Fully discrete Heterogeneous Multiscale Method for parabolic problems with multiple spatial and temporal scales

2025-05-06 0 0 785.47KB 22 页 10玖币
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Fully discrete Heterogeneous Multiscale
Method for parabolic problems with
multiple spatial and temporal scales∗
Daniel Eckhardt†Barbara Verfürth‡
Abstract. The aim of this work is the numerical homogenization of a parabolic problem with several time
and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate
one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we
give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and
the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.
Key words. multiscale method; numerical homogenization; parabolic problem; time-space multiscale coeffi-
cient; a priori error estimates
AMS subject classifications. 65M60, 65M15, 65M12, 35K15, 80M40
1. Introduction
Problems with multiple spatial and temporal scales occur in a variety of different phenomena and
materials. Prominent examples are saltwater intrusion, storage of radioactive waste products
or various composite materials ([14,7,18]). These examples all have in common that both
macroscopic and microscopic scales occur. Consequently, they are particularly challenging from
a numerical point of view. However, from the application point of view, it is often sufficient
to know a description of the macroscopic properties. Therefore, it is quite relevant to develop
a method that includes all small-scale effects without having to calculate them simultaneously.
This is the main component of (numerical) homogenization.
In this work we are interested in the following parabolic problem
∂u
∂t −∇·at, x, t
2,x
∇u=f,
with initial and boundary condtions. The precise setting is given further below. at, x, t
2,x
is
called the time-space multiscale coefficient and represents physical properties of the considered
material. If we use standard finite element and time stepping methods, we obtain sufficiently
good solutions only for small time steps and fine grids as the following example illustrates.
∗Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under VE 1397/2-1.
Major parts of this work were accomplished while BV was affiliated with Karlsruher Institut für Technologie.
†Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie, Englerstr. 2, D-
76131 Karlsruhe
‡Institut für Numerische Simulation, Universität Bonn, Friedrich-Hirzebruch-Allee 7, D-53115 Bonn
1
arXiv:2210.04536v1 [math.NA] 10 Oct 2022
(a) L2-Error with respect to grid width
hand at time t= 1.(b) Illustration of u
Example 1.1.Let Ω = (0,1) and T= 1. Furthermore we consider
a(t, x, s, y) = 3 + cos(2πy) + cos2(2πs).
Let the initial condition be u(0, x) = 0 for all x∈Ω. Figure 1a shows the error in the L2-norm
with respect to the numerical and a reference solution. The reference solutions were calculated
using finite elements with grid width h= 10−6and the implicit Euler method time step size
τ= 1/100. The theory yields an expected quadratic order of convergence. However, this occurs
here only for small grid sizes. More precisely, the error converges only when h<, see Figure
1a. Similar observations can be made for the time step, where one even needs τ < 2in general.
The reason is that uis highly oscillatory in space and time, see Figure 1b.
To tackle the outlined challenges, various multiscale methods have been proposed. Focusing on
approaches for parabolic space-time multiscale problems, examples include generalized multiscale
finite element methods [11], non-local multicontinua schemes [17], high-dimensional (sparse)
finite element methods [26], an approach based on an appropriate global coordinate transform
[22], a method in the spirit of the Variational Multiscale Method and the Localized Orthogonal
Decomposition [20] as well as optimal local subspaces [24,25]. As already mentioned, we consider
locally periodic problems in space and time here. Hence, we employ the Heterogeneous Multiscale
Method (HMM), first induced by E and Enquist [12], see also the reviews [1,3]. The HMM
has been successfully applied to various time-dependent problems such as (nonlinear) parabolic
problems [2,4,5,6], time-dependent Maxwell equations [13,15,16] or the heat equation for
lithium ion batteries [28]. We use the finite element version of the HMM, but note that other
discretization types such as discontinuous Galerkin schemes are generally possible as well.
The present contribution is inspired by [21], which considers the same parabolic model problem
and analyzes a semi-discrete HMM for it. Precisely, the microscopic cell problems are solved
analytically in [21]. Our main contribution is to propose a suitable discretization of these cell
problems and to show rigorous error estimates for the resulting fully discrete HMM. A particular
challenge for the estimate is to balance the order of the mesh size and the time step on the one
hand and the period on the other hand. Further, we illustrate our theoretical results with
numerical experiments and thereby underline the applicability of the method.
The paper is organized as follows. In Section 2, we introduce the setting and present the main
homogenization results. In Section 3, we derive the fully discrete finite element heterogeneous
multiscale method. The error of the macroscopic discretization is estimated in Section 4 and
the error arising from the microscopic modeling is investigated in Section 5. Finally, numerical
results are presented in Section 6.
2
2. Setting
In this section, we present our model problem and the associated homogenization results. Through-
out the paper, we use standard notation on function spaces, in particular the Lebesgue space
L2, the Sobolev spaces H1and H1
0, as well as Bochner spaces for time-dependent functions. We
denote the L2-scalar product (w.r.t to space) by h·,·i0and the L2-norm by k·k0. Furthermore,
we mark by #spaces of periodic functions. Let X#(Ω) be such a space for an arbitrary Ω⊂Rd,
then the subspace X#,0(Ω) ⊂X#(Ω) consist of all functions whose integrals over Ωis 0.
