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Coupling of Continuous and Hybridizable Discontinuous Galerkin Methods: Application to Conjugate Heat Transfer Problem

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Abstract

A coupling strategy between hybridizable discontinuous Galerkin (HDG) and continuous Galerkin (CG) methods is proposed in the framework of second-order elliptic operators. The coupled formulation is implemented and its convergence properties are established numerically by using manufactured solutions. Afterwards, the solution of the coupled Navier–Stokes/convection–diffusion problem, using Boussinesq approximation, is formulated within the HDG framework and analysed using numerical experiments. Results of Rayleigh–Bénard convection flow are also presented and validated with literature data. Finally, the proposed coupled formulation between HDG and CG for heat equation is combined with the coupled Navier–Stokes/convection diffusion equations to formulate a new CG–HDG model for solving conjugate heat transfer problems. Benchmark examples are solved using the proposed model and validated with literature values. The proposed CG–HDG model is also compared with a CG–CG model, where all the equations are discretized using the CG method, and it is concluded that CG–HDG model can have a superior computational efficiency when compared to CG–CG model.

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Acknowledgements

This work was supported by the Erasmus Mundus Joint Doctorate SEED project (European Commission, 2013-1436/001-001-EMJD). The work of the Carlos Tiago is part of the research activity carried out at Civil Engineering Research and Innovation for Sustainability (CERIS) and has been partially financed by Fundação para a Ciência e a Tecnologia (FCT) in the framework of project UID/ECI/04625/2013. The third author also wishes to acknowledge the DAFOH2 project (Ministerio de Economia y Competitividad, MTM2013-46313-R) and the Catalan government (Generalitat de Catalunya, 2009SGR875).

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Appendix: Definition of Elemental Matrices

Appendix: Definition of Elemental Matrices

In this appendix, the elemental matrices that arise from coupled CG–HDG formulation for heat equation and HDG method for coupled Navier–Stokes/convection–diffusion equations are defined. All the variables presented in this section are the elemental variables. Variable \(\varvec{L}^{(e)}\) is a second-order tensor and it is represented as a column vector, \(\left[ l_{11}\, l_{12}\, l_{21}\, l_{22}\right] ^{(e)T}\), in the numerical computations.

The independent variables \((\varvec{L}^{(e)},\varvec{u}^{(e)},p^{(e)},\theta ^{(e)},\varvec{q}^{(e)},\hat{\varvec{u}}^{(e)},\rho ^{(e)},\hat{\theta }^{(e)})\) over each element, \(\varOmega ^e\), can be approximated as follows,

$$\begin{aligned} \begin{array}{ll} \varvec{L}^{(e)}(\varvec{\xi }) = \varvec{\uppsi }_L(\varvec{\xi }) \mathbf {L}^{(e)}, \quad \varvec{u}^{(e)}(\varvec{\xi }) = \varvec{\uppsi }_u(\varvec{\xi }) \mathbf {u}^{(e)}, \quad p^{(e)}(\varvec{\xi }) = \varvec{\uppsi }_p(\varvec{\xi }) \mathbf {p}^{(e)} &{}\,\,\,\,\,\,\text {in} \,\,\,\,\,\,\varOmega ^e, \\ \theta ^{(e)} (\varvec{\xi }) = \varvec{\uppsi }_{\theta }(\varvec{\xi })\varvec{\uptheta }^{(e)}, \quad \varvec{q}^{(e)}(\varvec{\xi }) = \varvec{\uppsi }_{q}(\varvec{\xi })\mathbf {q}^{(e)}&{}\,\,\,\,\,\,\text {in} \,\,\,\,\,\,\varOmega ^e, \\ \hat{\varvec{u}}^{(e)}(\xi ) = \varvec{\uppsi }_{\hat{u}}(\xi ) \hat{\mathbf {u}}^{(e)}, \quad \hat{\theta }^{(e)}(\xi ) = \varvec{\uppsi }_{\hat{\theta }}(\xi )\varvec{\hat{\uptheta }}^{(e)}&{}\,\,\,\,\,\,\text {on} \,\,\,\,\,\,\partial \varOmega ^e, \end{array} \end{aligned}$$
(35)

