Discretization of the Wave Equation Using Continuous Elements in Time and a Hybridizable Discontinuous Galerkin Method in Space

Abstract

We provide an error analysis of two methods for time stepping the wave equation. These are based on the Hybridizable Discontinuous Galerkin (HDG) method to discretize in space, and the continuous Galerkin method to discretize in time. Two variants of HDG are proposed: a dissipative method based on the standard numerical flux used for elliptic problems, and a non-dissipative method based on a new choice of the flux involving time derivatives. The analysis of the fully discrete problem is based on simplified arguments using projections rather than explicit interpolants used in previous work. Some numerical results are shown that illuminate the theory.

Appendix

1.1 Projection Estimate on the Element Boundaries

In this section we give a proof of the projection estimate on the element boundaries (4c). Our arguments rest on those used in [4] to prove the corresponding interior estimates (4a) and (4b).

We start with a slight modification of Lemma A.2 of [4].

Lemma 6

Let $K \in \mathcal{T _h}, \tau$ as introduced in Sect. 2.1, and suppose

$$\begin{aligned} p \in \mathcal{P }_k^\perp (K) := \{ \phi \in \mathcal{P }_k(K) \;|\; (\phi , w)_K = 0 \; for \,all\, w \in \mathcal{P }_{k-1}(K) \} \end{aligned}$$

satisfies

$$\begin{aligned} \langle \tau p, \phi \rangle _{\partial K} = b(\phi ) \qquad for\, all\, \phi \in P_k^\perp (K), \end{aligned}$$

where $b: \mathcal{P }_k^\perp (K) \rightarrow \mathbb R$ is linear. Then

$$\begin{aligned} \Vert \tau p \Vert _{\partial K} \le C h_K^{1/2} \Vert b\Vert , \end{aligned}$$

where $\Vert b\Vert$ denotes the operator norm of $b$ with respect to the $L^2$-norm on $\mathcal{P }_k^\perp (K)$.

Proof

Denoting by $F$ the edge/face of $K$ at which $\tau = \tau _K^{\max }$,

$$\begin{aligned} \Vert \tau p \Vert _{\partial K}^2 = \langle \tau p, \tau p \rangle _{\partial K} \le \tau _K^{\max } \langle \tau p, p \rangle _{\partial K} = \tau _K^{\max } b(p) \le \tau _K^{\max } \Vert b\Vert \Vert p\Vert _K. \end{aligned}$$

Using the estimate

$$\begin{aligned} \Vert p\Vert _K \le C h_K^{1/2} \Vert p\Vert _F \qquad \text{ for } \text{ all } p \in \mathcal{P }_K^\perp (K), \end{aligned}$$

which has been shown in Lemma A.1 of [4], gives

$$\begin{aligned} \Vert \tau p \Vert _{\partial K}^2 \le \tau _K^{\max } \Vert b\Vert C h_K^{1/2} \Vert p\Vert _F = \Vert b\Vert C h_K^{1/2} \Vert \tau _K^{\max } p\Vert _F \le C h_K^{1/2} \Vert b\Vert \Vert \tau p\Vert _{\partial K}. \end{aligned}$$

$\square$

The following proposition should be compared to Proposition A.2 of [4].

Proposition 1

Suppose $k \ge 0$, and let $K\in \mathcal{T _h}$ and $\tau$ as introduced in Sect. 2.1. Then,

$$\begin{aligned} \Vert \tau (\varPi _W w - w)\Vert _{\partial K} \le C h_K^{1/2} \bigl ( h_K^{\ell _{\varvec{z}}} |{{\mathrm{\nabla \cdot }}}{\varvec{z}}|_{H^{\ell _{\varvec{z}}}(K)} + \tau _K^{\max } h_K^{\ell _w} |w|_{H^{\ell _w+1}(K)} \bigr ) \end{aligned}$$

for $\ell _w, \ell _{\varvec{z}}\in [0,k]$.

Proof

Denoting $\delta ^w := \varPi _W w - w_k$, where $w_k$ is the $L^2$-projection of $w$ onto $\mathcal{P }_k(K)$, we have

$$\begin{aligned} \Vert \tau (\varPi _W w - w) \Vert _{\partial K} \le \Vert \tau (w - w_k) \Vert _{\partial K} + \Vert \tau \delta ^w \Vert _{\partial K}. \end{aligned}$$

Applying a trace inequality and the approximation properties of the $L^2$-projection the first term can be estimated by

$$\begin{aligned}&\Vert \tau (w - w_k) \Vert _{\partial K} \le \tau _K^{\max } \Vert w - w_k \Vert _{\partial K} \le \tau _K^{\max } h_K^{-1/2} \left( \Vert w - w_k\Vert _K + h_K |w - w_k|_{H^1(K)} \right) \\&\quad \le C \tau _K^{\max } h_K^{\ell _w+1/2} |w|_{H^{\ell _w+1}(K)}. \end{aligned}$$

To estimate the second term we recall that on each element $K \in \mathcal{T _h}$ the component $\varPi _W w$ satisfies

$$\begin{aligned} (\varPi _W w, \phi )_K&= (w, \phi )_K \qquad \qquad \text{ for } \text{ all } \phi \in P_{k-1}(K),\\ \langle \tau \varPi _W w, \phi \rangle _{\partial K}&= ({{\mathrm{\nabla \cdot }}}{\varvec{z}}, \phi )_K + \langle \tau w, \phi \rangle _{\partial K} \qquad \text{ for } \text{ all } \phi \in P_k^\perp (K) \end{aligned}$$

(see Proposition A.1 of [4]). This implies that $\delta ^w \in \mathcal{P }_k^\perp$ and

$$\begin{aligned} \langle \tau \delta ^w, \phi \rangle _{\partial K} = b_w(\phi ) + b_{\varvec{z}}(\phi )\qquad \text{ for } \text{ all } \phi \in \mathcal{P }_k^\perp (K), \end{aligned}$$

where $b_w(\phi ) := \langle \tau (w-w_k), \phi \rangle _{\partial K}$ and $b_{\varvec{z}}(\phi ) := ({{\mathrm{\nabla \cdot }}}{\varvec{z}}, \phi )_K$. It has been shown in the proof of Proposition A.2 in [4] that

$$\begin{aligned} \Vert b_w\Vert \le C \tau _K^{\max } h_K^{\ell _w} |w|_{H^{\ell _w+1}(K)} \quad \text{ and } \quad \Vert b_{\varvec{z}}\Vert \le C h_K^{\ell _{\varvec{z}}} |{{\mathrm{\nabla \cdot }}}{\varvec{z}}|_{H^{\ell _{\varvec{z}}}(K)}. \end{aligned}$$

Since by Lemma 6,

$$\begin{aligned} \Vert \tau \delta ^w \Vert _{\partial K} \le C h_K^{1/2} ( \Vert b_w\Vert + \Vert b_{\varvec{z}}\Vert ), \end{aligned}$$

this ends the proof. $\square$