A direct discontinuous Galerkin finite element method for convection-dominated two-point boundary value problems

Abstract

The direct discontinuous Galerkin (DDG) finite element method, using piecewise polynomials of degree k ≥ 1 on a Shishkin mesh, is applied to convection-dominated singularly perturbed two-point boundary value problems. Consistency, stability and convergence of order k (up to a logarithmic factor) are proved in an energy-type norm appropriate to the method and problem. The results are robust, i.e., they hold uniformly for all values of the singular perturbation parameter. Numerical experiments confirm the theoretical convergence rate.

Appendices

Appendix 1

In this Appendix, we collect some standard results from finite element analysis. The three lemmas that follow are valid under the hypotheses stated for each lemma (i.e., they are not specifically for Shishkin meshes).

Lemma 10 (inverse inequality)

[1, Lemma 4.5.3]. LetD ⊂ [0, 1] be partitioned by a quasiuniform mesh of diameter d and$\mathbb {V}$bea finite-dimensional subspace of${W^{l}_{p}}(D)\cap {W^{m}_{q}}(D)$, where 1 ≤ p ≤∞,1 ≤ q ≤∞and 0 ≤ l ≤ m. Then, there exists a constant C such that forall$v\in \mathbb {V}$, one has

$$ \begin{array}{@{}rcl@{}} \|v\|{~}_{m,p,D}\leq C d^{l-m+1/p-1/q} \|v\|{~}_{l,q,D}. \end{array} $$

Lemma 11 (Bramble-Hilbert lemma)

[3, Theorem 4.1.3] Suppose$D \subset \mathbb {R}$isan interval. For non-negative integer kand 0 ≤ q ≤∞, let L be a continuous linear functional on the Sobolev spaceW^{k+ 1, q}(D) with the property thatL(w) = 0 for all$w\in \mathbb {P}_{k}(D)$. Then, there is a constantC(D) such that

$$ \begin{array}{@{}rcl@{}} |Lv|\leq C(D)\|L\|{~}_{k+1,q,D}^{*}|v|{~}_{k+1,q,D}, \end{array} $$

where$\|L\|{~}_{k+1,q,D}^{*}$is the norm inthe dual space ofW^{k+ 1, q}(D).

Lemma 12

[3, Theorem 3.1.2] Let D and$\overline {D}$betwo subintervals of$\mathbb {R}$. Let$F:\overline {D} \to D$bean invertible affine mapping defined by$F(\overline {x})=B \overline {x}+b$with$D = F(\overline {D})$. Letv ∈ W^{m, p}(D) for some integerm ≥ 0 and somep ∈ [1, ∞]. Then, the function$\overline {v} = v\circ F \in W^{m,p}(\overline {D})$and$|\overline {v}|{~}_{m,p,\overline {D}}\lesssim |B|{~}^{m-1/p} |v|{~}_{m,p,D}$forallv ∈ W^{m, p}(D). Analogously, one has$|v|{~}_{m,p,D} \lesssim |B|{~}^{-m+1/p} |\overline {v}|{~}_{m,p,\overline {D}}$forall$\overline {v}\in W^{m,p}(\overline {D})$.

Appendix 2

1.1 Proof of Lemma 5

We prove this lemma for an arbitrary general mesh. The interval I_j is affine equivalent to ω := [− 1, 1] through the linear mapping

$$ \begin{array}{@{}rcl@{}} x=\frac{x_{j-1}+x_{j}}{2} + \frac{h_{j}}{2}\overline{x}\ \text{ for } \overline{x}\in\omega. \end{array} $$

For m = 0, 1,..., let L_m be the m^th Legendre polynomial on the interval [− 1, 1] (see [11, Chapter 3.3] for the properties of Legendre polynomials). Let L_{j, m} be L_m mapped to the interval I_j, i.e., $L_{j,m}(x)=L_{m}(\overline {x})$. Each function w ∈ L²(I_j) can be expanded as a sum of Legendre polynomials: $w={\sum }_{j=0}^{\infty }w_{j,m}L_{j,m}$, where

$$ \begin{array}{@{}rcl@{}} w_{j,m}:=\frac{2m-1}{2}{\int}_{x_{j-1}}^{x_{j}}w(x)L_{j,m}(x) dx \text{ for } \ m=0, 1,... \end{array} $$

