}+\mu \left(\right|{\xi}_{h}^{l+1}{|}_{1,2}^{2}+{J}_{h}^{\sigma}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1}))\\ \phantom{\rule{1em}{0ex}}\le {\parallel {\xi}_{h}^{l}\parallel}_{2}^{2}+\mu \left(\right|{\xi}_{h}^{l}{|}_{1,2}^{2}+{J}_{h}^{\sigma}({\xi}_{h}^{l},{\xi}_{h}^{l}))+2c\tau \{\frac{{\tau}^{2}}{2{\nu}_{1}}+\frac{{\nu}_{1}}{2}{\parallel {\xi}_{h}^{l+1}\parallel}_{2}^{2}+\frac{{h}^{2(p+1)}}{2{\nu}_{2}}+\frac{{\nu}_{2}}{2}{\parallel {\xi}_{h}^{l+1}\parallel}_{2}^{2}\\ \phantom{\rule{2em}{0ex}}+\mu (\frac{{\tau}^{2}}{{\nu}_{3}}+\frac{{h}^{2(p)}}{{\nu}_{4}}+\frac{{\nu}_{3}+{\nu}_{4}}{2}\left(\right|{\xi}_{h}^{l+1}{|}_{1,2}^{2}+{J}_{h}^{\sigma}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})))\\ \phantom{\rule{2em}{0ex}}+\mu (\frac{{\tau}^{2}}{2{\nu}_{5}}+\frac{{h}^{2(p)}}{2{\nu}_{6}}+\frac{{\nu}_{5}+{\nu}_{6}}{2}{J}_{h}^{\sigma}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1}))\\ \phantom{\rule{2em}{0ex}}+\frac{{\nu}_{7}+{\nu}_{8}+{\nu}_{9}+{\nu}_{10}}{2}\mu \left(\right|{\xi}_{h}^{l+1}{|}_{1,2}^{2}+{J}_{h}^{\sigma}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1}))\\ \phantom{\rule{2em}{0ex}}+\frac{1}{\mu {\nu}_{7}}{\parallel {\xi}_{h}^{l+1}\parallel}_{2}^{2}+\frac{{h}^{2(p+1)}}{\mu {\nu}_{8}}+\frac{1}{\mu {\nu}_{9}}{\parallel {\xi}_{h}^{l}\parallel}_{2}^{2}+\frac{{\tau}^{2}}{\mu {\nu}_{10}}\}.\end{array}$

(57)

If we take into account that

${J}_{h}^{\sigma}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})\le |{\xi}_{h}^{l+1}{|}_{1,2}^{2}+{J}_{h}^{\sigma}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1})$, the previous inequality can be rewritten as

$\begin{array}{r}(1-c\tau ({\nu}_{12}+\frac{2}{\mu {\nu}_{7}})){\parallel {\xi}_{h}^{l+1}\parallel}_{2}^{2}+(1-c\tau \sum _{i=3}^{10}{\nu}_{i})\mu \left(\right|{\xi}_{h}^{l+1}{|}_{1,2}^{2}+{J}_{h}^{\sigma}({\xi}_{h}^{l+1},{\xi}_{h}^{l+1}))\\ \phantom{\rule{1em}{0ex}}\le (1+\frac{2c\tau}{{\nu}_{9}\mu}){\parallel {\xi}_{h}^{l}\parallel}_{2}^{2}+\mu \left(\right|{\xi}_{h}^{l}{|}_{1,2}^{2}+{J}_{h}^{\sigma}({\xi}_{h}^{l},{\xi}_{h}^{l}))+c\tau q(\tau ,h,\mu ),\end{array}$

(58)

where we denoted

${\nu}_{12}={\nu}_{1}+{\nu}_{2}$ and

$\begin{array}{rcl}q(\tau ,h,\mu )& =& (\frac{1}{{\nu}_{1}}+\frac{2\mu}{{\nu}_{3}}+\frac{\mu}{{\nu}_{5}}+\frac{2}{\mu {\nu}_{10}}){\tau}^{2}+(\frac{2}{{\nu}_{4}}+\frac{1}{2{\nu}_{6}})\mu {h}^{2p}\\ +(\frac{1}{{\nu}_{2}}+\frac{2}{{\nu}_{8}\mu}){h}^{2(p+1)}.\end{array}$

(59)

Let us now introduce the so-called energy norm

$\u2980{v}_{h}\u2980=\sqrt{{|{v}_{h}|}_{1,2}^{2}+{J}_{h}^{\sigma}({v}_{h},{v}_{h})}$

(60)

and the norm

${\parallel {v}_{h}\parallel}_{\mu}=\sqrt{{\parallel {v}_{h}\parallel}_{2}^{2}+\mu \u2980{v}_{h}{\u2980}^{2}}.$

(61)

Denoting

${C}_{L}=c\cdot max\{{\nu}_{12}+\frac{2}{\mu {\nu}_{7}},{\sum}_{i=3}^{10}{\nu}_{i}\}$ and

${C}_{R}=\frac{2c\tau}{\mu {\nu}_{9}}$ from (58), it follows

$(1-\tau {C}_{L}){\parallel {\xi}_{h}^{l+1}\parallel}_{\mu}^{2}\le (1+{C}_{R}){\parallel {\xi}_{h}^{l}\parallel}_{\mu}^{2}+c\tau q(\tau ,h,\mu ).$

(62)

In order to finish our estimates, we require a fulfillment of the following technical assumption.

