Temporal Error Analysis for the Limit System

In this section we sketch an error analysis for the limit system. In line with our general focus on the temporal aspects of the problem we only discuss a temporal semi-discretization of the limit equation with a space-continuous fluid problem.

For the equation forq0on the macro scale we will use the Runge–Kutta 2 (RK2) method from eq. (5.5), for the periodic fluid problem on the micro scale the Crank–Nicolson scheme is employed.

The error analysis is non-standard due to the coupling of micro and macro scales and the discretization of the time-periodic fluid equation. While the presented analysis is new, the basic principles of micro and macro coupling are also discussed e.g. in the context of the heterogeneous multiscale method [E+07]. To focus on the non-standard aspects of the error analysis, we will simplify the micro problem to a Stokes equation and make assumptions on the regularity of the solution and limit right-hand side g₀ which are expected for the error analysis, likely to hold, but are not verified. This is the sense in which this is merely a sketch of the error analysis. Constants occurring in the rest of this section, either explicitly or implicitly in the ≲-notation, will be independent of the discretization parameters but may depend on the limit solution and the data.

Simplified Limit System and Discretization

For simplicity we only study a Stokes equation with time-periodic right-hand side, Strouhal and Reynolds numbers equal to 1, just as in the discussion of the mean correction iterations, cf.eq. (5.11). Specifically, we denote byv_π(·; ˆq) forqˆ∈Qin this section the solution to

∂svπ(s; ˆq)−P∆vπ(s; ˆq) =P f(s; ˆq), divv_π(s; ˆq) = 0

)

in(0,1)×Ωˆq (5.42a) with Helmholtz projectionP and 1-periodic right-hand side f(·; ˆq). The solu-tionvπ(·; ˆq)must be1-periodic,

vπ(0; ˆq) =vπ(1; ˆq) in Ωˆq (5.42b) and satisfy the homogeneous Dirichlet boundary values

v_π(s) = 0 on(0,1)×∂Ω_qˆ. (5.42c) The calculations carried out in the following will work the same way with pressure boundary conditions. We make various regularity assumptions in the

following which require regularity of ∂Ωqˆ, f and g0 greater than those previ-ously assumed, e.g. differentiability ofg0. Such assumptions are natural for an error analysis but are not verified here.

The limit equation forq0is assumed to be just as eq. (5.3a), i.e.

q^′₀(τ) =g0(q0(τ)) (5.43a) with initial valueq₀(0) := 0and right-hand side

g0(ˆq) :=γ(1−q)ˆ

whereσ_WS,π(·; ˆq)is the wall shear stress of the simplified Stokes equation and Γ_ˆq⊂∂Ω_qˆhas positive measure and is sufficiently regular.

An approximation of the right-hand sideg₀ is evaluated twice per macro step in the RK2 discretization of the limit equation, we denote these approxi-mations by˜g₀⁽ⁱ⁾fori∈ {1,2}. The analysis will show that the use of different micro discretizations is more efficient and we denote byM⁽ⁱ⁾∈Nthe number of micro time steps per period with step sizek⁽ⁱ⁾:= 1/M⁽ⁱ⁾and byTOL⁽ⁱ⁾>0 a periodicity tolerance parameter, see ineq. (5.46) below. For notational sim-plicity will however often omit the indexiand just writeM,k andTOL.

The temporal discretization of the simplified Stokes equation is a Crank–

Nicolson scheme analogous to the one used for the computations, wherev_π^m(ˆq) form= 1, . . . , M is a solution to

For the RK2 discretization of the limit equation there must hold for n= 1, . . . ,N that

qⁿ₀ =q₀ⁿ⁻¹+κ˜g₀⁽²⁾(q₀ⁿ⁻¹+^κ₂g˜₀⁽¹⁾(q₀ⁿ⁻¹)) (5.45a) with q⁰₀ = 0. The numbers 1 and 2 here refer to the order of evaluation. We will mainly investigate the discretization ofg₀ fromeq. (5.7),

˜ whereσ˜_WS,π^m is the wall shear stress of the time-discrete, but only approxima-tively periodic solutionv˜^m_π(ˆq). This discretization ofg0will be beneficial since

the resulting velocity error can be easily estimated for the Crank–Nicolson scheme, we will later also discuss the corresponding estimates for other dis-cretizations ofg0.