Furthermore, the L²(Ω) projection defined in (17) can be expressed as

$$ \begin{array}{@{}rcl@{}} Pw(x)|{~}_{I_{j}}=\sum\limits_{m=0}^{k}w_{j,m}L_{j,m}(x) \text{ for } x\in I_{j} \text{ and }1\leq j\leq N. \end{array} $$

Then, the mapped projection is

$$ \begin{array}{@{}rcl@{}} \overline{P}\overline{w}(\overline{x})=\sum\limits_{m=0}^{k}w_{j,m}L_{m}(\overline{x}) \text{ for } \overline{x}\in \omega \text{ and }1\leq j\leq N. \end{array} $$

It follows that for any polynomial $v\in \mathbb {P}^{k}(\omega )$, one has $\overline {P}v=v$.

By Lemma 12, we have

$$ \begin{array}{@{}rcl@{}} |Pw-w|{~}_{m,p,I_{j}} &=&\left|\sum\limits_{j=k+1}^{\infty}w_{j,m}L_{j,m}(x)\right|{~}_{m,p,I_{j}}\\ &\lesssim& h_{j}^{-m+\frac{1}{p}}\left|\sum\limits_{j=k+1}^{\infty}w_{j,m}L_{m}(\overline{x})\right|{~}_{m,p,\omega}\\ &= &h_{j}^{-m+\frac{1}{p}}|(\overline{P}- I)\overline{w}|{~}_{m,p,\omega}. \end{array} $$

But Lemma 11 gives

$$ \begin{array}{@{}rcl@{}} |(\overline{P}- I)\overline{w}|{~}_{m,p,\omega} \lesssim C\|\overline{P}- I\|{~}_{k+1,q,\omega}^{*}|\overline{w}|{~}_{k+1,q,\omega}\ \text{ for }m\leq k+1, \end{array} $$

where $C\|\overline {P}- I\|{~}_{k+1,q,\omega }^{*}$ is a constant independent of $\overline {w}$. Using these results and again invoking Lemma 12, we get

$$ \begin{array}{@{}rcl@{}} |Pw-w|{~}_{m,p,I_{j}}\lesssim h_{j}^{k+1-m+1/p-1/q}|w|{~}_{k+1,q,\omega} . \end{array} $$

Hence, taking p = q = 2, we get (18b) for m ≤ k + 1; taking p = ∞ and q = 2, we get (18c) for m ≤ k + 1.

Finally, to prove (18a), by (18b), we have

$$ \begin{array}{@{}rcl@{}} \|Pw\|{~}_{m,I_{j}}&\lesssim& \sum\limits_{i=0}^{m}|Pw-w|{~}_{i,I_{j}}+ \|w\|{~}_{m,I_{j}}\\ &\lesssim& \sum\limits_{i=0}^{m}(h_{j})^{m-i}|w|{~}_{m,I_{j}} + \|w\|{~}_{m,I_{j}}\\ &\lesssim& \|w\|{~}_{m,I_{j}}. \end{array} $$

1.2 Proof of Lemma 6

The next inequality, which follows from (14), will be used frequently:

$$ \begin{array}{@{}rcl@{}} {\int}_{\tau}^{1}e^{-2\beta x/\varepsilon}dx=\frac{\varepsilon}{2\beta}\left( N^{-2\sigma}-e^{-2\beta/\varepsilon}\right)\lesssim \varepsilon N^{-2\sigma}. \end{array} $$

(22)

From (13), (16), (22), and Lemma 5, for ℓ ∈ {0, 1}, one has

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{N/2}\|PE-E\|{~}^{2}_{\ell,I_{j}}&\lesssim& \sum\limits_{j=1}^{N/2} h^{2k+2-2\ell}|E|{~}^{2}_{k+1,I_{j}}\\ &\lesssim& \left( \varepsilon N^{-1} \ln N\right)^{2k+2-2\ell} \varepsilon^{-2k-2} {\int}_{0}^{\tau} e^{-2\beta x/\varepsilon} dx\\ &\lesssim& \varepsilon^{1-2\ell} \left( N^{-1}\ln N\right)^{2k+2-2\ell}. \end{array} $$

