Stability Analysis of Discontinuous Galerkin Approximations to the Elastodynamics Problem

Abstract

We consider semi-discrete discontinuous Galerkin approximations of both displacement and displacement-stress formulations of the elastodynamics problem. We prove the stability analysis in the natural energy norm and derive optimal a-priori error estimates. For the displacement-stress formulation, schemes preserving the total energy of the system are introduced and discussed. We verify our theoretical estimates on two and three dimensions test problems.

Appendix: Proof of Lemma 1 and Lemma 2

In this appendix we collect the proofs of the auxiliary Lemmas 1 and 2, used in the stability analysis.

Proof

(Proof of Lemma 1) The proof goes along the same lines as in the continuous case with subtle modifications to obtain bounds independent of h. Estimate (31) follows from the Cauchy–Schwarz inequality together with the lower bound of the mass density (2). To show estimate (32), we proceed similarly to get

$$\begin{aligned} \left| \int _0^t \langle \mathbf{c}_{22}\mathbf{g}_\tau , \varvec{\sigma }^{h}\,\mathbf{n}\rangle _{{{\mathcal {F}}^{N}_h}} \, d\tau \right| \le \mathrm {D}_{*}^{-1/2} \int _0^t \Vert \mathbf{g}_\tau \Vert _{1/2, {\varGamma }_N} \Vert \mathbf{c}_{22}^{1/2} \varvec{\sigma }^{h}\mathbf{n}\Vert _{0,{{\mathcal {F}}^{N}_h}}\, d\tau \;. \end{aligned}$$

(50)

Next, we notice that for each $t\in [0,T]$, the map $\mathbf{g}(t)$ belongs to $\mathbf{H}^{1/2}({\varGamma }_N)$. The inverse trace theorem [1] guarantees that the trace operator has a continuous right inverse operator, say $\mathfrak {T}: \mathbf{H}^{1/2}({\varGamma }_N) \longrightarrow \mathbf{H}^{1}({\varOmega })$. Hence, taking into account the scaling of the parameter $\mathbf{c}_{22}$ and using the trace inequality (15) we have

$$\begin{aligned} \begin{aligned} \left\| \mathbf{c}_{22}^{1/2}\mathbf{g}\right\| _{0,F}^{2}=c_2 h_F k^{-2} \{{\mathcal {D}}\}^{-1} \Vert \mathbf{g}\Vert _{0,F}^{2} \lesssim c_2 k^{-2} \mathrm {D}_{*}^{-1}\Vert \mathbf{g}\Vert _{1,K}^{2}&\forall \, F\in {{\mathcal {F}}^{N}_h}, F \subset \partial K\;, \end{aligned} \end{aligned}$$

where, with an abuse of notation, we have denoted by $\mathbf{g}=\mathfrak {T}\mathbf{g}$ the extension of $\mathbf{g}$. Summing over all $F\in {{\mathcal {F}}^{N}_h}$ and using the continuity of the operator $\mathfrak {T}$ we get

$$\begin{aligned} \left\| \mathbf{c}_{22}^{1/2} \mathbf{g}_\tau \right\| ^{2}_{0,{{\mathcal {F}}^{N}_h}} \lesssim \sum _{K \in \mathcal {T}_h} \mathrm {D}_{*}^{-1}\Vert \mathbf{g}_\tau \Vert _{1,K}^{2} = \mathrm {D}_{*}^{-1}\Vert \mathbf{g}_\tau \Vert ^{2}_{1,{\varOmega }} \lesssim \mathrm {D}_{*}^{-1}\Vert \mathbf{g}_\tau \Vert _{1/2, {\varGamma }_N}^{2}. \end{aligned}$$

(51)

Substitution of the above estimate in (50) gives (32). To prove (33), we use integration by parts formula (8) with $\mathbf{w}=\mathbf{g}$ and $\mathbf{z}=\mathbf{u}^h$, together with triangle and Jensen’ inequality to get

$$\begin{aligned} \left| \int _0^t \langle \mathbf{g}, \mathbf{u}^{h}_{\tau }\rangle _{{{\mathcal {F}}^{N}_h}} \, d\tau \right| \le \left| \langle \mathbf{g}_{0},\mathbf{u}^{h}_0\rangle _{{{\mathcal {F}}^{N}_h}}\right| +\left| \langle \mathbf{g}, \mathbf{u}^{h}\rangle _{{{\mathcal {F}}^{N}_h}}\right| + \int _0^t \left| \langle \mathbf{g}_{\tau } , \mathbf{u}^{h}\rangle _{{{\mathcal {F}}^{N}_h}}\right| \,d\tau . \end{aligned}$$

(52)