We assume that there is a periodicity toleranceTOL=TOL⁽ⁱ⁾>0 which guarantees that

k˜v^m_π(ˆq)−v_π^m(ˆq)kH²(Ω_qˆ)≤TOL (5.46) for m= 1, . . . , M and qˆ∈ Q. For the analysis such a criterion is reasonable, but the periodic solver will not guarantee such a bound, in particular since the periodic solution is unknown.

Macro Scale Error

The error is split into the errors due to this inexact approximation of the right-hand side g0 and the normal error analysis for the RK2 scheme. Subtracting eqs. (5.43a) and(5.45a)forq0(τ)andq₀ⁿ, the erroreⁿ :=q0(τⁿ)−qⁿ₀ satisfies the recursion

eⁿ=eⁿ⁻¹+ Z τⁿ

τⁿ⁻¹

g0(q0(τ))dτ−κ˜g₀⁽²⁾(q₀ⁿ⁻¹+^κ₂g˜₀⁽¹⁾(q₀ⁿ⁻¹)) and we split the right-hand side as follows

eⁿ=eⁿ⁻¹+ Z τⁿ

τⁿ⁻¹

g0(q0(τ))dτ−κg0(q0(τⁿ⁻¹) +^κ₂g0(q0(τⁿ⁻¹))) +κg0(q0(τⁿ⁻¹) +^κ₂g0(q0(τⁿ⁻¹)))−κg0(q₀ⁿ⁻¹+^κ₂g0(q₀ⁿ⁻¹))

+κg0(q₀ⁿ⁻¹+^κ₂g0(q₀ⁿ⁻¹))−κ˜g⁽²⁾₀ (q₀ⁿ⁻¹+^κ₂˜g⁽¹⁾₀ (q₀ⁿ⁻¹)). (5.47) The second and third terms on the right of eq. (5.47) are the standard local truncation error for the RK2 method and of order O(κ³), verified e.g. using Taylor polynomials, assuming thatq0 is sufficiently regular. The term on the second line of eq. (5.47)is estimated by

κ|g0(q0(τⁿ⁻¹) +^κ₂g0(q0(τⁿ⁻¹)))−g0(qⁿ₀⁻¹+^κ₂g0(qⁿ₀⁻¹))|

≲κ|eⁿ⁻¹+^κ₂g₀(q₀(τⁿ⁻¹))−^κ₂g₀(q₀ⁿ⁻¹)|

≲κ|eⁿ⁻¹|+κ²|eⁿ⁻¹|.

The final line ofeq. (5.47) measures the inexact solution of the micro problem and is split up as

κ|g0(q₀ⁿ⁻¹+^κ₂g0(q₀ⁿ⁻¹))−g˜₀⁽²⁾(q₀ⁿ⁻¹+^κ₂g˜₀⁽¹⁾(q₀ⁿ⁻¹))|

≤κ|g0(qⁿ₀⁻¹+^κ₂g0(q₀ⁿ⁻¹))−g0(q₀ⁿ⁻¹+^κ₂g˜₀⁽¹⁾(q₀ⁿ⁻¹))| +κ|g0(q₀ⁿ⁻¹+^κ₂˜g₀⁽¹⁾(q₀ⁿ⁻¹))−g˜₀⁽²⁾(q₀ⁿ⁻¹+^κ₂g˜₀⁽¹⁾(q₀ⁿ⁻¹))|

≲κkg0−g˜₀⁽²⁾kC⁰(Q)+κ²|g0(qⁿ₀⁻¹)−˜g₀⁽¹⁾(q₀ⁿ⁻¹)|, using the Lipschitz continuity ofg0, and consequently

κ|g₀(qⁿ₀⁻¹+^κ₂g₀(qⁿ₀⁻¹))−˜g⁽²⁾₀ (qⁿ₀⁻¹+^κ₂˜g⁽¹⁾₀ (qⁿ₀⁻¹))|

≲κkg₀−g˜⁽²⁾₀ kC⁰(Q)+κ²kg₀−g˜⁽¹⁾₀ kC⁰(Q).