(23)

By (18a) and (22), for ℓ ∈ {0, 1}, one has

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=N/2+1}^{N}\|PE\|{~}^{2}_{\ell,I_{j}} &\!\lesssim\!&\!\sum\limits_{j=N/2+1}^{N}\|E\|{~}^{2}_{\ell,I_{j}} \!\lesssim\! \varepsilon^{-2\ell}{\int}_{\tau}^{1}e^{-2\beta x/\varepsilon}dx \!\lesssim\! \varepsilon^{1-2\ell} N^{-2\sigma}. \end{array} $$

(24)

Using (13) and Lemma 5, for ℓ ∈ {0, 1}, we get

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{N/2}\|PS-S\|{~}_{\ell,I_{j}}^{2} &\lesssim& \sum\limits_{j=1}^{N/2} h^{2k+2-2\ell} |S|{~}_{k+1,I_{j}}^{2} \lesssim \left( \varepsilon N^{-1}\ln N\right)^{2k+2-2\ell} \end{array} $$

(25)

and

$$ \begin{array}{@{}rcl@{}}\sum\limits_{j=N/2+1}^{N}\|PS-S\|{~}_{\ell,I_{j}}^{2} &\lesssim& \sum\limits_{j=N/2+1}^{N} H^{2k+2-2\ell} |S|{~}_{k+1,I_{j}}^{2} \lesssim N^{-2k-2+2\ell}. \end{array} $$

(26)

Note that $\|Pu-u\|{~}_{\ell ,I_{j}}\lesssim \|PE-E\|{~}_{\ell ,I_{j}}+\|PS-S\|{~}_{\ell ,I_{j}}$. Then by (23)–(26), for ℓ ∈ {0, 1}, one has

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{N/2}\|Pu-u\|{~}_{\ell,I_{j}}^{2} &\lesssim &\varepsilon^{1-2\ell}\left( N^{-1}\ln N\right)^{2k+2-2\ell} +\left( \varepsilon N^{-1}\ln N\right)^{2k+2-2\ell}\\ &\lesssim& \varepsilon^{1-2\ell}\left( N^{-1}\ln N\right)^{2k+2-2\ell}\end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=N/2+1}^{N}|Pu-u|{~}_{\ell,I_{j}}^{2} &\lesssim \varepsilon^{1-2\ell} N^{-2\sigma}+N^{-2k-2+2\ell}. \end{array} $$

This completes the proof.

1.3 Proof of Lemma 7

For m ∈ {0, 1, 2}, Lemma 5 gives (cf. proof of (23) above)

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{N/2}h_{j}^{2m-1}\|PE-E\|{~}^{2}_{m,\infty,I_{j}} &\lesssim& \sum\limits_{j=1}^{N/2}h^{2k}|E|{~}^{2}_{k+1,I_{j}} \lesssim \varepsilon^{-1} \left( N^{-1}\ln N\right)^{2k},\\ \sum\limits_{j=1}^{N/2}h_{j}^{2m-1}\|PS-S\|{~}^{2}_{m,\infty,I_{j}} &\lesssim& \sum\limits_{j=1}^{N/2}h^{2k}|S|{~}^{2}_{k+1,I_{j}} \lesssim \left( \varepsilon N^{-1}\ln N\right)^{2k},\\ \sum\limits_{j=N/2+1}^{N}h_{j}^{2m-1}\|PS-S\|{~}^{2}_{m,\infty,I_{j}} &\lesssim& \sum\limits_{j=N/2+1}^{N}H^{2k}|S|{~}^{2}_{k+1,I_{j}} \lesssim N^{-2k}. \end{array} $$

We have now proved (20a), (20b), and (20c).

