Numerics

Discretization

We use a discontinuous Galerkin (dG) discretization in space. This is motivated by the stability of this scheme for the advection dominated equation in the fluid domain as ε→ 0, but the use of discontinuous elements also allows the use a single dG function to represent both lumen and wall concentrations.

Furthermore, the interface condition can be easily formulated in the framework of dG jumps and averages, allowing a simple implementation in existing dG code. We refer to [DE12] for an exhaustive introduction into discontinuous Galerkin methods. A similar dG discretization is analyzed in [CGJ13;CGJ16]

with nonlinear coupling conditions. A conforming finite element discretization of a similar equation is analyzed in [QVZ02b] using domain decomposition methods to solve fluid and structure problems separately.

For simplicity we assume that Ωf and Ωs are polygonal domains using a linear interpolation of the continuous deformation field for the interface, even though e.g. isoparametric approaches would be better suited for the curved ge-ometry [BS08]. LetΩh,f be a triangulation ofΩf in the sense thatΩh,f consists of disjoint, open, non-degenerate simplices K ∈ Ωh,f with S

K∈Ω_h,fK = Ωf

and no vertex of any simplex lies in the interior of an edge of another sim-plex. SimilarlyΩ_h,s denotes a triangulation ofΩ_sandΩ_h:= Ωh,f ∪Ω_h,s. The triangulations are assumed to match at the permeable interfaceΓ.

We denote the set of all edgese⊂∂K of allK ∈Ω_h,f byE_h,f and define analogously E_h,s and write E_h for the set of edges in both E_h,f and E_h,s. Edges shared by both E_h,f and E_h,s but not lying on the permeable part of the interface are assumed to be distinct in the sense of a disjoint union. We assume that any edge on the domain boundaries∂Ωf and∂Ωs lies entirely in

∂iΩf, ∂oΩf, ∂wΩf, ∂Ωs\Γ or Γ. We writeE_h^int for the set of interior edges which excludes, by convention, interface edges which we denote by E_h^Γ. The exterior edges are denoted by E_h^ext := Eh\(E_h^int∪E_h^Γ) and subdivided into inflow edgesE_h,fⁱⁿ :={e∈Eh,f |e⊂∂iΩf}and Neumann-type boundary edges E_h,f^N :={e∈Eh,f |e⊂∂oΩf ∪∂wΩf}. The set of exterior boundary edges of E_h,sis denoted byE_h,s^N :={e∈E_h,s|e⊂∂Ω_s\Γ}. We writeE^N_h :=E_h,f^N ∪E_h,s^N for the set of all Neumann-type boundary edges. Exterior edges may lie on (∂Ω_f∩∂Ω_s)\Γ but are, by the disjoint union construction mentioned above, assumed to be adjacent only to one element. For every interior or interface edge e∈E^int_h ∪E^Γ_h we choose an orientation by denoting the two elements sharing that edge byK⁺andK⁻∈Ωhwith outward normal unit vectorsn:=n⁺ and n⁻, respectively. For exterior edges, n⁺ is the outward normal unit vector.

We write

Ch:={ch: Ωh→R|ch|_K ∈P¹(K) ∀K∈Ωh}

for the space of dG functions of first order. As common for discontinuous Galerkin methods we define average and jump of a dG function ch∈Ch over edges. We writec⁺_h andc⁻_h for the continuous extension ofch|_K+, respectively ch|_K−, toe. Jump and average ofch overeare then defined by

JchK:=c⁺_hn⁺+c⁻_hn⁻, {ch}:= 1

2 c⁺_h +c⁻_h

one∈E_h^int∪E_h^Γ

for interior and interface edges. For exterior edges we define

Jc_hK:=c_hn⁺, {c_h}:=c_h one∈E_h^ext.

The gradient of a dG function is not defined over the whole domain, but we use the notation∇hchfor the broken gradient defined as theL²function satisfying

∇hch|_K := ∇(ch|_K) for eachK ∈ Ωh. For vector-valued dG functions, such as ∇hch, the definition of jump and average are analogous, using the scalar product with the normal vector for the definition of the jump.

