In order to remedy the suboptimal error estimates that would result from using (
2.2) and (
2.5) directly, we follow the approach proposed in [
2] by introducing the
reconstruction \({\widehat{U}}_m\) of
\(U |_{I_{m}}\) which is defined over each closed interval
\(\bar{I}_m\),
\(1 \le m \le M\), by
$$\begin{aligned} \begin{aligned} {\widehat{U}}_m(t) :=\pi _m U(t^-_{m-1}) + \int _{t_{m-1}}^t \Pi _m^{r_m} \mathcal F(s,U) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.2)
This formulation of the reconstruction
\({\widehat{U}}_m\) is the same as that introduced in [
2] for the cG time stepping scheme, and in [
17] for the dG time stepping scheme, but with the initial value projected onto the space
\(H_m\) for convenience; it was also used in [
1,
16] for the Crank-Nicolson method (i.e., the cG method with
\(r_m=1\)). For
\(t \in I_m\),
\(1 \le m \le M\), we define the
error \(e(t) := u(t) - U(t) \in H\) where
U is either the
hp-cG solution from (
2.1) or the
hp-dG solution from (
2.3). Since we will be dealing with the reconstruction
\({\widehat{U}}_m\), it will also be necessary to introduce the
reconstruction error given by
\(\widehat{e}_m(t) := u(t) -{\widehat{U}}_m(t) \in H\),
\(t \in \bar{I}_m\),
\(1 \le m \le M\). We will proceed with the error analysis by first proving an
\(L^{\infty }\)-error bound for
\(\widehat{e}_m\).
To formulate the error equation, we begin by substituting (
1.1) into the definition of
\(\widehat{e}_m\), viz.,
$$\begin{aligned} \begin{aligned} \widehat{e}_m(t) = u(t_{m-1}) - {\widehat{U}}_m(t) + \int _{t_{m-1}}^t \mathcal F(s,u) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.3)
Adding and subtracting additional terms yields
$$\begin{aligned} \begin{aligned} \widehat{e}_m(t) = \widehat{e}_m(t_{m-1}) + \mathcal {R}(t) + \int _{t_{m-1}}^t \mathcal F(s,u) -\mathcal F(s,{\widehat{U}}_m) \,\text {d}s, \end{aligned} \end{aligned}$$
(3.4)
where
\(\mathcal {R}\) denotes the
residual given by
$$\begin{aligned} \begin{aligned} \mathcal {R}(t) := {\widehat{U}}_{m}(t_{m-1})- {\widehat{U}}_m(t)+ \int _{t_{m-1}}^t \mathcal F(s,{\widehat{U}}_m) \,\text {d}s, \qquad t \in {I}_m. \end{aligned} \end{aligned}$$
(3.5)
Using the triangle inequality and Bochner’s Theorem implies that
$$\begin{aligned} \begin{aligned} \Vert {\widehat{e}}_m(t)\Vert _H \le \Vert \widehat{e}_m(t_{m-1})\Vert _H+ \Vert \mathcal {R}(t)\Vert _H+\int _{t_{m-1}}^t \Vert \mathcal F(s,u) - \mathcal F(s,{\widehat{U}}_m)\Vert _H \,\text {d}s. \end{aligned} \end{aligned}$$
(3.6)
Moreover, applying the local
H-Lipschitz estimate (
3.1) together with the monotonicity of
\(\mathcal {L}\) yields
$$\begin{aligned} \begin{aligned} \Vert {\widehat{e}}_m(t)\Vert _H \le \Vert \widehat{e}_m(t_{m-1})\Vert _H+ \eta ^{{\mathtt {res}}}_m+\int _{t_{m-1}}^t \mathcal {L}(s,\Vert \widehat{e}_m\Vert _H+\Vert {\widehat{U}}_m\Vert _H,\Vert \widehat{U}_m\Vert _H) \Vert \widehat{e}_m\Vert _H \,\text {d}s, \end{aligned} \end{aligned}$$
(3.7)
for
\(t \in \bar{I}_m\). Here,
\(\displaystyle \eta ^{{\mathtt {res}}}_m\) denotes the
residual estimator given by
\(\displaystyle \eta ^{{\mathtt {res}}}_m := \Vert \mathcal {R}\Vert _{L^{\infty }(I_m;H)}\).