Combining these estimates foreq. (5.47)leads to

|eⁿ|≲(1 +κ+κ²)|eⁿ⁻¹|+κ³+κkg0−g˜⁽²⁾₀ kC⁰(Q)+κ²kg0−g˜₀⁽¹⁾kC⁰(Q)

and Gronwall’s inequality hence implies that

|eⁿ|≲κ²+κkg₀−˜g⁽¹⁾₀ kC⁰(Q)+kg₀−g˜⁽²⁾₀ kC⁰(Q) (5.48) with constant depending on the solution, properties ofg₀and the slow timeT. Inequality (5.48)shows that the (micro) error in the evaluation of˜g₀⁽²⁾must be of orderO(κ²), while the evaluation ofg˜⁽¹⁾₀ must be only of orderO(κ)for an

g₀⁽ⁱ⁾ fromeq. (5.45b) we can rewrite both functions usingg from the fast-slow system, given by

for sufficiently regular velocity field v: ˆΩ_q → R^d with associated wall shear stressσ_WS(v). The micro discretization error can then be expressed as

g0(ˆq)−g˜₀⁽ⁱ⁾(ˆq) =

We split this expression into integration error, discretization error and period-icity error: The second line ofineq. (5.49)describes the error for the midpoint integration rule, which is of second order. Thus for sufficiently regularv_π(·; ˆq)there holds

|

The third line of ineq. (5.49) is, by Lipschitz continuity of g and the H² -dependence on the velocity, estimated by

XM m=1

k|g(ˆq,¹₂(vπ(s^m; ˆq) +vπ(s^m−1; ˆq)))−g(ˆq,¹₂(v^m_π(ˆq) +v_π^m−1(ˆq)))|

≲ XM m=1

kk¹₂(w^m_π(ˆq) +w^m_π⁻¹(ˆq))kH²(Ωqˆ)

where we writew_π^m(ˆq) :=v_π(s^m; ˆq)−v_π^m(ˆq) for the error between continuous and time-discrete periodic solutions. Before investigating this velocity error, we estimate the last line ofineq.(5.49)using our periodicity tolerance condition from ineq. (5.46), which yields

XM m=1

k|g(ˆq,¹₂(v_π^m(ˆq) +v_π^m−1(ˆq)))−g(ˆq,¹₂(˜v_π^m(ˆq) + ˜v^m_π⁻¹(ˆq)))|

≲ XM m=1

k kv^m_π(ˆq)−˜v^m_π(ˆq)kH²(Ω_ˆq)+kv^m_π⁻¹(ˆq)−v˜_π^m−1(ˆq)kH²(Ω_ˆq)

≲TOL.

Combining these estimates for the right-hand side ofineq. (5.49)we get

|g0(ˆq)−˜g⁽ⁱ⁾₀ (ˆq)|≲k²+TOL+ XM m=1

kk¹₂(w_π^m(ˆq) +w^m_π⁻¹(ˆq))kH²(Ωqˆ). (5.50) The final part of the error analysis will be the estimate for the velocity error w^m_π(ˆq). Subtracting eq. (5.44)fromeq. (5.42) we arrive at the error equation

w^m_π(ˆq)−w^m_π⁻¹(ˆq)−^k₂P∆(w^m_π(ˆq) +w^m_π⁻¹(ˆq)) =r^m(ˆq), divw_π^m(ˆq) = 0,

)

inΩqˆ

form= 1, . . . , M with residual r^m(ˆq) :=

Z s^m s^m−1

∂svπ(s; ˆq)ds−^k₂(∂svπ(s^m; ˆq) +∂svπ(s^m−1; ˆq)).

The error is 1-periodic, w_π⁰(ˆq) =w^M_π (ˆq), andw_π^m= 0on∂Ω_qˆ.

Testing the momentum equation with−P∆¹₂(w^m_π(ˆq) +w^m_π⁻¹(ˆq))and inte-grating in space yields, with some standard estimates, that

kw_π^m(ˆq)k²H¹(Ω_qˆ)− kw^m_π⁻¹(q)k²H¹(Ω_qˆ)+kkP∆¹₂(w_π^m(ˆq) +w^m_π⁻¹(ˆq))k²L²(Ω_qˆ)

≲k⁴k∂ttvπk²L²(s^m−1,s^m;L²(Ω_qˆ))

using that the residual corresponds to the local error for the trapezoidal in-tegration scheme. Summing over m = 1, . . . , M, the estimate k · kH²(Ωqˆ) ≲ kP∆· kL²(Ω_qˆ)under our domain regularity assumptions and the periodicity of w^m_π(ˆq)yields that

