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An Advection-Robust Hybrid High-Order Method for the Oseen Problem

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Abstract

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer \(k\geqslant 0\), the discrete velocity unknowns are vector-valued polynomials of total degree \(\leqslant \, k\) on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree \(\leqslant \,k\) on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree \(\leqslant \,(k+1)\), a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity–pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element T of diameter \(h_T\) contributes to the discretization error with an \(\mathcal {O}(h_{T}^{k+1})\)-term in the diffusion-dominated regime, an \(\mathcal {O}(h_{T}^{k+\frac{1}{2}})\)-term in the advection-dominated regime, and scales with intermediate powers of \(h_T\) in between. Numerical results complete the exposition.

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Authors and Affiliations

  1. INRIA, Laboratoire Jean Dieudonné, CNRS, Université Côte d’Azur, Nice, France

    Joubine Aghili

  2. Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, Montpellier, France

    Daniele A. Di Pietro

Authors
  1. Joubine Aghili
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  2. Daniele A. Di Pietro
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Correspondence to Daniele A. Di Pietro.

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The work of the second author was supported by Agence Nationale de la Recherche Grant HHOMM (ANR-15-CE40-0005).

Flux Formulation

Flux Formulation

In this section we reformulate the discrete problem in terms of numerical fluxes, and show that local momentum and mass balances hold. Let a mesh element \(T\in \mathcal {T}_{h}\) be fixed, and define the boundary difference space

$$\begin{aligned} \underline{\varvec{D}}_{\partial T}^k:=\left\{ \underline{\varvec{\alpha }}_{\partial T} = (\varvec{\alpha }_F)_{F\in \mathcal {F}_{T}}\; : \;\varvec{\alpha }_F\in \mathbb {P}^{k}(F)^d \text{ for } \text{ all } F\in \mathcal {F}_{T} \right\} . \end{aligned}$$

We introduce the boundary difference operator \(\underline{\varvec{\varDelta }}_{\partial T}^{k}:\underline{\varvec{U}}_{T}^{k}\rightarrow \underline{\varvec{D}}_{\partial T}^k\) such that, for all \(\underline{\varvec{v}}_{T}\in \underline{\varvec{U}}_{T}^{k}\),

$$\begin{aligned} \underline{\varvec{\varDelta }}_{\partial T}^{k}\underline{\varvec{v}}_{T}:=(\varvec{v}_{F}-\varvec{v}_{T|F})_{F\in \mathcal {F}_{T}}. \end{aligned}$$

The following result was proved in the scalar case in [48, Proposition 3].

Proposition 2

(Reformulation of the viscous stabilization bilinear form) Let an element \(T\in \mathcal {T}_{h}\) be fixed, and let \(\{\mathrm {s}_{\nu ,T}\; : \;T\in \mathcal {T}_{h}\}\) denote a family of viscous stabilization bilinear forms that satisfy assumptions (S1)–(S3) in Remark 2, and which depend on their arguments only via the difference operators defined by (12). Then, for all \(T\in \mathcal {T}_{h}\) and all \(\underline{\varvec{w}}_{T},\underline{\varvec{v}}_{T}\in \underline{\varvec{U}}_{T}^{k}\) it holds that

$$\begin{aligned} \mathrm {s}_{\nu ,T}(\underline{\varvec{w}}_{T},\underline{\varvec{v}}_{T}) = \mathrm {s}_{\nu ,T}(\underline{\varvec{w}}_{T},(\varvec{0},\underline{\varvec{\varDelta }}_{\partial T}^{k}\underline{\varvec{v}}_{T})). \end{aligned}$$
(67)

The reformulation (67) of the viscous stabilization term prompts the following definition: For all \(T\in \mathcal {T}_{h}\), the boundary residual operator \(\underline{\varvec{R}}_{\partial T}^k:\underline{\varvec{U}}_{T}^{k}\rightarrow \underline{\varvec{D}}_{\partial T}^k\) is such that, for all \(\underline{\varvec{w}}_{T}\in \underline{\varvec{U}}_{T}^{k}\),

$$\begin{aligned} \underline{\varvec{R}}_{\partial T}^k\underline{\varvec{w}}_{T}=(\varvec{R}_{TF}^k\underline{\varvec{w}}_{T})_{F\in \mathcal {F}_{T}} \end{aligned}$$

satisfies

$$\begin{aligned} -\sum _{F\in \mathcal {F}_{T}}(\varvec{R}_{TF}^k\underline{\varvec{w}}_{T},\varvec{\alpha }_F)_F = \mathrm {s}_{\nu ,T}(\underline{\varvec{w}}_{T},(\varvec{0},\underline{\varvec{\alpha }}_{\partial T}))\qquad \forall \underline{\varvec{\alpha }}_{\partial T}\in \underline{\varvec{D}}_{\partial T}^k. \end{aligned}$$
(68)