Next, by the inverse inequality of Lemma 10 and (24), one has

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=N/2+1}^{N} h_{j}^{2m-1}\|PE\|{~}^{2}_{m,\infty,I_{j}} &\lesssim \sum\limits_{j=N/2+1}^{N}H^{-2}\|PE\|{~}^{2}_{0,I_{j}}\lesssim \varepsilon N^{-2\sigma+2}. \end{array} $$

Finally, (13) yields

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=N/2+1}^{N} h_{j}^{2m-1}\|E\|{~}^{2}_{m,\infty,I_{j}} &\lesssim& H^{2m-2}\varepsilon^{-2m}\sum\limits_{j=N/2+1}^{N} He^{-2\beta x_{j-1}/\varepsilon}\\ &\lesssim& N^{2-2m}\varepsilon^{-2m}{\int}_{\tau}^{1}e^{-2\beta x/\varepsilon}dx\\ &\lesssim& \varepsilon^{1-2m} N^{2-2\sigma-2m}. \end{array} $$

This completes the proof.

1.4 Proof of Lemma 6

By Lemma 7 with m = 0, since σ = k + 1, we obtain

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=N/2+1}^{N} h_{j}^{-1}\|PE-E\|{~}_{0,\infty,I_{j}}^{2} &\lesssim& \sum\limits_{j=N/2+1}^{N}H^{-1}\left( \|PE\|{~}_{0,\infty,I_{j}}^{2}+\|E\|{~}_{0,\infty,I_{j}}^{2}\right)\\ &\lesssim& \varepsilon N^{-2k}. \end{array} $$

(27)

But

$$ \begin{array}{@{}rcl@{}} \|Pu-u\|{~}_{0,\infty,I_{j}}\lesssim \|PE-E\|{~}_{0,\infty,I_{j}}+\|PS-S\|{~}_{0,\infty,I_{j}}\ \text{ for all }j. \end{array} $$

(28)

Thus, Lemma 7 (with σ = k + 1) and (27) give

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{N/2}h_{j}^{-1}\|Pu-u\|{~}_{0,\infty,I_{j}}^{2}&\lesssim &\varepsilon^{-1}\left( N^{-1}\ln N\right)^{2k}+\left( \varepsilon N^{-1}\ln N\right)^{2k}\\ &\lesssim& \varepsilon^{-1}\left( N^{-1}\ln N\right)^{2k}\end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=N/2+1}^{N}h_{j}^{-1}\|Pu-u\|{~}_{0,\infty,I_{j}}^{2} &\lesssim \varepsilon^{} N^{-2k}+N^{-2k}\lesssim N^{-2k}. \end{array} $$

Hence,

$$ \begin{array}{@{}rcl@{}} &&{\kern-3.7pc}\sum\limits_{j=0}^{N}\frac{\beta_{0}}{h_{j}}[Pu-u]_{j}^{2} \\ &\lesssim& \sum\limits_{j=1}^{N/2}h^{-1}\|Pu-u\|{~}_{0,\infty,I_{j}}^{2} + h^{-1}\|Pu-u\|{~}_{0,\infty,I_{N/2+1}}^{2} \\ &\qquad& +\sum\limits_{j=N/2+1}^{N}H^{-1}\|Pu-u\|{~}_{0,\infty,I_{j}}^{2}\\ &\lesssim &\sum\limits_{j=1}^{N/2}h^{-1}\|Pu-u\|{~}_{0,\infty,I_{j}}^{2} + \frac{H}{h}\sum\limits_{j=N/2+1}^{N}H^{-1}\|Pu-u\|{~}_{0,\infty,I_{j}}^{2}\\ &\lesssim& \varepsilon^{-1}\left( N^{-1}\ln N\right)^{2k} +\left( \varepsilon\ln N\right)^{-1} \left( N^{-2k}\right)\\ &\lesssim& \varepsilon^{-1}\left( N^{-1}\ln N\right)^{2k}. \end{array} $$