With the notation for jumps and averages the condition on the permeable interface can be formulated as

Jε∇cεK= 0, {ε∇cε}+εξJcεK= 0 onΓ. (6.61)

The discretization of the advection-diffusion equation in Ωf is standard with a symmetric interior penalty Galerkin method and upwinding for the advective term, see [DE12] for an introduction to these methods. A symmetric interior penalty method is also employed for the diffusion-reaction equation in Ω_s. Using the representation from eqs. (6.61)of the interface condition it can be checked by a lengthy but elementary calculation that the following equation for the discrete solutionc_h,ε∈Ch is consistent:

Z

Ω

ε∇hch,ε· ∇hϕhdx− Z

Ωf

ch,εvf· ∇hϕhdx+ Z

Ωs

ε ch,εϕhdx

− X

e∈E_h^int∪Eⁱⁿ_h,f

Z

e

{ε∇hch,ε}JϕhK+Jch,εK· {ε∇hϕh} −εη

h_eJch,εK·JϕhKdo

+ X

e∈Eh,f\(E_h^Γ∪Eⁱⁿ_h,f)

Z

e

{ch,ε}vf·JϕhK+¹₂|vf·n⁺|Jch,εK·JϕhKdo

+ X

e∈E_h^Γ

Z

e

εξJc_h,εK·Jϕ_hKdo

= X

e∈Eⁱⁿ_h,f

Z

e

−cⁱⁿ_fε∇ϕ·n⁺−v_f·n⁺cⁱⁿ_fϕ+ηε

h_ecⁱⁿ_fϕdo

for all ϕ_h ∈ Ch where η > 0 is a penalty parameter andh_e :=diam(e). The inflow boundary condition is only enforced weakly. We refer to [CGJ13;CGJ16]

for theoretical results for a dG discretization of a similar nonlinear problem.

The discretization for the limit system is very similar, given as solution

ch,0∈Ch to

for all ϕh∈Ch. Apart from the obvious changes due to the missing diffusion in Ωf in the limit problem, note that the interface term only contains the test functionϕh,s from the structure domain. The solution onΩh,f is in particular decoupled fromΩh,s, allowing a more efficient numerical solution.

For the instationary problem an implicit Euler discretization was chosen for both theεand limit system for simplicity. An adaptive step size control as discussed in [HV03] with fixed minimal step size was implemented to reduce the cost after the transient initial phase. We remark that while κ denoted the macro step size in Chapter 5we will keep our notation from the previous theoretical results and refer to the macro step size by name.

The following numerical calculations were carried out using EniCS v2018.1 [Aln+15;LWH12;Log+12;Aln+14;Kir04;ALM12].

Numerical Test Case

To observe the effect of increasing stenosis on the quality of the approximation ofcεbyc0we will use a family of 2D fluid domains as defined inChapter 5, with complementary structure domains, with a growth parameter q ∈Q := [0,1).

No growth actually occurs, so qis a fixed parameter.

Forq= 0both fluid and structure domains are rectangular and given by Ω0,f := (−^L₂,^L₂)×(0,1), Ω0,s:= (−^L₂,^L₂)×(−0.3,0)

with L:= 5. For q∈Qwe define Ωq,f and Ωq,s through a transformation of those rectangular domains, just as in Chapter 4 and Chapter 5. Let η be a bump function as inExample 4.1.2, i.e.

η(s) :=

3.7e-6 1.2e-4

1e-5

Figure 6.2: Area of triangles for a discretization of Ωq,f ∪Ωq,s for q = 0.3.

Logarithmic color scale.

with plaque length L₀ := 3.5. Φ_q is a linear interpolation in y between the fixed boundaryy= 1inΩ0,f, respectivelyy=−0.3inΩ0,s, and the graph of a bump-shaped function at the interfacey= 0. Then define fluid and structure domains by

Ωq,f := Φq(Ω0,f), Ωq,s:= Φq(Ω0,s).

Note thatΩf,qcorresponds toΩqas defined inChapter 5. In the following there will always hold Ω_f = Ω_q,f and Ω_s = Ω_q,s for some q ∈ Q. The boundaries are just as shown inFigure 6.1at the beginning of this section with permeable interfaceΓ₀:= (−^L₂⁰,^L₂⁰)×{0}spanning the whole section which gets deformed andΓ_q:= Φq(Γ₀).