Therefore, we only need to estimate the inner product $|\langle \mathbf{g}, \mathbf{u}^{h}\rangle _{{{\mathcal {F}}^{N}_h}}|$, where the first argument could be either $\mathbf{g}_0$, $\mathbf{g}$ or $\mathbf{g}_{\tau }$. Applying Hölder’s inequality, the trace inequality (16) and inequality (17) with $\omega =F\in {{\mathcal {F}}^{N}_h}$ gives

$$\begin{aligned} \begin{aligned} \left| \int _{F} \mathbf{g}\mathbf{u}^{h} ds \right|&\le \Vert \mathbf{g}\Vert _{L^{q}(F)} \Vert \mathbf{u}^{h}\Vert _{L^{p}(F)} \lesssim \Vert \mathbf{g}\Vert _{L^{q}(F)} h^{-1/p} \Vert \mathbf{u}^{h}\Vert _{\varvec{W}^{1,p}(K)} \\&\lesssim \Vert \mathbf{g}\Vert _{L^{q}(F)} h^{-1/p} h^{d(\frac{1}{p}-\frac{1}{2})}\Vert \mathbf{u}^{h}\Vert _{1,K} = \Vert \mathbf{g}\Vert _{L^{q}(F)} h^{\frac{2d-2 -dp}{2p}} \Vert \mathbf{u}^{h}\Vert _{1,K}\;, \end{aligned} \end{aligned}$$

where, for any $F\in {{\mathcal {F}}^{N}_h}$, K is the only element in ${\mathcal {T}}_h$ such that $F \subset \partial K$. Setting now $p=(2d-2)/d$ (whose conjugate is $q=\frac{(2d-2)}{(d-2)}$) the above inequality becomes

$$\begin{aligned} \left| \int _{F} \mathbf{g}\mathbf{u}^{h} ds \right| \lesssim \Vert \mathbf{g}\Vert _{L^{q}(F)} \Vert \mathbf{u}^{h}\Vert _{1,K}. \end{aligned}$$

(53)

Notice that $q=\infty$ for $d=2$ and $q=4$ for $d=3$. Using that F is a $(d-1)$ dimensional object and using the continuity of the Sobolev embedding $H^{1}(F) \longrightarrow L^{q}(F)$, [1], we have

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{g}\Vert _{\mathbf{L}^{q}(F)} \lesssim \Vert \mathbf{g}\Vert _{1, F}&\forall \, \mathbf{g}\in \mathbf{H}^{1}(F),&q=\frac{(2d-2)}{(d-2)}. \end{aligned} \end{aligned}$$

Substituting the above bound in (53) and summing over all faces $F \in {\mathcal {F}}_h^N$, gives

$$\begin{aligned} \begin{aligned} \left| \langle \mathbf{g}, \mathbf{u}^{h}\rangle _{{{\mathcal {F}}^{N}_h}}\right| \lesssim&\Vert \mathbf{g}\Vert _{1,{\varGamma }_N}\left( \Vert \mathbf{u}^{h}\Vert _{0,\mathcal {T}_h}^{2} + \left| \mathbf{u}^{h} \right| _{1,\mathcal {T}_h}^{2} \right) ^{1/2}. \end{aligned} \end{aligned}$$

(54)

Applying the discrete Poincaré and Korn inequalities [8, 9], and the bound in (5), we have

$$\begin{aligned} \Vert \mathbf{u}^{h}\Vert _{0,\mathcal {T}_h}^{2} + |\mathbf{u}^{h}|_{1,\mathcal {T}_h}^{2}&\lesssim \Vert \varvec{\varepsilon }{(\mathbf{u}^{h})}\Vert _{0,\mathcal {T}_h}^{2} + \sum _{F\in {{\mathcal {F}}^{o}_h}\cup {{\mathcal {F}}^{D}_h}} \left\| h_F^{-1/2}[\![ \mathbf{u}^{h}]\!]\right\| _{0,F}^{2} \lesssim \mathrm {D}_{*}^{-1} \left\| \mathbf{u}^{h} \right\| _{a}^{2}. \end{aligned}$$

Finally, substituting the above estimate in (54) yields

$$\begin{aligned} \begin{aligned} \left| \langle \mathbf{g}, \mathbf{u}^{h}\rangle _{{{\mathcal {F}}^{N}_h}}\right| \lesssim&\Vert \mathbf{g}\Vert _{1,{\varGamma }_N}\mathrm {D}_{*}^{-1} \left\| \mathbf{u}^{h} \right\| _{a}. \end{aligned} \end{aligned}$$