This inequality and Lemma 6 (with σ = k + 1) yield finally

$$ \begin{array}{@{}rcl@{}} \|Pu-u\|{~}_{DG}^{2} &=& \varepsilon\left( |Pu-u|{~}_{1}^{2} +\sum\limits_{j=0}^{N}\frac{\beta_{0}}{h_{j}}[Pu-u]_{j}^{2}\right) +\|Pu-u\|{~}_{0}^{2}\\ &\lesssim& \varepsilon\left[\varepsilon^{-1}\left( N^{-1}\ln N\right)^{2k} + \varepsilon^{-1}N^{-2k-2} + N^{-2k}\right] \\ &\qquad& + \varepsilon \left( N^{-1}\ln N\right)^{2k+2}+N^{-2k-2} \\ &\lesssim& \left( N^{-1}\ln N\right)^{2k}. \end{array} $$

1.5 Proof of Lemma 9

The definition (5) gives $\widehat {(Pu-u)}|{~}_{i} = (Pu-u)^{\prime }(x_{i})$ for i ∈ {0, N} and

$$ \widehat{(Pu-u)}|{~}_{j} = \frac{\beta_{0}}{h_{j}}[Pu-u]_{j} +\{(Pu-u)^{\prime}\}_{j} +\beta_{1} h_{j}[(Pu-u)^{\prime\prime}]_{j} $$

for j = 1, 2,…, N − 1. Now, two Cauchy-Schwarz inequalities show that

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=0}^{N}&&|\widehat{(Pu-u)}_{j}[ v_{h} ]_{j}+[Pu-u]_{j}\{v_{h}^{\prime}\}_{j}| \\ &&\lesssim \left[\sum\limits_{j=0}^{N}\left( \frac{\beta_{0}}{h_{j}}[Pu-u]_{j}^{2}+h_{j}\{(Pu-u)^{\prime}\}_{j}^{2} +\frac{{\beta_{1}^{2}}}{\beta_{0}} {h_{j}^{3}}[(Pu-u)^{\prime\prime}]_{j}^{2}\right)\right]^{1/2} \\ &&\qquad\times \left\{ \left[\sum\limits_{j=0}^{N}\frac{\beta_{0}}{h_{j}}[ v_{h} ]_{j}^{2}\right]^{1/2} +\left[\sum\limits_{j=0}^{N}\frac{h_{j}}{\beta_{0}}\{v_{h}^{\prime}\}_{j}^{2} \right]^{1/2} \right\}. \end{array} $$

(29)

We shall estimate the various terms in (29).

Lemma 6 yields

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=0}^{N}\frac{\beta_{0}}{h_{j}}[Pu-u]_{j}^{2}\lesssim \varepsilon^{-1}\left( N^{-1}\ln N \right)^{2k}. \end{array} $$

(30)

From Lemma 7 and σ = k + 1, one obtains

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=0}^{N}h_{j}\{(PE-E)^{\prime} \}_{j}^{2} &\lesssim& \sum\limits_{j=1}^{N/2}h^{}\|PE-E\|{~}_{1,\infty,I_{j}}^{2} + \sum\limits_{j=N/2+1}^{N}H^{}\|PE-E\|{~}_{1,\infty,I_{j}}^{2}\\ &\lesssim& \varepsilon^{-1}\left( N^{-1}\ln N \right)^{2k}+ \varepsilon N^{-2k} +\varepsilon^{-1}N^{-2k-2} \\ &\lesssim& \varepsilon^{-1}\left( N^{-1}\ln N \right)^{2k}. \end{array} $$

(31)

As E^″∈ C[0, 1], we see that

$$ \begin{array}{@{}rcl@{}} &&{\kern-3.8pc}\sum\limits_{j=0}^{N} \frac{{\beta_{1}^{2}}}{\beta_{0}} (h_{j})^{3}[(PE-E)^{\prime\prime}]_{j}^{2}\\ &\lesssim& \sum\limits_{j=1}^{N/2+1}h^{3}\|PE-E\|{~}_{2,\infty,I_{j}}^{2}. \\ &\qquad&+\sum\limits_{j=N/2+1}^{N}H^{3}\left( \lim\limits_{x\rightarrow x_{j}^{+}}(PE-E)^{\prime\prime}(x) -\lim\limits_{x\rightarrow x_{j}^{-}}(PE-E)^{\prime\prime}(x) \right)^{2}\\ &\lesssim& \sum\limits_{j=1}^{N/2+1}h^{3}\|PE-E\|{~}_{2,\infty,I_{j}}^{2}+\sum\limits_{j=N/2+1}^{N}H^{3}\|PE\|{~}_{2,\infty,I_{j}}^{2}\\ &\lesssim& \varepsilon^{-1}\left( N^{-1}\ln N\right)^{2k} + \varepsilon N^{-2k}\\ &\lesssim& \varepsilon^{-1}\left( N^{-1}\ln N\right)^{2k}, \end{array} $$