**Assumption (T)**

(T1) *There exists* $\theta \in (0,1)$ *such that* $0<\tau <\theta /{C}_{L}$.

If assumption (T) is fulfilled, then $\tau <\frac{1}{{C}_{L}}\le \frac{\mu}{\mu {\nu}_{12}+2/{\nu}_{7}}\le \frac{{\nu}_{7}}{2}\mu $. Thus we can also reformulate assumption (T) so that $\tau =\mathcal{O}(\mu )$.

Thus, let us assume that assumption (T) holds, then

${\parallel {\xi}_{h}^{l+1}\parallel}_{\mu}^{2}\le B{\parallel {\xi}_{h}^{l}\parallel}_{\mu}^{2}+\frac{c\tau}{1-\tau {C}_{L}}q(\tau ,h,\mu )$

(63)

with

$B=\frac{1+\tau {C}_{R}}{1-\tau {C}_{L}}=1+\tau \frac{{C}_{L}+{C}_{R}}{1-\tau {C}_{L}}\le exp(\tau \frac{{C}_{L}+{C}_{R}}{1-\tau {C}_{L}})$. Consequently,

${\parallel {\xi}_{h}^{l}\parallel}_{\mu}^{2}\le {B}^{l}{\parallel {\xi}_{h}^{0}\parallel}_{\mu}^{2}+\frac{{B}^{l}-1}{B-1}\frac{c\tau}{1-\tau {C}_{L}}q(\tau ,h,\mu )$

(64)

and since

$B-1=\tau \frac{{C}_{L}+{C}_{R}}{1-\tau {C}_{L}}$, we have

$\frac{c\tau}{(B-1)(1-\tau {C}_{L})}=\frac{c}{{C}_{L}+{C}_{R}}$,

*i.e.*,

$\begin{array}{rcl}{\parallel {\xi}_{h}^{l}\parallel}_{\mu}^{2}& \le & {B}^{l}{\parallel {\xi}_{h}^{0}\parallel}_{\mu}^{2}+\frac{({B}^{l}-1)c}{{C}_{L}+{C}_{R}}q(\tau ,h,\mu )\\ \le & exp\left(l\tau \frac{{C}_{L}+{C}_{R}}{1-\tau {C}_{L}}\right)({\parallel {\xi}_{h}^{0}\parallel}_{\mu}^{2}+\frac{c}{{C}_{L}+{C}_{R}}q(\tau ,h,\mu ))\\ \le & \tilde{C}exp\left(T\frac{{C}_{L}+{C}_{R}}{1-\tau {C}_{L}}\right)(\mu {h}^{2p}+{h}^{2(p+1)}+\frac{\mu {\nu}_{7}{\nu}_{9}}{2({\nu}_{7}+{\nu}_{9})}q(\tau ,h,\mu )),\end{array}$

(65)

where we used a straightforward estimate ${\parallel {\xi}_{h}^{0}\parallel}_{\mu}^{2}\le \tilde{C}(\mu {h}^{2p}+{h}^{2(p+1)})$. We can notice that due to the presence of the factor *μ* in front of the function $q(\tau ,h,\mu )$ on the left-hand side of (65), we lost *μ* in denominators of $q(\tau ,h,\mu )$.

Now we are ready to formulate the main theorem.

**Theorem 1** *Let assumptions* (M), (H), (R)

*and* (T)

*be satisfied*,

*then there exists a constant* $C=C(\mu )$ *such that* *where* $\u2980\cdot \u2980$ *is defined by* (60).

*Proof* Since

${\parallel {e}_{h}^{l}\parallel}_{\mu}\le {\parallel {\xi}_{h}^{l}\parallel}_{\mu}+{\parallel {\eta}_{h}^{l}\parallel}_{\mu}$, the statement of the theorem comes from (65) and the fact that

${\parallel {\eta}_{h}^{l}\parallel}_{\mu}\le \tilde{c}({h}^{p+1}+\mu {h}^{p})$. Then we set

$C(\mu )=\tilde{C}exp\left(T\frac{{C}_{L}+{C}_{R}}{1-\theta}\right)\frac{{\nu}_{7}{\nu}_{9}}{{\nu}_{7}+{\nu}_{9}}max\{\frac{1}{{\nu}_{j}},j\in \{1,2,\dots ,6,8,10\}\}.$

(68)

□

**Remark 3** Theorem 1 implies that the error of our method is $\mathcal{O}({h}^{p}+\tau )$ in both energy and ${L}^{2}$-norm. However, as we will see in the next section, the error estimate in the ${L}^{2}$-norm is suboptimal with respect to *h*.

**Remark 4** The dependency *C* on *μ* in the expression (68) (choice of *θ* depends on *μ*) can be removed by applying the so-called continuous mathematical induction mentioned in [19]. This is useful namely in the cases when convection terms dominate, *i.e.*, $\mu \to 0+$. Consequently, in these cases assumption (T) can be weakened to a CFL-like condition $\tau =\mathcal{O}({h}^{\alpha})$ for suitable $\alpha >0$.