Forq= 0we will use the Poiseuille-like flow fields v0(x, y) =v0(y^κ(1−y)^κ,0)^T in Ω0,f

with κ ∈ N0 as advection velocity, v = v0. Such velocity fields were found in Example 6.2.14 to satisfy Assumption 6.2.12 such that our theory with vanishing velocities is applicable. The advection field forq∈Qisv=vq where vq is the inverse Piola transformation ofv0fromeq. (4.5), preserving the vector field’s solenoidality and vanishing behavior near the interface, i.e.

v_q(x, y) := (P_q⁻¹v₀)(x, y) in Ω_q,f.

Following the model by Yang et. al. we assume thatcⁱⁿ_f ≡1. As a conse-quence the fluid concentration in the limit solution is known a priori,c0,f ≡1, such that only the structure concentrationc0,s must be determined.

For the instationary problem we setc⁰_ε,s :=c⁰_s := 0. On the fluid domain we set either c⁰_ε,f := 0 or c⁰_ε,f := 1 to investigate the occurrence of boundary layers if the initial value and c_0,f disagrees, as predicted inTheorem 6.3.5.

The computational meshes were generated by triangulating the fluid and structure domains, delimited by the exterior boundaries and∂Ω_q,f∩∂Ω_q,susing Gmsh [GR09]. The triangulation was manually refined around the interface to better resolve the boundary layer. This meshing approach guarantees an even triangulation in both subdomains, in contrast to the method from Chapter 5 where a (structured) reference mesh forq= 0was deformed byΦq. We present results for q ∈ {0,0.3,0.6} with similar observed behavior for other values of q. The size of mesh elements forq= 0.3 is displayed inFigure 6.2. This mesh consisted of 438 390 degrees of freedom, 280 566 in the fluid and 157 824 in the structure domain, the other meshes had a similar total number of degrees of freedom with decomposition in fluid and structure nodes depending onq.

Results for the Stationary Problem

Figure 6.3shows the solutioncεof the stationary problem forq= 0andκ= 0.

Forκ∈ {0, . . . ,3} the error between cε andc0 is in good agreement with the theoretical results fromTheorem 6.2.11forκ= 0andTheorem 6.2.17forκ >0 as evident fromFigure 6.4, noting that sincec_0,f ≡1the improved estimates from ineqs. (6.9), respectively ineqs. (6.25), hold. In Figure 6.4 and in the following k · k²_H¹

0(Ωf) = k∇ · k²L²(Ω_f). The numerical order of convergence is directly compared to the theoretical predictions inFigure 6.4(b), showing only slight under-estimation of the predicted order of convergence for the L²(Ωf) error for largeκ.

For stenosed domains with q > 0 similar good agreement with the theo-retical results was found, cf. Figure 6.6. Only for κ = 0 the H₀¹(Ωf) errors deviate from the predicted results, presumably due to numerical errors tangen-tial to the permeable interface, most pronounced at the beginning and end of the bump as shown in Figure 6.7. For κ = 0 the simulation was in general most sensitive to the mesh refinement and quality near the interface, which is unsurprising given that the advection is affecting the fluid concentration near the interface with full force forκ= 0, an unrealistic configuration. Note that it is not apparent that the discrete solution possesses the sameH₀¹(Ω_f)stability as the continuous solution, which rested onAssumption 6.2.2.

Results for the Instationary Problem

As mentioned above we use an implicit Euler discretization for the temporal discretization of both εand limit system on a time interval ofI = (0,T)with T = 1. A step size control is employed to reduce the cost after the initial transient phase, with a minimal macro step size of 0.1·ε. For simplicity of implementation the step size control is only employed for theεsystem and the solution of the limit equation which is compared to this solution employs the same temporal discretization. This is not considered a problem since the step size for the pure diffusion problem onΩsin the limit equation is not a bottleneck for the accuracy. Figure 6.8 shows the step sizes used for our computation.

The spatial discretization, including the underlying mesh, is identical to the stationary problem but only the domain with q= 0is investigated.

InFigure 6.9the error for the instationary problem withκ= 0andc⁰_ε,f = 0, i.e. an initial value not satisfying the stronger condition from ineq. (6.43), is plotted in various space-time norms to test the results from Theorem 6.3.5.

These numerical results for the fully discrete equation mostly agree with the predictions from Theorem 6.3.5, with the exception that the L²(I, H₀¹(Ωf)) error is of order ε^1/4 even though the boundary layer is included, instead of ε⁰ as predicted. Furthermore, the order of convergence for the norms inΩ_s is with approximatelyε^0.58 slightly better than the predictedε^1/2.