XM m=1

kk¹₂(w^m_π(ˆq) +w_π^m−1(ˆq))k²H²(Ω_qˆ)≲k⁴. (5.51)

Just as in the continuous theory a temporal L^∞estimate is not directly avail-able with periodic boundary conditions in time. This finishes the estimate for the micro error since

XM m=1

kk¹₂(w^m_π(ˆq) +w_π^m−1(ˆq))kH²(Ω_qˆ)≤ XM m=1

kk¹₂(w_π^m(ˆq) +w^m_π⁻¹(ˆq)k²H²(Ωˆq)

!¹₂

and thus we can conclude ineq. (5.50) that

|g0(ˆq)−˜g⁽ⁱ⁾₀ (ˆq)|≲k²+TOL. (5.52) The error estimate from ineq. (5.51) as a time-discrete L² integral of av-erages is somewhat natural for the Crank–Nicolson scheme, see [SR20]. The discretization ofg₀fromeq. (5.7)was chosen for this analysis since it naturally leads to such averages in the velocity error. We briefly discuss the estimates for the discretization scheme ofg₀fromeq. (5.6), which is used in the computa-tions, for the remainder of this section. Such discretization leads to the micro error estimate

|g0(ˆq)−˜g⁽ⁱ⁾₀ (ˆq)|≲k²+TOL+ XM m=1

kkw_π^m(ˆq)kH²(Ω_qˆ)

and a velocity error estimate in this form is not readily available for the Crank–

Nicolson scheme. One strategy to derive such a result is using a time-discrete version of the (temporal) Poincaré inequality

kv_π(ˆq)−v_π(ˆq)kL²(0,1;H²(Ω_qˆ)) ≲k∂_sv_π(ˆq)kL²(0,1;H²(Ω_qˆ))

wherevπ(ˆq)is the temporal average of the periodic solutionvπ(·; ˆq). By testing the error momentum equation by ¹_k(w_π^m(ˆq)−w_π^m−1(ˆq))one can derive anO(k² )-estimate for the discrete time-derivative. By summing the error momentum equation over m = 1, . . . , M one can derive an O(k²)-estimate for the time-discrete average. Together with the Poincaré inequality this yields the desired estimate for the velocity error. We note that this approach requires much more regularity for the continuous solution to compensate for the mismatch between the spatial regularity of the time-derivative and the solution itself.

Overall Error Estimate

To finish the error estimate we must combine the macro error estimate from ineq. (5.48),

|eⁿ|≲κ²+κkg₀−˜g⁽¹⁾₀ kC⁰(Q)+kg₀−g˜₀⁽²⁾kC⁰(Q), and the micro error estimate fromineq. (5.52),

|g0(ˆq)−˜g⁽ⁱ⁾₀ (ˆq)|≲k²+TOL.

Distinguishing now between the micro parameters for the two evaluations˜g⁽ⁱ⁾₀ , i∈ {1,2}, this yields the following result.

Theorem 5.4.1. Under suitable regularity assumptions the error between the solution q0 of eq.(5.43) and its temporal semi-discretizationqⁿ₀ using the RK2 method on the macro scale, eq.(5.45), and the Crank–Nicolson scheme for the fluid problem on the micro scale,eq.(5.44), using in particular the approxima-tions of g0 fromeq.(5.45b), satisfies the error estimate

n=1,...,Nmax |q₀(τⁿ)−q₀ⁿ|≲κ²+κ k⁽¹⁾

2

+TOL⁽¹⁾

+

k⁽²⁾ 2

+TOL⁽²⁾. Here k⁽¹⁾ and k⁽²⁾ are the step sizes of the two micro discretizations corre-sponding to the two evaluations of the right-hand side g₀ in the RK2 scheme.

Similarly, TOL⁽¹⁾ and TOL⁽²⁾ are the assumed tolerances for the approxima-tively periodic solution, as specified in ineq.(5.46).

The choice of parameters for the micro discretization as k⁽¹⁾≈√

κ, TOL⁽¹⁾ ≈κ, k⁽²⁾≈κ, TOL⁽²⁾≈κ², (5.53) to be understood asymptotically in κ, hence yields the overall balanced error

max

n=1,...,N|q0(τⁿ)−qⁿ₀|≲κ². (5.54) Application to the Limit System

The computations for RK2 in the limit system presented at the beginning of this chapter were carried out with a balanced error according toTheorem 5.4.1.

