Inhomogeneous Dirichlet and initial data

So far, we considered the incompressible Navier–Stokes system with homogeneous Dirichlet data.

In the following, we extend the scope to problems involving inhomogeneous boundary conditions.

In the context of POD-Galerkin modeling based on adaptive finite element snapshots, the main challenge is to find suitable continuous extensions of the Dirichlet data, which need to be sub-tracted from the snapshots before a POD basis is computed. For the derivation of a velocity POD-Galerkin model, we must ensure that these continuous extensions fulfill the correct weak divergence-free property.

Inhomogeneous problem setting

We extend (4.44) to the case of inhomogeneous Dirichlet boundary conditions by introducing Dirichlet boundary data y_D : [0, T]×∂Ω → R. The resulting problem reads as follows: find a velocity field ˘y and a pressure field p such that

˘

y_t+ (˘y· ∇)˘y−Re⁻¹∆˘y+∇p=f in (0, T]×Ω, (4.57a)

∇ ·y˘= 0 in [0, T]×Ω, (4.57b)

˘

y =y_D on [0, T]×∂Ω, (4.57c)

˘

y(0,·) =y0 in Ω. (4.57d)

4.6 POD model order reduction with space-adapted snapshots for incompressible flows 115

where∇ ·y₀ = 0 in Ω andy₀ =y_D(0,·) on∂Ω. In the following, we derive a homogenized version of this problem, which provides a foundation for the subsequent finite element discretization and reduced-order modeling.

Homogenized problem setting

We assume that the boundary functiony_D is sufficiently smooth such that it can be continuously extended by a function g: [0, T]×Ω¯ →Rwith g(t,·)|_∂Ω =y_D(t,·) for t∈[0, T]. The regularity requirements ong are such that a unique weak solution exists, see [154]. We provide a concrete choice ofg by computation.

We homogenize (4.57) by subtracting the boundary function from the inhomogeneous velocity solution such that the homogeneous velocity field is given byy= ˘y−g. Substituting ˘y in (4.57), we obtain the following homogenized problem: find a velocity y and a pressurep such that

y_t+ (y· ∇)y+ (g· ∇)y+ (y· ∇)g−Re⁻¹∆y+∇p (4.58a)

=f −(g· ∇)g+Re⁻¹∆g−gt in (0, T]×Ω, (4.58b)

∇ ·y= − ∇ ·g in [0, T]×Ω, (4.58c)

y= 0 on [0, T]×∂Ω, (4.58d)

y(0,·) =y₀−g in Ω, (4.58e)

where∇ ·y0 = 0 in Ω andy0=g(0,·) on ∂Ω.

In order to derive a time-discrete weak form of the homogenized problem, we implement the time integrals involving the Dirichlet data using the right-sided rectangle rule, which evaluates the Dirichlet data at the new time instance. For ease of notation, we define f^j := f(tj) and g^j := g(t_j) for j = 0, . . . , n. Consequently, the time-discrete weak form of the homogenized problem consists in finding sequences y¹, . . . , yⁿ ∈ H₀¹(Ω) and p¹, . . . , pⁿ ∈ L²₀(Ω), for given y⁰=y0−g⁰ withy0 ∈H_div(Ω), such that

y^j−y^j−1

∆tj

, v

L²(Ω)

+c(y^j, y^j, v) +c(g^j, y^j, v) +c(y^j, g^j, v) +a(y^j, v) +b(v, p^j) (4.59a)

= hf^j, vi_H−1(Ω),H¹₀(Ω)−c(g^j, g^j, v)−a(g^j, v)−

g^j−g^j−1

∆t_j , v

L²(Ω)

∀v∈H₀¹(Ω), (4.59b)

b(y^j, q) = −b(g^j, q) ∀q ∈L²₀(Ω).

(4.59c) We apply an adaptive finite element method. Let us denote byV_h⁰the velocity finite element space associated with the initial mesh T_h⁰. For a given initial condition (y_h⁰, v)_L²_(Ω) = (y₀−g⁰, v)_L²_(Ω) for all v∈V_h⁰ withy0 ∈Hdiv(Ω) the fully discrete homogenized Navier–Stokes problem reads as follows: find y_h¹ ∈V_h¹, . . . , yⁿ_h ∈V_hⁿ and p¹_h ∈Q¹_h, . . . , pⁿ_h ∈Qⁿ_h such that

y_h^j −y_h^j−1

∆t_j , v

!