The boundary layer estimates require knowledge ofτ0 which by eq. (6.42) depends on an unknown constant µ0. The boundary layer behavior for the L²(Ωf) norm for κ = 0 and c⁰_ε,f ≡ 1 is depicted in Figure 6.10, making it apparent that for ε = 0.1 the boundary layer with transient behavior spans across the whole time interval. We set µ0 = 1/8 such that τ0 ≈ −2εlnε, omittingη, lies behind the transient phase forε <0.1.

1.0

Figure 6.3: Solution c_ε of the stationary problem for κ = 0 and multiple values ofq andε.

1e-04 1e-03 1e-02 1e-01 1e+00

1e-05 1e-04 1e-03 1e-02 1e-01

‖ cε - c0‖

ε

L²(Ω_f) H¹₀(Ω_f) H¹(Ω_s)

ε^3/4 ε^1/4 ε^1/2

(a) Errors forκ= 0.

1e-04 1e-03 1e-02 1e-01 1e+00

1e-05 1e-03 1e-01

‖ cε - c0‖

ε κ = 1

ε^3/6 ε^1/6 ε^1/3

1e-05 1e-03 1e-01

ε κ = 2

ε^3/8 ε^1/8 ε^1/4

1e-05 1e-03 1e-01 ε

κ = 3

ε^3/10 ε^1/10 ε^1/5

(b) Errors forκ∈ {1,2,3}. Solid lines with the same meaning as in (a).

Figure 6.4: Error between the numerical approximations ofc_εandc₀for the stationary problem on Ωq for q = 0 for different norms with κ= 0, . . . ,3. The dashed lines show the predicted convergence rates fromineqs. (6.9)in Theorem 6.2.11.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 1 2 3

order

κ order of L²(Ω_f) error order of H¹₀(Ω_f) error order of H¹(Ω_s) error

3/(4+2κ) 1/(4+2κ) 1/(2+κ)

Figure 6.5: Numerically determined order of convergence inεfromFigure 6.4 for q = 0and different values of κin comparison with the pre-dicted orders fromTheorem 6.2.17.

Figure 6.11 complements Figure 6.9 by showing the errors from Theo-rem 6.3.5 which exclude the boundary layer, with τ0 as specified above. In contrast to the theoretical predictions theL^∞(I0, L²(Ωf))error is of the same order ε^3/4 as the L²(I0, L²(Ωf)) error, which itself is improved compared to ε^1/2forL²(I, L²(Ωf)), cf.Figure 6.9, as predicted by the theory. The exclusion of the boundary layer does not change the convergence for the H₀¹(Ω_f)error, which was already better than expected inFigure 6.9, as remarked above, but has the predicted order if the boundary layer is excluded. As mentioned in the discussion of Figure 6.10 the behavior of the error at ε = 0.1 should be ignored. Another choice ofµ₀ did not improve the convergence but the results deteriorate asµ₀→ ∞to those on the whole intervalI, as expected.

For the initial value c⁰_ε,f ≡ 1, which is a plausible choice in the model by Yang et. al., no boundary layer appears as predicted by the theory, cf.

Figure 6.12. The convergence behavior is as predicted in Theorem 6.3.5 with the exception of the error in L^∞(I, L²(Ωf)) which is of order ε^3/4 just as observed for c⁰_ε,f ≡0. As expected the exclusion of a boundary layer in the error norms does not improve the convergence in this case (not pictured).

Forκ= 1 the discrepancies between numerical and theoretical results in-crease. As Figure 6.13(a) shows the error in L²(I, L²(Ωf)) converges with rate ε^0.47 instead of ε^1/3 as predicted inTheorem 6.3.5, which is close to the order of convergence ε^3/6 =ε^1/2 predicted if the boundary layer is excluded, while the error in L²(I, H₀¹(Ω_f)) again behaves as predicted by our theory if the boundary layer is excluded. The convergence for the errors inΩ_s is again slightly better than predicted with ε^0.41 compared to ε^1/3. If the boundary layer is excluded, Figure 6.13(b), the behavior is very similar to κ= 0, with good agreement with the theoretical results except for theL^∞(I0, L²(Ωf)) er-ror which converges with the same rate as the temporalL²error. Similar mixed agreement with the theoretical results was observed forκ >1.