On the first interval, recalling that
\(U(t_0^-)=u_0=u(t_0)\), we can estimate
\(\Vert \widehat{e}_m(t_{m-1})\Vert _H\) directly by
\(\Vert \widehat{e}_1(t_0)\Vert _H = \eta _0^{{\mathtt {proj}}}\) where the
projection estimator \(\eta ^{{\mathtt {proj}}}_m\) is given by
$$\begin{aligned} \eta ^{{\mathtt {proj}}}_m := \left\| U(t_m^-) - \pi _{m+1}U(t_{m}^-)\right\| _H, \qquad m\ge 0. \end{aligned}$$
(3.8)
For later intervals, the unknown term
\(\Vert \widehat{e}_m(t_{m-1})\Vert _H\) needs to be replaced with the known term
\(\Vert \widehat{e}_{m-1}(t_{m-1})\Vert _H\). To this end, for
\(m \ge 1\), we have
$$\begin{aligned} \begin{aligned} \widehat{e}_{m+1}(t_{m}) - \widehat{e}_{m}(t_{m})&= {\widehat{U}}_{m}(t_{m}) - {\widehat{U}}_{m+1}(t_{m}) \\&= \pi _{m}U(t_{m-1}^-) - \pi _{m+1}U(t_{m}^-) + \int _{I_m} \Pi _{m}^{r_{m}} \mathcal F(s,U) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.9)
Recall that the
hp-cG method (
2.1) satisfies the strong form (
2.2) on
\(I_m\). Thus, we have that
$$\begin{aligned} \begin{aligned} {U}(t^-_{m}) =\pi _{m} U(t^-_{m-1}) + \int _{I_m} \Pi _{m}^{r_{m}-1} \mathcal F(s,U) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.10)
By the definition of the
\(L^2\)-projection an equivalent formulation of the above is
$$\begin{aligned} \begin{aligned} {U}(t^-_{m}) =\pi _{m} U(t^-_{m-1}) + \int _{I_m} \Pi _{m}^{r_{m}} \mathcal F(s,U) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.11)
Substituting this into (
3.9) implies that for the
hp-cG method we have
$$\begin{aligned} \begin{aligned} \widehat{e}_{m+1}(t_{m}) - \widehat{e}_{m}(t_{m}) = U(t_m^-) - \pi _{m+1}U(t_{m}^-). \end{aligned} \end{aligned}$$
(3.12)
For the
hp-dG method (
2.3), we have the strong form (
2.5) on
\(I_m\). Thus, it follows that
$$\begin{aligned} \begin{aligned} {U}(t^-_{m}) =U(t^+_{m-1}) + \int _{I_m} \Pi _{m}^{r_{m}} \mathcal F(s,U) \,\text {d}s - \int _{I_m} \left( \mathsf {L}^{r_m}_m\pi _m{[[U]]_{m-1}}\right) (s) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.13)
By definition of the lifting operator
\(\mathsf {L}^{r_m}_m\) from (
2.4) we arrive at
$$\begin{aligned} \begin{aligned} {U}(t^-_{m}) =U(t^+_{m-1})- \pi _m{[[U]]_{m-1}} + \int _{I_m} \Pi _{m}^{r_{m}} \mathcal F(s,U) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.14)
Equivalently,
$$\begin{aligned} \begin{aligned} {U}(t^-_{m}) = \pi _m U(t^-_{m-1}) + \int _{I_m} \Pi _{m}^{r_{m}} \mathcal F(s,U) \,\text {d}s. \end{aligned} \end{aligned}$$
(3.15)
Since this is the same form as for the
hp-cG method then for the
hp-dG method we also have that
$$\begin{aligned} \begin{aligned} \widehat{e}_{m+1}(t_{m}) - \widehat{e}_{m}(t_{m}) = U(t_m^-) - \pi _{m+1}U(t_{m}^-). \end{aligned} \end{aligned}$$
(3.16)
Applying the triangle inequality to (
3.12) and (
3.16) yields
$$\begin{aligned} \begin{aligned} \Vert \widehat{e}_{m+1}(t_{m})\Vert _H \le \Vert \widehat{e}_{m}(t_{m})\Vert _H + \eta ^{{\mathtt {proj}}}_{m}, \end{aligned} \end{aligned}$$
(3.17)
with
\(\eta ^{{\mathtt {proj}}}_m\) from (
3.8). Substituting this result into (
3.7) gives
$$\begin{aligned} \begin{aligned} \Vert {\widehat{e}}_m(t)\Vert _H \le \psi _m+\int _{t_{m-1}}^t \mathcal {L}\left( s,\Vert \widehat{e}_m\Vert _H+\Vert {\widehat{U}}_m\Vert _H,\Vert \widehat{U}_m\Vert _H\right) \Vert \widehat{e}_m\Vert _H \,\text {d}s, \end{aligned} \end{aligned}$$
(3.18)
where
\(\psi _m\) is chosen such that
$$\begin{aligned} \psi _m \ge {\left\{ \begin{array}{ll}\displaystyle \eta ^{{\mathtt {proj}}}_{m-1}+ \eta ^{{\mathtt {res}}}_m \qquad &{} \text {if } m = 1 \\ \displaystyle \Vert \widehat{e}_{m-1}(t_{m-1})\Vert _H + \eta ^{{\mathtt {proj}}}_{m-1}+ \eta ^{{\mathtt {res}}}_m \qquad &{} \text {if } m \ne 1 \end{array}\right. }. \end{aligned}$$
(3.19)
Finally, applying Gronwall’s inequality to (
3.18) for
\(t \in \bar{I}_m\),
\(1 \le m \le M\), yields the following result.