(32)

where we used Lemma 7 and σ = k + 1. Turning now to the contributions of the smooth component S of u to (29), by Lemma 7, one has

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=0}^{N}h_{j} &&\{(PS-S)^{\prime} \}_{j}^{2} +{h_{j}^{3}}[(PS-S)^{\prime\prime}]_{j}^{2} \\ &\lesssim& \sum\limits_{j=0}^{N} h_{j}\|PS-S\|{~}_{1,\infty,I_{j}}^{2}+\sum\limits_{j=0}^{N} {h_{j}^{3}}\|PS-S\|{~}_{2,\infty,I_{j}}^{2}\\ &\lesssim& N^{-2k}. \end{array} $$

(33)

Finally, an inverse inequality (Lemma 10) yields

$$ \sum\limits_{j=0}^{N} h_{j}\{v_{h}^{\prime}\}_{j}^{2} \le \sum\limits_{j=0}^{N} h_{j}\|v_{h}\|{~}_{1,\infty,I_{j}}^{2} \lesssim \sum\limits_{j=0}^{N} |v_{h}|{~}_{1,I_{j}}^{2}. $$

(34)

Substituting (30)–(34) into (29), for all v_h ∈ V_h, we get

$$ \sum\limits_{j=0}^{N}|\widehat{(Pu-u)}_{j}[ v_{h} ]_{j}+[Pu-u]_{j}\{v_{h}^{\prime}\}_{j}| \lesssim \varepsilon^{-1/2}\left( N^{-1}\ln N \right)^{k}\|v_{h}\|{~}_{E}. $$

Hence,

$$ \begin{array}{@{}rcl@{}} \left|A_{h}(u-Pu, v_{h} )\right| &\lesssim&\sum\limits_{j=1}^{N/2}{\int}_{I_j}\left|(u-Pu)^{\prime} v_{h}^{\prime}\right| dx +\sum\limits_{j=N/2+1}^{N}{\int}_{I_j}\left|(u-Pu)^{\prime} v_{h}^{\prime}\right| dx\\ &\qquad& +\sum\limits_{j=0}^{N}\left|\widehat{(Pu-u)}_{j}[ v_{h} ]_{j}+[Pu-u]_{j}\{v_{h}^{\prime}\}_{j}\right|\\ &\lesssim& \varepsilon^{-1/2}\left( N^{-1}\ln N \right)^{k}\|v_{h}\|{~}_{E} \ \text{ for all }v_{h}\in V_{h}, \end{array} $$

where we applied a Cauchy-Schwarz inequality to the integral terms then invoked Lemma 6.

Next, by (4b), we have

$$ B_{h}(u-Pu, v_{h}) = \sum\limits_{j=1}^{N} {\int}_{I_j} b(x)(u-Pu)v_{h}^{\prime} dx+\sum\limits_{j=0}^{N} \widetilde{b(u-Pu)}_{j}[ v_{h} ]_{j} $$

for all v_h ∈ V_h. Here, a Cauchy-Schwarz inequality gives

$$ \begin{array}{@{}rcl@{}} && \left| \sum\limits_{j=1}^{N} {\int}_{I_j} b(x)(u-Pu)v_{h}^{\prime} dx \right| \\ &\lesssim &\left( \sum\limits_{j=1}^{N/2}\|u-Pu\|{~}^{2}_{0,I_{j}}\right)^{1/2} \left( \sum\limits_{j=1}^{N/2}| v_{h} |{~}^{2}_{1,I_{j}}\right)^{1/2}\\ &\qquad& +\left( \sum\limits_{j=N/2+1}^{N} H^{-2}\|u-Pu\|{~}^{2}_{0,I_{j}}\right)^{1/2} \left( \sum\limits_{j=N/2+1}^{N} H^{2}| v_{h} |{~}^{2}_{1,I_{j}}\right)^{1/2}\\ &\lesssim& \varepsilon^{1/2}(N^{-1}\ln N)^{k+1}\| v_{h} \|{~}_{E}+N^{-k}\|v_{h}\|{~}_{0}\ \text{ for all }v_{h}\in V_{h}, \end{array} $$