where \(\varvec{\uppsi }_L(\varvec{\xi })\), \(\varvec{\uppsi }_u(\varvec{\xi })\), \(\varvec{\uppsi }_p(\varvec{\xi }),\varvec{\uppsi }_{\theta }(\varvec{\xi }),\varvec{\uppsi }_{q}(\varvec{\xi }),\varvec{\uppsi }_{\hat{u}}(\xi )\) and \(\varvec{\uppsi }_{\hat{\theta }}(\xi )\) are matrices that gather the approximation functions of respective unknowns, while \(\mathbf {L}^{(e)}\), \(\mathbf {u}^{(e)}\), \(\mathbf {p}^{(e)}, \varvec{\uptheta }^{(e)},\mathbf {q}^{(e)}, \hat{\mathbf {u}}^{(e)}\) and \(\hat{\varvec{\uptheta }}^{(e)}\) are the elemental nodal column vectors of gradient of velocity, velocity, pressure, temperature, flux, velocity trace and temperature trace, respectively. \(\varvec{\xi }\) and \(\xi \) represent the coordinate in the area and line reference domains, respectively. \(\hat{\mathbf {u}}^{(e)}\) contains the trace of velocity on each face of the element and it can be represented as \(\left[ \hat{\mathbf {u}}^{\mathbf {F}_{e1}} \ldots \hat{\mathbf {u}}^{\mathbf {F}_{en}}\right] ^T\), where \(\mathbf {F}_{ef}\) is the \(f^{th}\) face of \(e^{th}\) element. Here, \(n=3\) in the case of triangular elements, while \(n=4\) for quadrilateral elements. From now on explicit dependence on \(\varvec{\xi }\) and \(\xi \) will be omitted for the sake of simplicity. The approximation functions can be represented as follows,

$$\begin{aligned} \begin{array}{l} \varvec{\uppsi }_L = \begin{bmatrix} \varvec{\uppsi } &{} &{} &{}\\ &{} \varvec{\uppsi } &{} &{} \\ &{} &{} \varvec{\uppsi } &{} \\ &{} &{} &{} \varvec{\uppsi } \end{bmatrix}, \quad \varvec{\uppsi }_u =\varvec{\uppsi }_q = \begin{bmatrix} \varvec{\uppsi } &{} \\ &{} \varvec{\uppsi } \end{bmatrix}, \quad \varvec{\uppsi }_p =\varvec{\uppsi }_{\theta }=\varvec{\uppsi }, \\ \varvec{\uppsi }_{\hat{u}} = \begin{bmatrix} \varvec{\uppsi }_{\mathbf {F}_{e1}} &{} \quad \quad \ldots &{} \varvec{\uppsi }_{\mathbf {F}_{en}} &{} \\ &{} \varvec{\uppsi }_{\mathbf {F}_{e1}} \ldots &{} &{} \varvec{\uppsi }_{\mathbf {F}_{en}} \end{bmatrix}, \quad \varvec{\uppsi }_{\hat{\theta }} = \begin{bmatrix} \varvec{\uppsi }_{\mathbf {F}_{e1}} &{} \ldots &{} \varvec{\uppsi }_{\mathbf {F}_{en}} \end{bmatrix}, \end{array} \end{aligned}$$
(36)

where \(\varvec{\uppsi }\) is the matrix that gathers the shape functions associated to the nodes of the elements and \(\varvec{\uppsi }_{\mathbf {F}_{ef}}\) is the matrix collecting the shape functions associated to the nodes along the sides of the element.