1e-04

Figure 6.6: Error between c_ε and c₀ for the stationary problem for q ∈ {0.3,0.6} andκ∈ {0,1,2}. Compare withFigure 6.4forq= 0.

6e-7 magnifica-tion of the error near the permeable interface. Colored by error magnitude in logarithmic scale.

Figure 6.8: Macro step size used for our multiscale convergence computa-tions, as determined from the step size control.

1e-04 1e-03 1e-02 1e-01 1e+00

1e-05 1e-04 1e-03 1e-02 1e-01

‖ cε - c0‖

ε

L²(𝓘, L²(Ω_f)) L^∞(𝓘, L²(Ω_f)) L²(𝓘, H¹₀(Ω_f)) L^∞(𝓘, L²(Ω_s)) L²(𝓘, H¹(Ωs))

ε^1/2

ε^1/4 ε^0.58

Figure 6.9: Error betweenc_εandc₀ for the instationary problem withκ= 0 and c⁰_ε,f ≡ 0 for various norms including the boundary layer, together with some convergence orders.

1e-04 1e-03 1e-02 1e-01 1e+00 1e+01

1e-06 1e-05 0.0001 0.001 0.01 0.1 1

‖ cε(τ) - c0(τ) ‖f

τ

ε = 1e-01

ε = 1e-02 ε = 1e-03 ε = 1e-04 ε = 1e-05 τ_0,1e-01

τ_0,1e-01 τ_0,1e-02 τ_0,1e-02 τ_0,1e-03

τ_0,1e-03 τ_0,1e-04

τ_0,1e-04 τ_0,1e-05

τ_0,1e-05

Figure 6.10: Pointwise-in-time error betweencε and c0 for the instationary problem withκ = 0 and c⁰_ε,f ≡ 0 for the L²(Ωf) norm. The boundary layer timeτ0 withµ0:= 1/8for eachεis also plotted (dashed lines).

1e-04 1e-03 1e-02 1e-01 1e+00

1e-05 1e-04 1e-03 1e-02 1e-01

‖ cε - c0‖

ε

L²(𝓘₀, L²(Ω_f)) L^∞(𝓘0, L²(Ωf)) L²(𝓘₀, H¹₀(Ω_f))

ε^3/4 ε^1/4

Figure 6.11: Error betweencεandc0for the instationary problem withκ= 0 andc⁰_ε,f ≡0for various norms excluding the boundary layer for τ0 as depicted in Figure 6.10, together with some convergence orders.

1e-04 1e-03 1e-02 1e-01 1e+00

1e-05 1e-04 1e-03 1e-02 1e-01

‖ cε - c0‖

ε

L²(𝓘, L²(Ω_f)) L^∞(𝓘, L²(Ω_f)) L²(𝓘, H¹₀(Ω_f)) L^∞(𝓘, L²(Ω_s)) L²(𝓘, H¹(Ω_s))

ε^3/4

ε^1/4 ε^1/2

Figure 6.12: Error betweencεandc0for the instationary problem withκ= 0 andc⁰_ε,f ≡1for various norms, together with some convergence orders.

1e-03 1e-02 1e-01 1e+00

1e-05 1e-04 1e-03 1e-02 1e-01

‖ cε - c0‖

ε

L²(𝓘, L²(Ω_f)) L^∞(𝓘, L²(Ω_f)) L²(𝓘, H¹₀(Ω_f)) L^∞(𝓘, L²(Ω_s)) L²(𝓘, H¹(Ω_s))

ε^0.47

ε^1/6 ε^0.41

(a) Errors including the boundary layer.

1e-03 1e-02 1e-01 1e+00

1e-05 1e-04 1e-03 1e-02 1e-01

‖ cε - c0‖

ε

L²(𝓘₀, L²(Ω_f)) L^∞(𝓘₀, L²(Ω_f)) L²(𝓘₀, H¹₀(Ω_f))

ε^3/6 ε^1/6

(b) Errors excluding the boundary layer.

Figure 6.13: Error betweenc_ε andc₀ for the instationary problem with q= 0, κ = 1 and c⁰_ε,f ≡ 0 for various norms, together with some convergence orders.

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