A Phase-Space Discontinuous Galerkin Scheme for the Radiative Transfer Equation in Slab Geometry

2.1 Weak Formulation and Solvability

Performing the usual integration-by-parts, see, e.g., [6, 39], the weak formulation of (1.7)–(1.8) is as follows: find u ∈ V such that

with bilinear form a e : V × V → ℝ ,

Here, for ease of notation, we use the scattering operator P : L 2 ⁢ ( Ω ) → L 2 ⁢ ( Ω ) ,

Using the Cauchy–Schwarz inequality, we deduce that ∥ P ⁢ u ∥ L 2 ⁢ ( Ω ) ≤ ∥ u ∥ L 2 ⁢ ( Ω ) for u ∈ L 2 ⁢ ( Ω ) . Assuming

for some c > 0 , we therefore obtain that the bilinear form a e is V-elliptic, and, in view of the trace theorem, cf. (2.1), bounded. Similarly, for f ∈ L 2 ⁢ ( Ω ) and g ∈ L 2 ⁢ ( Γ ; μ ) , the right-hand side in (2.2) defines a bounded linear functional on V. Hence, there exists a unique weak solution u ∈ V of (2.2) by the Lax–Milgram lemma, see also [6], [39, Theorem 3.3] or [19, Section 5.3] for similar well-posedness statements.

If f ∈ L 2 ⁢ ( Ω ) , testing (2.2) with functions in C 0 ∞ ⁢ ( Ω ) shows that μ σ t ⁢ ∂ z ⁡ u has a weak μ ⁢ ∂ z -derivative in L 2 ⁢ ( Ω ) and (1.7) holds a.e. in Ω. In particular, μ σ t ⁢ ∂ z ⁡ u ∈ V and μ σ t ⁢ ∂ z ⁡ u has a trace. For v ∈ V , an integration by parts in (2.2) then shows that

Since the trace operator is surjective from V to L 2 ⁢ ( Γ ; μ ) (see [2, Theorem 2.9]), it follows that (1.8) holds in L 2 ⁢ ( Γ ; μ ) . We denote the space of solutions with data f ∈ L 2 ⁢ ( Ω ) and g ∈ L 2 ⁢ ( Γ ; μ ) by

3.1 Mesh and Broken Polynomial Spaces

In order to simplify the presentation, and subsequently the implementation, we consider quad-tree meshes [23] as follows. Let 𝒯 be a partition of Ω such that σ t is constant on each element K ∈ 𝒯 , and that

for illustration see Figure 1. We denote the local mesh size by h K = z K r - z K l .

Next, let us introduce some standard notation. Denote ℙ k the space of polynomials of one real variable of degree k ≥ 0 , and let the broken polynomial space V h be denoted by

with k z , k μ ≥ 0 . Here, ℙ k z + 1 ⊗ ℙ k μ denotes the tensor product of ℙ k z + 1 and ℙ k μ . Moreover, let V ⁢ ( h ) ≔ V + V h . By ℱ h v i we denote the set of interior vertical faces, that is for any F ∈ ℱ h v i there exist two disjoint elements

such that z F = z 1 r = z 2 l and F = { z F } × ( ( μ 1 l , μ 1 r ) ∩ ( μ 2 l , μ 2 r ) ) . Accordingly, we denote by ℱ h v b the set of vertical boundary faces and by ℱ h v the set of all vertical faces in the mesh. For F ∈ ℱ h v i we define the jump and the average of v ∈ V h by

and

In order to take into account local variations in the mesh size and diffusion coefficient 1 σ t , we furthermore define the dimensionless quantity

where h K i , i ∈ { 1 , 2 } , denotes the local mesh size of the element K i in z-direction. For an interior face F ∈ ℱ h v i with F = { z F } × ( μ F b , μ F t ) , which is shared by two elements K F i ∈ 𝒯 , i = 1 , 2 , as above, let us introduce the sub-elements

We note that the inclusion in (3.3) can be strict in the case of hanging nodes, see for instance Figure 1.

Combining Lemma 1 with common inverse inequalities, cf. [8, Sect. 4.5], i.e., for any k ≥ 0 there exists a constant C i ⁢ e ⁢ ( k ) such that

we obtain the following discrete trace lemma.

3.2 Derivation of the DG Scheme

In order to extend the bilinear form defined in (2.3) to the broken space V h , we denote with ∂ z h the broken derivative operator such that

for u h , v h ∈ V h . In view of (2.2), let us then introduce the bilinear form

which is defined on V ⁢ ( h ) . Note that a e and a h e coincide on V. In order to obtain a consistent bilinear form, a h e needs to be modified. We follow [16, Chapter 4] to determine the required modification. Choosing w ∈ V * h ≔ V * + V h and v h ∈ V h , integration by parts in z shows that

where we used the identity [ [ μ σ t ⁢ ∂ z h ⁡ w ⁢ v h ] ] = { { μ σ t ⁢ ∂ z h ⁡ w } } ⁢ [ [ v h ] ] + [ [ μ σ t ⁢ ∂ z h ⁡ w ] ] ⁢ { { v h } } in the last step, see [16, p. 123]. Hence, for any solution u ∈ V * to (1.7)–(1.8) and v ∈ V h we have that

Since [ [ μ σ t ⁢ ∂ z h ⁡ u ] ] = 0 for all F ∈ ℱ h v i by z-continuity of the flux of u ∈ V * , we arrive at the identity

Hence, a consistent bilinear form is given by

which, for V * h ≔ V * + V h , is well-defined on V * h × V h . Using that [ [ u ] ] = 0 on F ∈ ℱ h v i for any u ∈ V , we arrive at the following symmetric and consistent bilinear form:

which is again well-defined on V * h × V h . We note that the summation over the vertical faces on the boundary Γ is included in the term 〈 u , v 〉 in a h e . The stabilized bilinear form is then defined on V * h × V h by

with D F , σ defined in (3.2) and with positive penalty parameter α F > 0 , which will be specified below. Since [ [ u ] ] = 0 on any F ∈ ℱ h v i and u ∈ V , it follows that a h is consistent, i.e., for u ∈ V * it holds that

The discrete variational problem is formulated as follows: Find u h ∈ V h such that

3.3 Analysis

For the analysis of (3.7), let us introduce mesh-dependent norms

(3.8)

In order to show discrete stability and boundedness of a h , we will use the following auxiliary lemma.

The auxiliary lemma allows to bound the consistency terms in a h , which gives discrete stability of a h .

Discrete stability implies that the scheme (3.7) is well-posed, cf. [16, Lemma 1.30].

To proceed with an abstract error estimate, we need the following boundedness result.