Some notation used to represent the element matrices in case of both HDG and CG is given as follows,

$$\begin{aligned} \begin{array}{l} \tilde{\varvec{\nabla }} \equiv \begin{bmatrix} \dfrac{\partial \phantom {x_1}}{\partial x_1} &{}\dfrac{\partial \phantom {x_1}}{\partial x_2} &{} &{}\\ &{} &{} \dfrac{\partial \phantom {x_1}}{\partial x_1} &{} \dfrac{\partial \phantom {x_1}}{\partial x_2} \end{bmatrix}, \quad \tilde{\mathbf {N}} \equiv \begin{bmatrix} n_1 &{} n_2 &{} &{}\\ &{} &{} n_1 &{} n_2 \end{bmatrix}. \end{array} \end{aligned}$$
(37)

The definition of elemental matrices corresponding to the standard local problem of HDG in the case of coupled CG–HDG formulation for heat Eq. (13) are presented as follows,

$$\begin{aligned} \begin{array}{lllllll} \mathbf {A}^{(e)}_{qq} &{}=&{} \left( \varvec{\uppsi }_q^T,k_D^{-1}\varvec{\uppsi }_q\right) _{\varOmega ^e}, &{}\quad &{} \mathbf {A}^{(e)}_{q\theta } &{}=&{} -\left( (\tilde{\varvec{\nabla }}^T \varvec{\uppsi }_q)^T,\varvec{\uppsi }_{\theta }\right) _{\varOmega ^e},\\ \mathbf {A}^{(e)}_{q\hat{\theta }} &{}=&{} \langle (\varvec{n}^T\,\varvec{\uppsi }_q )^T,\varvec{\uppsi }_{\hat{\theta }} \rangle _{\partial \varOmega ^e}, &{}\quad &{} \mathbf {A}^{(e)}_{\theta q} &{}=&{} \left( \varvec{\uppsi }_{\theta }^T,(\tilde{\varvec{\nabla }}^T \varvec{\uppsi }_q)^T\right) _{\varOmega ^e}, \\ \mathbf {A}^{(e)}_{\theta \theta } &{}=&{} \langle \varvec{\uppsi }_{\theta }^T,\tau \varvec{\uppsi }_{\theta } \rangle _{\partial \varOmega ^e},&{}\quad &{} \mathbf {A}^{(e)}_{\theta \hat{\theta }} &{}=&{} - \langle \varvec{\uppsi }_{\theta }^T,\tau \varvec{\uppsi }_{\hat{\theta }} \rangle _{\partial \varOmega ^e}. \end{array} \end{aligned}$$
(38)

The matrices of global system of HDG and CG in system (13) are defined as,

$$\begin{aligned} \begin{array}{lllllll} \mathbf {A}^{(e)}_{\hat{\theta }\hat{\theta }} &{}=&{} - \langle \varvec{\uppsi }_{\hat{\theta }}^T,\tau \varvec{\uppsi }_{\hat{\theta }} \rangle _{\partial \varOmega ^e}, &{}\quad &{} \mathbf {A}^{(e)}_{\hat{\theta }\theta } &{}=&{} \langle \varvec{\uppsi }_{\hat{\theta }}^T,\tau \varvec{\uppsi }_{\theta } \rangle _{\partial \varOmega ^e}, \\ \mathbf {A}^{(e)}_{\hat{\theta }q} &{}=&{} \langle \varvec{\uppsi }_{\hat{\theta }}^T, (\varvec{n}^T\,\varvec{\uppsi }_q ) \rangle _{\partial \varOmega ^e}, &{}\quad &{} \mathbf {K}^{(e)}_{\theta \theta } &{}=&{} \left( (\varvec{\nabla }\varvec{\uppsi }^C_{\theta } )^T, k_C \varvec{\nabla }\varvec{\uppsi }^C_{\theta } \right) _{\varOmega ^e}. \end{array} \end{aligned}$$
(39)

where \(\varvec{\uppsi }^C_{\theta }\) corresponds to the matrix that has the shape functions of the temperature, \(\theta _C\), in \(\varOmega _C\). Finally, the matrices that arise from the coupling of HDG and CG on the interface, \(\varGamma _I\), in the Eq. (13) can be expressed as follows,

$$\begin{aligned} \begin{array}{lllllll} \mathbf {B}^{(e)}_{\theta \theta } &{}=&{} -\langle \varvec{\uppsi }_{\theta }^T,\tau \varvec{\uppsi }^C_{\theta } \rangle _{\partial \varOmega ^e \cap \varGamma _I}, &{}\,\,\,\,\,\,&{} \mathbf {B}^{(e)}_{q\theta } &{}=&{} \langle (\varvec{n}^T\,\varvec{\uppsi }_q )^T, \varvec{\uppsi }^C_{\theta } \rangle _{\partial \varOmega ^e \cap \varGamma _I}\\ \mathbf {B}^{(e)}_{\theta q} &{}=&{} -\mathbf {B}^{(e)T}_{q\theta } &{} &{} &{} &{} \end{array} \end{aligned}$$
(40)

