A Space-Time Hybridizable Discontinuous Galerkin Method for Linear Free-Surface Waves

Abstract

We present and analyze a novel space-time hybridizable discontinuous Galerkin (HDG) method for the linear free-surface problem on prismatic space-time meshes. We consider a mixed formulation which immediately allows us to compute the velocity of the fluid. In order to show well-posedness and to obtain a priori error estimates, our space-time HDG formulation makes use of weighted inner products. We perform an a priori error analysis in which the dependence on the time step and spatial mesh size is explicit. The analysis results are supported by numerical examples.

Appendix A: HDG with BDF2 Time Stepping

In this section we present a discretization for the linear free-surface problem in which we combine an HDG spatial discretization with BDF2 time stepping. The discretization in this appendix is inspired by the DG discretization presented in [43].

First, we introduce a variable ${\varvec{q}} = -\nabla \phi$ and write Eq. (2c) as

$$\begin{aligned} {\varvec{q}} + \nabla \phi&= 0&\text {in } \varOmega , \end{aligned}$$

(58a)

$$\begin{aligned} \nabla \cdot {\varvec{q}}&= 0&\text {in } \varOmega , \end{aligned}$$

(58b)

$$\begin{aligned} {\varvec{q}} \cdot {\varvec{n}}&= \partial _{t}^2\phi&\text {on } \varGamma _S, \end{aligned}$$

(58c)

$$\begin{aligned} {\varvec{q}} \cdot {\varvec{n}}&= 0&\text {on } \varGamma _N. \end{aligned}$$

(58d)

Next, let ${\mathcal {T}}^{\varOmega } := \left\{ K\right\}$ denote a triangulation of the domain $\varOmega \subset {\mathbb {R}}^2$. The boundary and length measure of an element K are denoted by, respectively, $\partial K$ and $h_K$. Furthermore, we denote by ${\mathbb {E}}_h^0$ the set of interior edges of ${\mathcal {T}}^{\varOmega }$, by ${\mathbb {E}}_h^{\partial }$ we denote the set of boundary edges, and we let ${\mathbb {E}}_h = {\mathbb {E}}_h^0\cup {\mathbb {E}}_h^{\partial }$. Additionally, we denote by ${\mathbb {E}}_h^{S}$ the set of edges that lie on the free-surface boundary.

To define the HDG method we require the following finite element spaces:

$$\begin{aligned} W_h^p&:= \left\{ w\in L^2(\varOmega ) \, : w|_K\in P^p(K),\,\forall K\in {\mathcal {T}}_h\right\} , \\ {\varvec{V}}_h^p&:= \left\{ {\varvec{v}}\in \left[ L^2(\varOmega )\right] ^2 \, : {\varvec{v}}|_K\in \left[ P^p(K)\right] ^2,\,\forall K\in {\mathcal {T}}_h\right\} , \\ M_h^p&:= \left\{ \mu \in L^2({\mathbb {E}}_h)\, : \mu |_e\in P^p(e),\,\forall e\in {\mathbb {E}}_h\right\} , \end{aligned}$$

where $P^p(D)$ denote the space of polynomials of degree at most p on a domain D.

For functions ${\varvec{f}}$ and ${\varvec{g}}$ in $\left[ L^2(D)\right] ^2$, we denote $\left( {\varvec{f}},{\varvec{g}}\right) _D = \int _D{\varvec{f}}\cdot {\varvec{g}}\hbox {d}D$, where $D\subset {\mathbb {R}}^2$. If f and g are functions in $L^2(D)$, we denote $\left( f,g\right) _D = \int _D fg\,\hbox {d}D$ if $D\subset {\mathbb {R}}^2$, and $\langle f,g \rangle _D = \int _D fg \hbox {d}s$ if $D \subset {\mathbb {R}}$. Moreover, we use the following notation

$$\begin{aligned} \left( w,v\right) _{{\mathcal {T}}^{\varOmega }}&= \sum _{K\in {\mathcal {T}}^{\varOmega }}\left( w,v\right) _K,&\langle w,v \rangle _{\partial {\mathcal {T}}^{\varOmega }}&= \sum _{K\in {\mathcal {T}}^{\varOmega }}\langle w, v \rangle _{\partial K}, \\ \langle w, v \rangle _{{\mathbb {E}}_h}&= \sum _{e\in {\mathbb {E}}_h}\langle w, v\rangle _{e},&\langle w, v \rangle _{{\mathbb {E}}_h^S}&= \sum _{e\in {\mathbb {E}}_h^S}\langle w, v\rangle _{e}. \end{aligned}$$

The semidiscrete HDG discretization for eq. (58) is now given by: Find $\left( {\varvec{q}}_h, \phi _h, \lambda _h\right) \in {\varvec{V}}_h^p \times W_h^p \times M_h^p$ such that

$$\begin{aligned} -\left( {\varvec{q}}_h, {\varvec{v}}_h\right) _{{\mathcal {T}}^{\varOmega }} + \left( \phi _h, \nabla \cdot {\varvec{v}}_h\right) _{{\mathcal {T}}^{\varOmega }} - \langle \lambda _h, {\varvec{v}}_h \cdot {\varvec{n}}\rangle _{\partial {\mathcal {T}}^{\varOmega }}&= 0, \\ \left( w_h,\nabla \cdot {\varvec{q}}_h\right) _{{\mathcal {T}}^{\varOmega }} + \tau \langle w_h, \phi _h \rangle _{\partial {\mathcal {T}}^{\varOmega }} - \tau \langle w_h, \lambda _h \rangle _{\partial {\mathcal {T}}^{\varOmega }}&= 0, \\ -\langle {\varvec{q}}_h \cdot {\varvec{n}}, \mu _h \rangle _{{\mathbb {E}}_h} - \tau \langle \phi _h, \mu _h \rangle _{{\mathbb {E}}_h} + \tau \langle \lambda _h, \mu _h \rangle _{{\mathbb {E}}_h}&= \langle \partial ^2_t\phi _h ,\mu _h \rangle _{{\mathbb {E}}_h^S}, \end{aligned}$$

for all $\left( {\varvec{v}}_h, w_h, \mu _h\right) \in {\varvec{V}}_h^p \times W_h^p \times M_h^p$. We follow the identical approach introduced in [43] to discretize the $\partial ^2_t\phi _h$ and to obtain an approximation to the wave height $\zeta _h$ by a second order backward differentiation formula (BDF2). We therefore do not present the time discretization here.