Theorem 3

(Flux formulation) Under the assumptions of Proposition 2, denote by \((\underline{\varvec{u}}_{h},p_h)\in \underline{\varvec{U}}_{h,0}^{k}\times P_{h}^{k}\) the unique solution of problem (31) and, for all \(T\in \mathcal {T}_{h}\) and all \(F\in \mathcal {F}_{T}\), define the numerical normal trace of the momentum flux as

$$\begin{aligned} \varvec{\varPhi }_{TF} :=\varvec{\varPhi }_{TF}^{\mathrm{cons}} + \varvec{\varPhi }_{TF}^{\mathrm{stab}} \end{aligned}$$

with consistency and stabilization contributions given by, respectively,

$$\begin{aligned} \varvec{\varPhi }_{TF}^{\mathrm{cons}} :=-\nu \nabla (\varvec{r}_{T}^{k+1}\underline{\varvec{u}}_{T})\varvec{n}_{TF} + \beta _{TF}\varvec{u}_{T} + p_T\varvec{n}_{TF},\qquad \varvec{\varPhi }_{TF}^{\mathrm{stab}} :=\varvec{R}_{TF}^k\underline{\varvec{u}}_{T} + \beta _{TF}^-(\varvec{u}_{T}-\varvec{u}_{F}). \end{aligned}$$

Then, for all \(T\in \mathcal {T}_{h}\) the following local balances hold: For all \(\varvec{v}_{T}\in \mathbb {P}^{k}(T)^d\) and all \(q_T\in \mathbb {P}^{k}(T)\),

$$\begin{aligned}&\nu (\nabla (\varvec{r}_{T}^{k+1}\underline{\varvec{u}}_{T}),\nabla \varvec{v}_{T})_T - (\varvec{u}_{T},(\varvec{\beta }{\cdot }\nabla )\varvec{v}_{T})_T + \mu (\varvec{u}_{T},\varvec{v}_{T})_T - (p_T,\nabla {\cdot }\varvec{v}_{T})_T\nonumber \\&\quad + \sum _{F\in \mathcal {F}_{T}}(\varvec{\varPhi }_{TF},\varvec{v}_{T})_F = (\varvec{f}_{},\varvec{v}_{T})_T,\end{aligned}$$
(69a)
$$\begin{aligned}&(D_{T}^{k}\underline{\varvec{u}}_{T},q_T)_T = 0, \end{aligned}$$
(69b)

where \(p_T:=p_{h|T}\) and, for any interface \(F\in \mathcal {F}_{h}^{\mathrm{i}}\) such that \(F\subset \partial T_1\cap \partial T_2\) for distinct mesh elements \(T_1,T_2\in \mathcal {T}_{h}\), the numerical traces of the flux are continuous in the sense that

$$\begin{aligned} \varvec{\varPhi }_{T_1F} + \varvec{\varPhi }_{T_2F}=\varvec{0}. \end{aligned}$$
(70)

Proof

(i) Local momentum balance. Let \(\underline{\varvec{v}}_{h}\in \underline{\varvec{U}}_{h,0}^{k}\) be fixed. Expanding \(\mathrm {a}_{\nu ,h}\) according to its definition (11) then using, for all \(T\in \mathcal {T}_{h}\), the definition (9) of \(\varvec{r}_{T}^{k+1}\underline{\varvec{v}}_{T}\) with \({\varvec{w}}=\varvec{r}_{T}^{k+1}\underline{\varvec{u}}_{T}\) and the definition (68) of the boundary residual operator with \(\underline{\varvec{w}}_{T}=\underline{\varvec{u}}_{T}\) and \(\underline{\varvec{\alpha }}_{\partial T}=\underline{\varvec{\varDelta }}_{\partial T}^k\underline{\varvec{v}}_{T}\), we can write

$$\begin{aligned}&\mathrm {a}_{\nu ,h}(\underline{\varvec{u}}_{h},\underline{\varvec{v}}_{h}) \\&\quad = \sum _{T\in \mathcal {T}_{h}}\left( \nu (\nabla (\varvec{r}_{T}^{k+1}\underline{\varvec{u}}_{T}),\nabla \varvec{v}_{T})_T - \sum _{F\in \mathcal {F}_{T}}(-\nu \nabla (\varvec{r}_{T}^{k+1}\underline{\varvec{u}}_{T}) + \varvec{R}_{TF}^k\underline{\varvec{u}}_{T},\varvec{v}_{F}-\varvec{v}_{T})_F \right) , \end{aligned}$$

where the viscous stabilization was reformulated using (67) then (68). In a similar way, expanding \(\mathrm {a}_{\varvec{\beta },\mu ,h}\) then, for all \(T\in \mathcal {T}_{h}\), \(\varvec{G}_{\varvec{\beta },T}^{k}\underline{\varvec{v}}_{T}\) according to their respective definitions (17) and (16), we have that