The balancing from eq. (5.53) is valid only asymptotically as κ → 0 and no constants are known. For the computations presented in e.g. Figure 5.6 the following relations were employed

M⁽¹⁾= 6dκ^−1/2e,TOL⁽¹⁾= 10⁻²κ, M⁽²⁾= 6dκ⁻¹e,TOL⁽²⁾ = 10⁻²κ². (5.55) These factors were chosen experimentally such that the solutions for the bal-anced parameters are of similar quality as the “RK2 (unbalbal-anced)” solutions depicted inFigure 5.6 withTOL⁽¹⁾=TOL⁽²⁾= 10⁻⁷ andM⁽¹⁾=M⁽²⁾= 40.

The benefit of choosing a lower tolerance and larger step size forg˜₀⁽¹⁾ can be estimated using the numbers provided for the RK2 discretization of the limit system with κ= 0.2 from Figure 5.9. For this problem a total of 1602 time steps were used for the evaluation of ˜g⁽¹⁾₀ and 2910 time steps for ˜g₀⁽²⁾. Assuming thatg˜₀⁽¹⁾requires as many steps asg˜⁽²⁾₀ without adaptivity, the total number of time steps required for the limit system would increase by ≈29%

from 4512 to 5820.

Multiscale Analysis of a

Permeable Membrane Model

Departing from the previous three chapters we now discuss another aspect of the model by Yang et. al. from Section 2.3 by investigating the equations for monocytes in the fluid (lumen) and macrophages in the structure (wall), coupled through the permeable endothelium. These equations also feature processes on differing scales, with fast advection compared to diffusion and reaction, as visible from the magnitude of the nondimensional parameters from Table 2.4and reflected in the scaling byεin eqs. (6.1)below.

This chapter is organized as follows. After introduction of the problem and review of the existing literature we investigate the singular limit convergence for a series of simplified problems. Starting with a stationary equation we show qualitative convergence to the limit equation, then quantify this convergence under further assumptions on the velocity field, first in the case of non-vanishing advection velocities, then for Poiseuille-flow like fields which vanish on the boundary of the domain. These results are then carried over to the instationary case. The final part of this chapter discusses a numerical realization of the equations and the agreement between theoretical and numerical results.

∂_wΩ_f

∂_iΩ_f ∂_oΩ_f

Γ Ωs

Ω_f

Figure 6.1: Example geometry for the concentration equations in two dimen-sions. The boundary of Ωs is only divided into Γ and ∂Ωs\Γ.

Even though(∂Ωs\Γ)∩∂wΩf 6=∅in this example, we use∂wΩf

only in the context of the fluid problem.

139

6.1 Problem Description

We begin with a description of the geometry, an example of which is sketched in Figure 6.1. We assume thatΩf,Ωs⊂R^d with d∈ {2,3}are disjoint domains with Lipschitz boundaries and ∂Ω_f ∩∂Ω_s 6= ∅. As in Chapter 4 the fluid domain’s boundary is disjointly subdivided into inflow boundary∂_iΩ_f, outflow boundary∂_oΩ_f, the wall∂_wΩ_f and the permeable interfaceΓ⊂∂Ω_f∩∂Ω_s. In contrast to the previous chaptersΓis not assumed to be part of∂wΩ_f, leading to the decomposition

∂Ω_f=∂iΩ_f∪∂wΩ_f∪∂oΩ_f∪Γ.

We assume that only ∂_wΩ_f and Γ are on the shared boundary withΩ_s, such that (∂iΩ_f ∪∂oΩ_f)∩∂Ω_s = ∅, and that Γ, ∂iΩ_f and ∂oΩ_f have non-zero Lebesgue surface measure. On the structure domain we only distinguish be-tween the permeable interfaceΓand the remainder

∂Ω_s= Γ∪(∂Ω_s\Γ).

Even though∂_wΩ_f∩∂Ω_smay be non-empty we will not use∂_wΩ_fin the context of the structure problem.