The definition of elemental matrices that arise from Navier–Stokes equations are already provided in our previous work [38]. The non-linear matrices from the local problem (28) and global problem (29) are defined as follows,

$$\begin{aligned} \begin{array}{lll} \mathbf {C}^{(e)}_{\theta \theta } (\varvec{u}) &{}=&{} - \langle \varvec{\uppsi }_{\theta ,1}^T,u_1 \varvec{\uppsi }_{\theta } \rangle _{\partial \varOmega ^e} - \langle \varvec{\uppsi }_{\theta ,2}^T,u_2 \varvec{\uppsi }_{\theta } \rangle _{\partial \varOmega ^e},\\ \mathbf {C}^{(e)}_{\theta \hat{\theta }} (\varvec{\hat{u}}) &{}=&{} \langle \varvec{\uppsi }_{\theta }^T, (\varvec{\hat{u}} \cdot \varvec{n}) \varvec{\uppsi }_{\hat{\theta }} \rangle _{\partial \varOmega ^e}, \\ \mathbf {C}^{(e)}_{\hat{\theta } \hat{\theta }} (\varvec{\hat{u}}) &{}=&{} \langle \varvec{\uppsi }_{\hat{\theta }}^T, (\varvec{\hat{u}} \cdot \varvec{n}) \varvec{\uppsi }_{\hat{\theta }} \rangle _{\partial \varOmega ^e}. \end{array} \end{aligned}$$
(41)

The tangent operators of the non-linear matrices already presented in (41) are given as,

$$\begin{aligned} \begin{array}{lll} \mathbf {C}^{(e)}_{\theta \theta T} (\theta ) &{}=&{} -\begin{bmatrix} \langle \varvec{\uppsi }_{\theta ,1}^T,\theta \, \varvec{\uppsi }_{\theta } \rangle _{\partial \varOmega ^e} &{} \langle \varvec{\uppsi }_{\theta ,2}^T,\theta \, \varvec{\uppsi }_{\theta } \rangle _{\partial \varOmega ^e} \end{bmatrix},\\ \mathbf {C}^{(e)}_{\theta \hat{\theta }T} (\theta ) &{}=&{} \begin{bmatrix} \langle \varvec{\uppsi }_{\theta }^T, \theta \, n_1\, \varvec{\uppsi }_{\hat{u}} \rangle _{\partial \varOmega ^e} &{} \langle \varvec{\uppsi }_{\theta }^T, \theta \, n_2\, \varvec{\uppsi }_{\hat{u}} \rangle _{\partial \varOmega ^e} \end{bmatrix}, \\ \mathbf {C}^{(e)}_{\hat{\theta } \hat{\theta }T} (\theta ) &{}=&{} \begin{bmatrix} \langle \varvec{\uppsi }_{\hat{\theta }}^T, \theta \, n_1\, \varvec{\uppsi }_{\hat{u}} \rangle _{\partial \varOmega ^e} &{} \langle \varvec{\uppsi }_{\hat{\theta }}^T, \theta \, n_2\, \varvec{\uppsi }_{\hat{u}} \rangle _{\partial \varOmega ^e} \end{bmatrix}. \end{array} \end{aligned}$$
(42)

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Paipuri, M., Tiago, C. & Fernández-Méndez, S. Coupling of Continuous and Hybridizable Discontinuous Galerkin Methods: Application to Conjugate Heat Transfer Problem. J Sci Comput 78, 321–350 (2019). https://doi.org/10.1007/s10915-018-0769-8

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