An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

C.1 Extensions for the H 1 -Space

We recall the transformation from K ̂ to 𝐾 for the H 1 -functions [9],

where ϕ : K ̂ → K is the parametric map from the master element K ̂ to the actual element 𝐾, u ′ the function on K ̂ , and 𝑢 this function shifted to 𝐾. First we define the linear extension of a function u ̂ ′ defined on ∂ K ̂ to the whole element K ̂ ,

Here ξ v are coordinates of vertices, ψ v , v = 1 , … , 8 , are the trilinear shape functions associated with vertices of the master element. Functions μ e extend in a bilinear way the functions defined on 12 edges,

while functions ν s extend linearly the functions defined on faces,

For location of vertices, edges, faces of the hexahedral element, see Figure 12.

Figure 12

Hexahedral element. Location of vertices, edges and faces (bottom, top, left, front, right, back).

To sum up, we first define the trilinear extension u ̂ v ′ of values of u ̂ ′ at vertices, then bilinear extensions u ̂ e ′ of the edge functions vanishing at their endpoints ( u ̂ ′ − u ̂ v ′ ) | e , and finally linear extension u ̂ s ′ of functions on faces ( u ̂ ′ − u ̂ v ′ − u ̂ e ′ ) | s which vanish on boundaries of faces.

We define the extension on 𝐾 by the transformation { u ̂ } e = { u ̂ ′ } e ∘ ϕ − 1 .

C.2 Extensions for the H ⁢ ( div ) -Functions

We recall the Piola transformation for the H ⁢ ( div ) -functions [9],

Here J = ∂ x / ∂ ξ is the Jacobian matrix of the map ϕ : K ̂ → K , and | J | = det J . From this definition, one can find that, for instance, for the face 1 of the element, we have (see [15])

where n ′ = ( 0 , 0 , − 1 ) , and analogously for the remaining 5 faces. This implies the rule for the transformation of the numerical flux f ̂ (interpreted as f ̂ = n ⋅ ( u ⁢ β − σ ) , cf. (2.4)) to the master element,

(for face 1, for the others accordingly). We define a vector-valued function on K ̂ whose trace of the normal component coincides with f ̂ ′ and its 𝑖-th component varies linearly with ξ i ,

with unit vectors of faces of K ̂ ,

and ν s being defined in (C.2).

The H ⁢ ( div ) projection-based interpolation operator I h ′ of order p = 2 acting on { f ̂ ′ } e results in a vector-valued polynomial from the Q ( 2 , 1 , 1 ) × Q ( 1 , 2 , 1 ) × Q ( 1 , 1 , 2 ) -space without the bubble contribution corresponding to the central node because of the construction (C.4): the 𝑖-th component of { f ̂ ′ } e is linear with respect to ξ i . We define the extension on element 𝐾 by shifting the extension on K ̂ with the Piola transformation,

D.1 H 1 -Interpolation Operator on K ̂ and on 𝐾

The interpolation procedure consists of four steps [9].

1. Vertex interpolant:

   u v ′ ⁢ ( ξ ) = ∑ u ′ ⁢ ( ξ v ) ⁢ ψ v ⁢ ( ξ ) ,

   where ψ v are the vertex trilinear shape functions and ξ v the locations of vertices v = 1 , … , 8 , Figure 12.

2. Edge interpolant:

   for all edges ⁢ e , we find ⁢ q e ∈ P 0 p ⁢ ( e ) such that ∫ e ∂ ∂ s ⁢ ( u ′ − u v ′ − q e ) ⁢ d ⁢ ϕ d ⁢ s ⁢ d s = 0 for all ⁢ ϕ ∈ P 0 p ⁢ ( e ) , and we set ⁢ u e ′ ⁢ ( ξ ) = ∑ e = 1 12 q e ⁢ ( s ) ⁢ μ e ⁢ ( ξ ) .

   Here P 0 p ⁢ ( e ) are polynomials of the order 𝑝 on the edge 𝑒 which vanish at its endpoints, 𝑠 is the one of Cartesian coordinates of K ̂ which parametrizes the edge 𝑒, functions μ e are defined in (C.1).

3. Face interpolant:

   for all faces ⁢ s , we find ⁢ q s ∈ Q 0 ( p , p ) ⁢ ( s ) such that ∫ s ∇ s ( u ′ − u v ′ − u e − q s ) ⋅ ∇ s ϕ ⁢ d ⁢ ξ = 0 for all ⁢ ϕ ∈ Q 0 ( p , p ) ⁢ ( s ) , and we set ⁢ u s ′ ⁢ ( ξ ) = ∑ s = 1 6 q s ⁢ ( s ) ⁢ ν s ⁢ ( ξ ) .

   Here Q 0 ( p , p ) ⁢ ( s ) are the bivariate polynomials of the order 𝑝 on a square face, vanishing on its boundary, 𝒔 are the two Cartesian coordinates of K ̂ parametrizing the face 𝑠, functions ν s are defined in(C.2).