We study the following simplified version of the non-dimensional equa-tions fromeqs. (2.21a)and (2.21b)for the monocytescε,f inside the fluid and macrophagescε,sin the structure, formulated on the slow timescale. Let

∂_τc_ε,f−∆c_ε,f +¹_εv_f· ∇c_ε,f = 0 inI×Ω_f, (6.1a)

∂τcε,s−∆cε,s+cε,s= 0 inI_×Ωs, (6.1b) with initial values cε,f(0) =c⁰_ε,f andcε,s(0) =c⁰_ε,s whose possible dependence onεis specified below, with exterior boundary conditions

cε,f =cⁱⁿ_f onI_×∂iΩf, (6.1c)

∂n_fcε,f = 0 onI×(∂oΩf∪∂wΩf), (6.1d)

∂n_scε,s= 0 onI_×(∂Ωs\Γ) (6.1e) and interface coupling conditions

∂n_fcε,f+ξ(cε,f−cε,s) = 0 onI_×Γ, (6.1f)

∂_nsc_ε,s+ξ(c_ε,s−c_ε,f) = 0 onI×Γ. (6.1g) We assume that ξ ≥0 is a bounded, measurable permeability function onΓ with ξ >0 on a set of non-zero Lebesgue surface measure. Furthermore, the transport velocityvf inside the lumen is assumed to satisfy divvf = 0in Ωf, vf ·nf = 0onΓ∪∂wΩf andvf·nf <0on∂iΩf.

The scaling of eqs. (6.1)agrees with that of the model of Yang et. al., cf.

Table 2.4, in such that diffusion, reaction and permeation are slow processes, whereas advection is fast. All ε-independent parameters were set to 1 for simplicity. In contrast to the model by Young et. al. the blood velocity in the lumen is assumed to be a given stationary field and any effects of fluid-structure interaction or growth are omitted. There is in particular no domain movement and thus no advection of macrophages inside the wall. Future research is

necessary to extend the present analysis to more accurate models, to include e.g. periodic velocities and a coupling to growth.

Equations (6.1)is a singularly perturbed system of partial differential equa-tions, since for ε → 0 the equation in the fluid domain becomes hyperbolic.

The equation in the wall is regular, but coupled to the fluid equation through the interface, which makes this problem non-standard.

For fixed ε = 1 a linear problem similar to eqs. (6.1) was analyzed in [QVZ02b]. An analysis with nonlinear coupling conditions is discussed in [CZ06] although for the case that one domain is contained in the other. None of these models study the limit asε→0.

There is a vast literature for singularly perturbed problems on a single domainΩ⊂R^d of the form

∂_τc_ε−∆c_ε+¹_εv· ∇c_ε= 0 inI×Ω and its stationary variant

−∆cε+¹_εv· ∇cε= 0 in Ω

with corresponding boundary and initial conditions, see e.g. the monographs [Lio73; Eck79; Goe+83] for an overview. The limit behavior depends fun-damentally on the velocity field v, specifically on the solution to the pure advection problem, which in the stationary case is given by

v· ∇c₀= 0 in Ω.

We assume that the velocity field is such that monocytes are transported out of the domain in finite time. We only investigate stationary velocity fields here, where such a condition prevents flow reversal, which is an important phenomenon in atherosclerosis. Nevertheless, we consider this as an artifact of the assumption that the flow is stationary, whereas finite exit times are physiologically essential. Under the assumption of finite exit times, formu-lated asAssumption 6.2.8below, [Eck79] gives integral norm estimates for the advection-diffusion problem similar to those we will employ for the permeable interface problem. [DEF74] uses a similar condition to estimate the smallest eigenvalue of the operator associated with the singularly perturbed advection-diffusion problem, although the velocity field is allowed to vanish at a point.

Since it has links to the averaging theory from the previous chapters, we briefly also discuss the setting wherevis a periodic velocity field with trajecto-ries which do not leave the domain. After a coordinate transformation along the characteristics of the unperturbed advection problem [Kro91; HKV95] could show that the solutions of such problems converge to an averaging-type limit with order O(ε) on a slow timescale of order O(1) [Kro91] or for all time [HKV95]. [Spi16] formally extends this method to more general scenarios us-ing multiscale expansions. In [SPV09; Ded+15] similar problems to [Kro91]

are investigated using Lie averaging techniques, focusing in particular on the spectral properties of the averaging limit to check for advection-enhanced dis-sipation, where the rate of decay of initial values is nonlinear inε, in contrast to the linear rate of decay expected from the diffusive term alone.

Im Dokument Temporal Multiscale Methods for a Model of Atherosclerosis (Seite 140-150)