L²(Ω)

+c(y_h^j, y^j_h, v) +c(g^j, y_h^j, v) +c(y_h^j, g^j, v) +a(y^j_h, v) +b(v, p^j_h) (4.60a)

= hf^j, vi_H−1(Ω),H₀¹(Ω)−c(g^j, g^j, v)−a(g^j, v)−

g^j−g^j−1

∆t_j , v

L²(Ω)

∀v∈V_h^j, (4.60b)

b(y_h^j, q) = −b(g^j, q) ∀q ∈Q^j_h.

(4.60c)

Lifting function

Based on (4.60), approximations to the inhomogeneous solutions ˘y(tj) of (4.57) are given by ˆ

y_h^j := y_h^j + g^j for j = 0, . . . , n. Regardless of the choice of lifting functions g⁰, . . . , gⁿ, we can guarantee that ˆy⁰_h fulfills the initial condition (4.57d) and ˆy_h¹, . . . ,yˆⁿ_h fulfill the Dirichlet condition (4.57c) by construction. Nevertheless, in order to solve (4.60) numerically, candidates of g⁰, . . . , gⁿ must be fixed, at least implicitly. Our approach to reduced-order modeling is not tied to a particular choice. In the following, we provide suitable candidates which can be realized without the need to modify usual finite element codes.

Note that for the velocity finite element spaces we have V_h^j ⊂ H₀¹(Ω). Thus, for the context of inhomogeneous Dirichlet conditions, we start by introducing the spaces V_D^j for j = 1, . . . , n, which denote the spaces spanned by the union of the finite element basis functions of V_h^j and the finite element basis functions associated with the corresponding Dirichlet boundary nodes.

We assume that in (4.60) the integrals involving g^j are approximated by a numerical quadrature which consists of substituting the Lagrange interpolation of g^j onto V_D^j and integrating the resulting piecewise polynomials exactly. We assume that by a finite number of refinements of any V_D¹, . . . , V_Dⁿ one can find a reference finite element space ˜VD such that V_D¹, . . . , V_Dⁿ ⊂ V˜D. Now, forj= 1, . . . , n, we define lifting functionsg^j as a sufficiently smooth continuous extension of the Dirichlet datay_D(t_j) into the domain Ω such thatg^j is zero at all nodes of the reference finite element space ˜V. This is equivalent to the standard approach of using an approximate Dirichlet lifting given by a Lagrange interpolation of the Dirichlet data onto the finite element space at the boundary and a subsequent continuous extension using the finite element space in the interior, because we have

(g^j =yD(t) at all Dirichlet nodes of V_D^j, g^j = 0 at all interior nodes ofV_D^j.

A disadvantage of the standard approach is that it implies a Dirichlet lifting which satisfies the boundary data only in an approximate sense. Our description, on the other hand, delivers an output which is exact at the boundary. In particular, we have





 ˆ

y^j_h=y^j_h at all interior nodes of ˜VD, ˆ

y^j_h=g^j at all Dirichlet nodes of ˜V_D, y^j_h= 0 at all Dirichlet nodes of ˜V_D.

This holds for all ˜VD which fulfill our assumptions, without the need to specify a concrete candidate of ˜VD during the adaptive finite element simulation. When the adaptive finite element simulation is finished andV_D¹, . . . , V_Dⁿare available, some ˜V_D can be computed by refinement and g^j can be evaluated at all nodal points of ˜V_D. Therefore, we are even able to formulate a finite element discretization of (4.59) on ( ˜V ,Q) using the same˜ g⁰, . . . , gⁿ as in (4.60). Moreover, we are able to solve (4.59) on subspaces of ( ˜V ,Q) using the same˜ g⁰, . . . , gⁿ as in (4.60).