4. Middle node interpolant:

   we find ⁢ q m ⁢ ( ξ ) ∈ Q 0 ( p , p , p ) ⁢ ( K ̂ ) such that ∫ K ̂ ∇ ( u ′ − u v ′ − u e ′ − u f ′ − q m ) ⋅ ∇ ϕ ⁢ d ⁢ ξ = 0 for all ⁢ ϕ ∈ Q 0 ( p , p , p ) ⁢ ( K ̂ ) ,

   where Q 0 ( p , p , p ) ⁢ ( K ̂ ) denotes trivariate polynomials of the order 𝑝 vanishing on the boundary of K ̂ .

The H 1 -interpolant for the trace u ̂ ′ boils down to the interpolant above without the middle node contribution,

We note that, just by the definitions above,

that is, the trace of the interpolant is equal to the interpolant of the trace. To define the interpolants on 𝐾, it is convenient to name the shifts of the H 1 -functions from K ̂ to 𝐾 and of their traces as follows:

Then we set

as mentioned in the introductory remarks. Finally, we can state the following property of interpolants on K ̂ and the extension of the trace,

(extension of the interpolant of the trace is equal to the interpolant of the extension of the trace). It is obvious as, for both expressions, we interpolate the same function on ∂ K ̂ . We formally prove the analogous but less obvious result for the H ⁢ ( div ) -interpolants and extensions. Because the extension and the interpolant on 𝐾 are defined as shifts of these operators from the master element K ̂ , property (D.1) holds on 𝐾, too. We also discuss this for the H ⁢ ( div ) case.

D.2 H ⁢ ( div ) -Interpolation Operator on K ̂ and on 𝐾

The procedure of interpolation on a master element K ̂ consists of two steps [9].

1. Interpolant on face 𝑓:

   we find ⁢ q f ∈ Q ( p − 1 , p − 1 ) ⁢ ( f ) such that ∫ f ( e f ′ ⋅ u ′ − q f ) ⁢ ϕ ⁢ d ⁢ s = 0 for all ⁢ ϕ ∈ Q ( p − 1 , p − 1 ) ⁢ ( f ) , and we set ⁢ u s ′ ⁢ ( ξ ) = ∑ f = 1 6 q f ⁢ ( s ) ⁢ ν f ⁢ ( ξ ) ⁢ e f ′ ,

   where 𝒔 are the two Cartesian coordinates of K ̂ parametrizing face 𝑓, functions ν f are defined in (C.2) and vectors e f ′ in (C.5).

2. Middle node interpolant: we find q m ∈ ( Q ( p , p , p ) ) n 0 3 such that

   (D.2) ∫ K ̂ ∇ ⋅ ( u ′ − u s ′ − q m ) ⁢ ∇ ⋅ ϕ ⁢ d ξ = 0 for all ⁢ ϕ ∈ ( Q ( p , p , p ) ) n 0 3 , ∫ K ̂ ( u ′ − u s ′ − q m ) ⋅ ∇ × ϕ ⁢ d ξ = 0 for all ⁢ ϕ ∈ ( Q ( p + 1 , p + 1 , p + 1 ) ) t 0 3 .

   Above, ( Q ( p , p , p ) ) n 0 3 denotes the vector-valued polynomials of the order 𝑝 in each variable whose normal component on ∂ K ̂ vanishes. On the other hand, ( Q ( p + 1 , p + 1 , p + 1 ) ) t 0 3 are the vector-valued polynomials of the order p + 1 with vanishing tangential component on ∂ K ̂ .

The interpolant for the numerical flux f ̂ ′ corresponds to the face contribution considered above:

We note that, like for the case of H 1 -functions, we have

as functions q f in (D.2)₁ and (D.3)₁ are identical. That is, the H ⁢ ( div ) -trace of the interpolant is equal to the interpolant of the H ⁢ ( div ) -trace of u ′ .

To define the interpolants on 𝐾, it is convenient to denote the shifts of the H ⁢ ( div ) -functions from K ̂ to 𝐾 and of their traces as follows:

We used here (2.14)₂ and (C.3) (for the remaining faces, we use adequate pairs of coordinates: ξ 1 , ξ 3 for faces 3, 5, ξ 2 , ξ 3 for faces 4, 6, and face 2 as face 1). Then we set

Finally, we mention the following property of interpolants on K ̂ and the extension of the trace:

(extension of the interpolant of the trace is equal to the interpolant of the extension of the trace). The proof is as follows. We evaluate the left-hand side of (D.5):

We evaluate the right-hand side of (D.5):

(we used here the fact that ν t = 1 on face 𝑡, see (C.2), and e i ′ ⋅ e j ′ = δ i ⁢ j ). By comparing (D.6)₁ and (D.7)₃, we observe that q t = p t on every face 𝑡. Therefore, the expressions in (D.6)₂ and (D.7)₄ are identical.

Property (D.5) is valid on the physical element, too,

To see this, let us express in (D.5) the interpolation operators by their counterparts on 𝐾, I ̂ h ′ = γ ̂ − 1 ⁢ I ̂ h ⁢ γ ̂ and I h ′ = γ − 1 ⁢ I h ⁢ γ ,

We note that the expression on the left-hand side of (D.9)₂ is, by the definitions (C.3) and (C.6) of H ⁢ ( div ) -extensions on 𝐾 (with notation (D.4)), equal to { a ̂ } e = { I ̂ h ⁢ f ̂ } e , while in the expression on the right-hand side, we have b = { f ̂ } e . Therefore, { I ̂ h ⁢ f ̂ } e = I h ⁢ { f ̂ } e .