Applying now the above estimate to each term in (52), we finally get

$$\begin{aligned} \begin{aligned} \left| \langle \mathbf{g}_{0} , \mathbf{u}^{h}_0\rangle _{{{\mathcal {F}}^{N}_h}} \right|&\lesssim \mathrm {D}_{*}^{-1} \left\| \mathbf{g}_{0} \right\| _{1,{\varGamma }_N} \left\| \mathbf{u}^{h}_0 \right\| _{a}, \\ \int _0^t \left| \langle \mathbf{g}_{\tau } , \mathbf{u}^{h}\rangle _{{{\mathcal {F}}^{N}_h}} \right| \,d\tau \lesssim&\int _0^t \mathrm {D}_{*}^{-1/2} \Vert \mathbf{g}_{\tau }\Vert _{1,{\varGamma }_N} \left\| \mathbf{u}^{h} \right\| _{a} \,d\tau .&\\ \left| \langle \mathbf{g}(t) , \mathbf{u}^{h}(t)\rangle _{{{\mathcal {F}}^{N}_h}} \right|&\lesssim \frac{ \mathrm {D}_{*}^{-1}}{\epsilon } \left\| \mathbf{g}(t) \right\| _{1,{\varGamma }_N}^2 + \epsilon \left\| \mathbf{u}^{h}(t) \right\| _{a}^2 , \end{aligned} \end{aligned}$$

where for the last term we have also used the arithmetic geometric inequality with $\epsilon >0$. Substitution of the above estimates into (52) completes the proof. $\square$

Proof

(Proof of Lemma  2) We start rewriting the second equation in (22) with $\varvec{\tau }={\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)$

$$\begin{aligned} \Vert \mathcal {D}^{1/2}\varvec{\varepsilon }{(\mathbf{u}^{h})}\Vert _{0,\mathcal {T}_h}^{2}= & {} ( \varvec{\varepsilon }(\mathbf{u}^h), {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h))_{\mathcal {T}_h}= ( {\mathcal {A}}\varvec{\sigma }^h , {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h))_{\mathcal {T}_h} \nonumber \\&+\langle \mathbf{c}_{22}[\![ \varvec{\sigma }^{h}]\!], [\![ {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)]\!]\rangle _{{{\mathcal {F}}^{o}_h}} +\langle \mathbf{c}_{22}(\varvec{\sigma }^{h}\mathbf{n}-\mathbf{g}), {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h) \,\mathbf{n}\rangle _{{{\mathcal {F}}^{N}_h}} \nonumber \\&+ \langle [\![ \mathbf{u}^h]\!], \{{\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)\}\rangle _{{{\mathcal {F}}^{o}_h}\cup {{\mathcal {F}}^{D}_h}}-\langle \{\mathbf{u}^{h}\}_{(1-\delta )}-\{\mathbf{u}^h\}, [\![ {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)]\!] \rangle _{{{\mathcal {F}}^{o}_h}},\qquad \quad \end{aligned}$$

(55)

Prior to estimate all terms on the right-hand side above, we note that Agmon’s (14) and inverse inequalities (18), and the definition of $\mathbf{c}_{22}$ give

$$\begin{aligned} \left\| \mathbf{c}_{22}^{1/2} [\![ {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)]\!] \right\| _{0,F} \lesssim \Vert {\mathcal {D}}^{1/2}\varvec{\varepsilon }(\mathbf{u}^h) \Vert _{0,K},\quad \left\| \mathbf{c}_{22}^{1/2} \{{\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)\}_{\delta } \right\| _{0,F} \lesssim \Vert {\mathcal {D}}^{1/2}\varvec{\varepsilon }(\mathbf{u}^h) \Vert _{0,K}\;. \end{aligned}$$

(56)

Using Cauchy–Schwarz inequality and the first estimate above, the first three terms in (55) can be bounded by

$$\begin{aligned} \left| ( {\mathcal {A}}\varvec{\sigma }^h , {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h))_{\mathcal {T}_h} \right|\le & {} \Vert \mathcal {A}^{1/2}\varvec{\sigma }^{h} \Vert _{0,\mathcal {T}_h} \Vert \mathcal {D}^{1/2}\varvec{\varepsilon }{(\mathbf{u}^{h})}\Vert _{0,\mathcal {T}_h}, \\ \left| \langle \mathbf{c}_{22}[\![ \varvec{\sigma }^{h}]\!], [\![ {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)]\!]\rangle _{{{\mathcal {F}}^{o}_h}} \right|\lesssim & {} \left\| \mathbf{c}_{22}^{1/2} [\![ \varvec{\sigma }^h]\!]\right\| _{0,{{\mathcal {F}}^{o}_h}} \Vert {\mathcal {D}}^{1/2}\varvec{\varepsilon }(\mathbf{u}^h) \Vert _{0,K} \\ \left| \langle \mathbf{c}_{22}(\varvec{\sigma }^{h}\mathbf{n}-\mathbf{g}), {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h) \,\mathbf{n}\rangle _{{{\mathcal {F}}^{N}_h}} \right|\lesssim & {} \left( \Vert \mathbf{c}_{22}^{1/2} [\![ \varvec{\sigma }^h]\!] \Vert _{0,{{\mathcal {F}}^{N}_h}} + \Vert \mathbf{c}_{22}^{1/2} \mathbf{g}\Vert _{0,{{\mathcal {F}}^{N}_h}} \right) \Vert {\mathcal {D}}^{1/2}\varvec{\varepsilon }(\mathbf{u}^h) \Vert _{0,K} . \end{aligned}$$