Remark 4.7. In principle, it is possible to impose a weak divergence-free constraint on the homogenized velocity finite element solution by using lifting functions which are computed such that b(g^j, q) = 0 for all q ∈ Q^j_h for j = 0, . . . , n. But this would require the solution of an additional stationary finite element problem for eachg^j. Moreover, this would not automatically imply a weak divergence-free property with respect to a reference pressure spaceQ. An alternative˜ to the implicit choice of the Dirichlet lifting function is its explicit choice at the level of the strong formulation (4.58). Disadvantages would be a possibly larger support of such a lifting function and the effort of actually finding a suitable function. Also in this case, it would be attractive to impose a strong divergence-free constraint on g^j, because this implies a weakly divergence-free homogenized velocity field y_h^j. Still, finding a suitable candidate may be challenging in general.

4.6 POD model order reduction with space-adapted snapshots for incompressible flows 117

Velocity reduced-order model for the inhomogeneous setting

In the following, we derive a reduced-order model for the velocity field, based on the semi-discretized problem (4.59). We introduce the projections Pg^V,Q according to Problem 4.5 by Problem 4.8. For given u∈X, sufficiently smoothg and given spaces V and Q, findP_g^V,Q(u) which solves

minv∈V

1

2kv−uk²_X subject to b(v+g, q) = 0 ∀q ∈ Q.

We use X =V =H₀¹(Ω) and Q=L²₀(Ω) in the following. By definition of this projection onto a divergence-free space, we have b(P_g^V,Qj (0) +g^j, q) = 0 for allq ∈ L²₀(Ω) and j = 0, . . . , n. In (4.59), we substituteg^j =g^j+P_g^V,Qj (0)−P_g^V,Qj (0) and reformulate the equations so that we can set ˜y^j =y^j−P_g^V,Qj (0). We obtain the following problem, which is equivalent to (4.59): for given

˜

y⁰ =y₀−g˜⁰ with y₀ ∈H_div(Ω) and ˜g^j =g^j+P_g^V,Qj (0) forj = 0, . . . , n, find ˜y¹, . . . ,y˜ⁿ ∈H₀¹(Ω) and p¹, . . . , pⁿ∈L²₀(Ω) such that

y˜^j−y˜^j−1

∆t_j , v

L²(Ω)

+c(˜y^j,y˜^j, v) +c(˜g^j,y˜^j, v) +c(˜y^j,g˜^j, v) +a(˜y^j, v) +b(v, p^j) (4.61a)

= hf^j, vi_H−1(Ω),H¹₀(Ω)−c(˜g^j,g˜^j, v)−a(˜g^j, v)−

g˜^j−g˜^j−1

∆tj

, v

L²(Ω)

∀v∈H₀¹(Ω), (4.61b)

b(˜y^j, q) = 0 ∀q ∈L²₀(Ω).

(4.61c) One can show that (4.60) is a discretization of (4.61) by replacing (H₀¹(Ω), L²₀(Ω)) with (V_h^j, Q^j_h) for j = 1, . . . , n in (4.61). Then, for the resulting solution holds ˜y^j = y_h^j −P^V

j h,Q^j_h

g^j if ˜g^j = g^j+P^V

j h,Q^j_h

g^j (0) forj= 0, . . . , n. In this way, (4.60) can be used to obtain approximate solutions of (4.61).

We base the model equation on a discretization of (4.61) using the pair of reference spaces ( ˜V ,Q) as test spaces together with ˜˜ g^j = g^j +P_g^{V ,˜}j^Q˜(0). The resulting solutions are approxima-tions to the soluapproxima-tions of (4.61) using the original pair of spaces (H₀¹(Ω), L²₀(Ω)). We have shown that the solutions of (4.61) using the original pair of spaces (H₀¹(Ω), L²₀(Ω)) are approximated by y_h^j −P^V

j h,Q^j_h

g^j (0) for j = 1, . . . , n resulting from (4.60). But these solutions are not weakly divergence-free with respect to the reference pair of spaces ( ˜V ,Q). Therefore, we have to modify˜ them.

Following the velocity-ROM approach a) in Section 4.6.3 we substitute y_h^j −P^V

j h,Q^j_h

g^j (0) by their approximations P_g^{V ,˜}_j^Q˜(y_h^j)−P_g^{V ,˜}_j^Q˜(0) for j= 1, . . . , n. Using nowP_g^{V ,˜}_j^Q˜(y_h^j)−P_g^{V ,˜}_j^Q˜ as snapshots in a POD yields POD basis functions

ψ_i ∈span(P_g¹(y¹_h)−P_g¹(0), . . . , P_gⁿ(y_hⁿ)−P_gⁿ(0))⊂V˜_div ∀i= 1, . . . , `<sub>y</sub> for some`y ≤n, which define a POD spaceV`:= span(ψ1, . . . , ψ`y)⊂V˜div.