(35)

by Lemma 6 and the inverse inequality $H^{2}| v_{h} |{~}^{2}_{1,I_{j}}\lesssim \|v_{h}\|{~}^{2}_{0,I_{j}}$ for j > N/2 (Lemma 10). The other term in B_h(u − Pu, v_h) is bounded by

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{j=0}^{N}\left|\widetilde{b(u-Pu)}_{j}[ v_{h} ]_{j} \right|\\ &\lesssim& \left( \sum\limits_{j=1}^{N/2}h_{j}^{}\widetilde{(u-Pu)}_{j}^{2}\right)^{1/2}\left( \sum\limits_{j=1}^{N/2}\frac{\beta_{0}}{h_{j}}[v_{h}]^{2}_{j}\right)^{1/2}\\ &\qquad&+\left( \sum\limits_{j=N/2+1}^{N}h_{j}^{-1}\widetilde{(u-Pu)}_{j}^{2}\right)^{1/2}\left( \sum\limits_{j=N/2+1}^{N}h_{j}^{}[v_{h}]^{2}_{j}\right)^{1/2}\\ \end{array} $$

$$ \begin{array}{@{}rcl@{}} &\lesssim& \left( h\sum\limits_{j=1}^{N/2}\|Pu-u\|{~}^{2}_{0,\infty,I_{j}}+h\|Pu-u\|{~}^{2}_{0,\infty,I_{N/2+1}}\right)^{1/2}\|v_{h}\|{~}_{E}\\ &\qquad&+\left( H^{-1}\sum\limits_{j=N/2+1}^{N}\|Pu-u\|{~}^{2}_{0,\infty,I_{j}}\right)^{1/2}\|v_{h}\|{~}_{0}\\ &\lesssim& \left[\varepsilon (N^{-1}\ln N)^{2k+2}+\varepsilon N^{-2k-2}\ln N\right]^{1/2}\| v_{h} \|{~}_{E}+N^{-k}\|v_{h}\|{~}_{0}\\ &\lesssim& \varepsilon^{1/2}(N^{-1}\ln N)^{k+1}\| v_{h} \|{~}_{E}+N^{-k}\|v_{h}\|{~}_{0}\ \text{ for all } v_{h}\in V_{h}, \end{array} $$

(36)

where we used Lemma 7 and the triangle inequality of (28) to estimate the Pu − u terms, while ${\sum }_{N/2+1}^{N}h_{j}^{}[v_{h}]^{2}_{j} \lesssim \|v_{h}\|{~}_{0}^{2}$ follows from the inverse inequality of Lemma 10. Adding (35) and (36), one gets

$$ \left|B_{h}(u-Pu, v_{h} )\right| \lesssim \varepsilon^{1/2}(N^{-1}\ln N)^{k+1}\| v_{h} \|{~}_{E}+N^{-k}\|v_{h}\|{~}_{0}\ \text{ for all } v_{h}\in V_{h}. $$

Finally, Lemma 6 gives

$$ \begin{array}{@{}rcl@{}} \left|C_{h}(u-Pu, v_{h} )\right| &= &\left| \sum\limits_{j=1}^{N}{\int}_{I_j} c(x)(u-Pu)v_{h}dx \right| \lesssim \|u-Pu\|{~}_{0} \|v_{h}\|{~}_{0} \\ &\lesssim& \left[\varepsilon^{1/2}(N^{-1}\ln N)^{k+1}+N^{-k-1} \right]\| v_{h} \|{~}_{0}\ \text{ for all } v_{h}\in V_{h}. \end{array} $$

This completes the proof of Lemma 9.