Finally, recalling the definition (23) of \(\mathrm {b}_h\) and (21) of the discrete divergence operator, we have that

$$\begin{aligned} \mathrm {b}_h(\underline{\varvec{v}}_{h},p_h) = \sum _{T\in \mathcal {T}_{h}}\left( -(p_h,\nabla {\cdot }\varvec{v}_{T})_T - \sum _{F\in \mathcal {F}_{T}}(p_T\varvec{n}_{TF},\varvec{v}_{F}-\varvec{v}_{T})_F \right) . \end{aligned}$$

Plugging the above expressions into (31a), we conclude that

$$\begin{aligned}&\sum _{T\in \mathcal {T}_{h}}\Bigg ( \nu (\nabla (\varvec{r}_{T}^{k+1}\underline{\varvec{u}}_{T}),\nabla \varvec{v}_{T})_T - (\varvec{u}_{T},(\varvec{\beta }{\cdot }\nabla )\varvec{v}_{T})_T + \mu (\varvec{u}_{T},\varvec{v}_{T})_T - (p_T,\nabla {\cdot }\varvec{v}_{T})_T \\&\qquad \quad - \sum _{F\in \mathcal {F}_{T}}(\varvec{\varPhi }_{TF},\varvec{v}_{F}-\varvec{v}_{T})_F \Bigg ) = (\varvec{f}_{},\varvec{v}_{h}). \end{aligned}$$

Selecting now \(\underline{\varvec{v}}_{h}\) such that \(\varvec{v}_{T}\) spans \(\mathbb {P}^{k}(T)^d\) for a selected mesh element \(T\in \mathcal {T}_{h}\) while \(\varvec{v}_{T'} = \varvec{0}\) for all \(T'\in \mathcal {T}_{h}\setminus \{T\}\) and \(\varvec{v}_{F} = \varvec{0}\) for all \(F\in \mathcal {F}_{h}\), we obtain the local momentum balance (69a). On the other hand, selecting \(\underline{\varvec{v}}_{h}\) such that \(\varvec{v}_{T} = \varvec{0}\) for all \(T\in \mathcal {T}_{h}\), \(\varvec{v}_{F}\) spans \(\mathbb {P}^{k}(F)^d\) for a selected interface \(F\in \mathcal {F}_{h}^{\mathrm{i}}\) such that \(F\subset \partial T_1\cap \partial T_2\) for distinct mesh elements \(T_1,T_2\in \mathcal {T}_{h}\), and \(\varvec{v}_{F'} = \varvec{0}\) for all \(F'\in \mathcal {F}_{h}\setminus \{F\}\) yields the flux continuity (70) after observing that \(\left( \varvec{\varPhi }_{T_1F}+\varvec{\varPhi }_{T_2F}\right) \in \mathbb {P}^{k}(F)^d\).

(ii) Local mass balance. We start by observing that (31b) holds in fact for all \(q_h\in \mathbb {P}^{k}(\mathcal {T}_{h})\), not necessary with zero mean value on \(\varOmega \). This can be easily checked using the definition (23) of \(\mathrm {b}_h\) and (21) of the discrete divergence to write

$$\begin{aligned} b_h(\underline{\varvec{u}}_{h},1)= & {} -\sum _{T\in \mathcal {T}_{h}}(D_{T}^{k}\underline{\varvec{u}}_{T},1)_T = -\sum _{T\in \mathcal {T}_{h}}\sum _{F\in \mathcal {F}_{T}}(\varvec{u}_{F}{\cdot }\varvec{n}_{TF},1)_F \\= & {} -\sum _{F\in \mathcal {F}_{h}}\sum _{T\in \mathcal {T}_{F}}(\varvec{u}_{F}{\cdot }\varvec{n}_{TF},1)_F = 0, \end{aligned}$$

where we have denoted by \(\mathcal {T}_{F}\) the set of elements that share F and the conclusion follows from the single-valuedness of \(\varvec{u}_{F}\) for any \(F\in \mathcal {F}_{h}^{\mathrm{i}}\) and the fact that \(\varvec{u}_{F}=\varvec{0}\) for any \(F\in \mathcal {F}_{h}^{\mathrm{b}}\). In order to prove the local mass balance (69b), it then suffices to take \(q_h\) in (31b) equal to \(q_T\) inside T and zero elsewhere. \(\square \)

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Aghili, J., Di Pietro, D.A. An Advection-Robust Hybrid High-Order Method for the Oseen Problem. J Sci Comput 77, 1310–1338 (2018). https://doi.org/10.1007/s10915-018-0681-2

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