In the time-discrete equation (4.61), we use the pair (V_{`</sub>,Q) as test and trial spaces. Consequently,˜ the continuity equation is fulfilled by construction. For the pressure term, we have b(v, p<sup>j</sup>) = 0 for all v ∈ V<sub>`} and all p^j ∈ Q. The resulting reduced-order model is given by the following set˜

of equations: for y_{`</sub><sup>0</sup> =y0 −˜g<sup>0</sup> with y0 ∈ Hdiv(Ω) and ˜g<sup>j</sup> =g<sup>j</sup>+P<sub>g</sub><sup>V ,˜</sup>j<sup>Q˜</sup>(0) for j = 0, . . . , n, find y<sub>`}¹, . . . , y_{`</sub><sup>n</sup>∈V<sub>`} such that

y_{`</sub><sup>j</sup>−y<sup>j−1</sup><sub>`}

∆t_j , v

!

L²(Ω)

+c(y^j_{`</sub>, y<sub>`}^j, v) +c(˜g^j, y_{`</sub><sup>j</sup>, v) +c(y<sub>`}^j,g˜^j, v) +a(y^j_`, v)

=hf^j, vi_H−1(Ω),H₀¹(Ω)−c(˜g^j,g˜^j, v)−a(˜g^j, v)−

g˜^j−g˜^j−1

∆tj

, v

L²(Ω)

∀v∈V_`.

(4.62)

Remark 4.9. Concerning the computational complexity, we have to additionally consider the projections P_g^{V ,˜}j^Q˜(0) for j = 0, . . . , n in comparison to the homogeneous case. Therefore, the solution of Problem 4.5 has to be computed n+ 1 times additionally to the projections of the homogeneous solutions y_h^j.

Following the velocity-ROM approach b) in Section 4.6.3, we first introduce a set of modified ho-mogeneous solutions ˆy^j−P^{V ,˜ Q˜}

g^j (0), forj= 1, . . . , n. These modified snapshots can be constructed using e.g. a Lagrange interpolation of the original homogeneous solutions y^j_h onto the reference space ˜V and an approximation of P_g^Vj^j,Qj(0) for j = 1, . . . , n. From these modified snapshots, we compute a POD basis ˆψ1, . . . ,ψˆ_{`y</sub> ∈ V˜. Note that these modes are in general not discrete divergence-free. Thus, they are then projected onto the space ˜V<sub>div</sub> by solving Problem 4.6 with g= 0. This leads to a divergence-free velocity POD spaceV` = span(ψ1, . . . , ψ`y)⊂V˜div. Replac-ing (H<sub>0</sub><sup>1</sup>(Ω), L<sup>2</sup><sub>0</sub>(Ω)) by the pair (V<sub>`},Q) in (4.61) leads to a reduced-order model of the form (4.62).˜ Velocity-pressure reduced-order model for the inhomogeneous setting

To derive a velocity-pressure reduced-order model of the homogenized problem (4.60), we require a suitable inf-sup stable pair of reduced spaces. Since the homogenization does not alter the bilinear form b(·,·), the inf-sup stability criterion stays the same. Therefore, we compute a pressure reduced spaceQ_{`</sub>and a velocity reduced spaceV<sub>`}like in Section 4.6.4, but using Lagrange interpolated velocity and pressure snapshots of (4.60) instead of (4.48). We derive a stable POD-Galerkin model from the time-discrete problem (4.59) by using the pair (V_{`</sub>, Q<sub>`}) as test and trial spaces.

We solve the reduced-order model for the POD approximations y_{`</sub><sup>1</sup>, . . . , y<sup>n</sup><sub>`} of the homogeneous velocity fields and the POD approximations p¹_{`</sub>, . . . , p<sup>n</sup><sub>`} of the pressure fields. Finally, y^j_` +g^j is a time-discrete reduced-order approximation of the velocity solution of the inhomogeneous problem.

Im Dokument Adaptivity in Model Order Reduction with Proper Orthogonal Decomposition (Seite 123-127)