To estimate the last two terms in (55), notice that $\mathbf{c}_{11}\mathbf{c}_{22}=O(1)$ since,

$$\begin{aligned} \mathbf{c}_{11}^{-1} = \left( c_1 h_F^{-1} k^{2} \{{\mathcal {D}}\}\right) ^{-1} = (c_1 c_2)^{-1} c_2 h_F k^{-2} \{{\mathcal {D}}\}^{-1} = (c_1 c_2)^{-1} \mathbf{c}_{22}. \end{aligned}$$

Then, the Cauchy Schwarz inequality and (56) give for the fourth term

$$\begin{aligned} \left| \langle [\![ \mathbf{u}^h]\!], \{{\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)\}\rangle _{{{\mathcal {F}}^{o}_h}\cup {{\mathcal {F}}^{D}_h}} \right|\lesssim & {} \left\| \mathbf{c}_{11}^{1/2} [\![ \mathbf{u}^h]\!]\right\| _{0,{{\mathcal {F}}^{o}_h}\cup {{\mathcal {F}}^{D}_h}} \; \Vert \mathbf{c}_{22}^{1/2} \{{\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)\} \Vert _{0,{{\mathcal {F}}^{o}_h}\cup {{\mathcal {F}}^{D}_h}}\nonumber \\\lesssim & {} \left\| \mathbf{c}_{11}^{1/2} [\![ \mathbf{u}^h]\!]\right\| _{0,{{\mathcal {F}}^{o}_h}\cup {{\mathcal {F}}^{D}_h}} \; \Vert {\mathcal {D}}^{1/2}\varvec{\varepsilon }(\mathbf{u}^h) \Vert _{0,{\varOmega }}. \end{aligned}$$

Analogously, the last term can be estimated using identity (13) and (56)

$$\begin{aligned} \left| - \langle \{\mathbf{u}^{h}\}_{(1-\delta )}-\{\mathbf{u}^h\}, [\![ {\mathcal {D}}\varvec{\varepsilon }(\mathbf{u}^h)]\!] \rangle _{{{\mathcal {F}}^{o}_h}} \right| \lesssim \left\| \mathbf{c}_{11}^{1/2} [\![ \mathbf{u}^{h}]\!] \right\| _{0,{{\mathcal {F}}^{o}_h}} \Vert {\mathcal {D}}^{1/2}\varvec{\varepsilon }(\mathbf{u}^h) \Vert _{0,{\varOmega }}. \end{aligned}$$

Finally, substituting all the estimates into (55) we obtain

$$\begin{aligned} \Vert \mathcal {D}^{1/2}\varvec{\varepsilon }{(\mathbf{u}^{h})}\Vert _{0,\mathcal {T}_h} \lesssim \Vert \mathcal {A}^{1/2}\varvec{\sigma }^{h} \Vert _{0,\mathcal {T}_h} + \left\| \mathbf{c}_{11}^{1/2} [\![ \mathbf{u}^h]\!]\right\| _{0,{{\mathcal {F}}^{o}_h}} + \left\| \mathbf{c}_{22}^{1/2} [\![ \varvec{\sigma }^h]\!] \right\| _{0,{{\mathcal {F}}^{N}_h}} + \left\| \mathbf{c}_{22}^{1/2} \mathbf{g}\right\| _{0, {{\mathcal {F}}^{N}_h}}, \end{aligned}$$

The proof is then concluded by arguing as in the proof of (32) in Lemma 1 (using estimate (51))

$$\begin{aligned} \left\| \mathbf{c}_{22}^{1/2} \mathbf{g} \right\| ^{2}_{0,{{\mathcal {F}}^{N}_h}} \lesssim \mathrm {D}_{*}^{-1}\Vert \mathbf{g}\Vert _{1/2, {\varGamma }_N}^{2}. \end{aligned}$$

$\square$