2.1. Model problem
Let Ω⊂Rdbe a bounded Lipschitz domain, T > 0the final time and Y:= (−1
2,1
2)d. We
consider the following parabolic problem
∂u
∂t −∇·(a(t, x)∇u) = f(t, x)x∈Ω, t ∈(0, T )
u(0, x) = u0(x)x∈Ω
u(t, x) = 0 x∈∂Ω, t ∈(0, T ),
(2.1)
where f∈L2((0, T ), L2(Ω)) and u0∈L2(Ω).ais the time-space multiscale coefficient as
introduced in Section 1 and is defined by the matrix-valued function a(t, x, s, y)∈C([0, T ]ׯ
Ω×
[0,1] ׯ
Y , Rd×d
sym). The function ais (0,1) ×Y-periodic with respect to sand y, furthermore it
is coercive and uniformly bounded, in particular this means that there are constants Λ, λ > 0,
such that for all ξ, η ∈Rd:
η·a(t, x, s, y)ξ≤Λ|η||ξ|und ξ·a(t, x, s, y)ξ≥λ|ξ|2
for all (t, x, s, y)∈[0, T ]×Ω×[0,1] ×Y. Further, we assume that ais Lipschitz continuous in t
and x.
2.2. Homogenized Problem
Analytical homogenization results for (2.1) were obtained in [8,26]. For ease of presentation,
we follow the traditional approach of asymptotic expansions here, but we emphasize that the
same results are obtained with the more recent approach of time-space multiscale convergence as
in [26], which is a generalization of two-scale convergence. Based on the multiscale asymptotic
expansion
u(t, x) = U0(t, x, t
2,x
) + U1(t, x, t
2,x
) + 2U2(t, x, t
2,x
) + . . . , (2.2)
it is shown that U0solves the homogenized problem
∂U0
∂t −∇·(A0∇U0) = fin (0, T )×Ω
U0= 0 on (0, T )×∂Ω
U0(0,·) = u0in Ω.
(2.3)
Here, the homogenized coefficient A0is defined by
Aij
0(t, x) = ˆ1
0ˆY
d
X
k=1
aik(t, x, s, y)δjk +∂χj
∂yk
(t, x, s, y)dyds, (2.4)
3
where δjk denotes the Kronecker delta. The function χi∈L2((0, T )×Ω×(0,1), H1
#,0(Y)) ∩
L2((0, T )×Ω, H1
#((0,1), H−1
#,0(Y))) solves the cell problem
∂χi
∂s − ∇y·(a(ei+∇yχi)) = 0 in (0,1) ×Y,
χi(t, x, s, ·)Y-periodic for all t, x, s,
χi(t, x, ·, y) (0,1)-periodic for all t, x, y.
(2.5)
Using these χi,U1in the asymptotic expansion (2.2) can be written as
U1(t, x, s, y) =
d
X
i=1
∂U0
∂xi
(t, x)χi(t, x, s, y).
[8, Chapter 2, Section 1.7] shows in Theorem 2.1 and Theorem 2.3 that
||u−U0−U1||L2((0,T ),H1
0(Ω)) →0for →0.
We call U0the homogenized solution.U0describes the macroscopic behavior of u, because U0
only depends on the macroscopic scale x.U1is called the first-order corrector.
Remark 2.1.A0is not symmetric in Rdfor d > 1in general since
Aij
0(t, x) = ˆ1
0ˆY
d
X
l=1
d
X
k=1
δilalk(t, x, s, y)(δjk +∂χj
∂yk
(t, x, s, y))dyds
=ˆ1
0ˆY
d
X
l=1
d
X
k=1
(δil +∂χi
∂yl
(t, x, s, y))alk(t, x, s, y)(δjk +∂χj
∂yk
(t, x, s, y))dyds
+ˆ1
0ˆY
χi(t, x, s, y)∂χj
∂s (t, x, s, y)dyds. (2.6)
The last term does not vanish in general, but it is zero for i=jdue to integration by parts and
the time-periodicity of χi.
In the following we reformulate A0in a way which we use to derive the discretized problem
later. We transform the reference cell (0,1) ×Yto a general cell (t, t +2)× {x0}+Iwith
I:=Y for t∈[0, T )and x0∈Ωfixed. Application of the chain and transformation rule allows
us to write
A0(t, x) = ˆ1
0ˆY
a(t, x, s, y)(Idd+Dyχ(t, s, y))dyds (2.7)
=1
2|I|ˆt+2
tˆ{x0}+I
at, x, s
2,y
Idd+Dyχt, x, s
2,y
dyds. (2.8)
3. The finite-element heterogeneous multiscale method
(FE-HMM)
Based on the results of Section 2, we want to compute an approximation of the homogenized
solution U0based on the Finite-Element Heterogeneous Multiscale Method (FE-HMM). In [21],
this method was already introduced, but it was assumed that the cell problems (2.5) could be
4
solved exactly/analytically. The main aim of this section is to introduce also the (microscopic)
discretization of the cell problems, allowing for a fully discrete method. Further, we also account
for the non-symmetry of A0. This leads to a slightly different formulation in comparison to [21]
where the symmetric part of A0was considered throughout. In the following, we will derive
the full method step by step, which is on the one hand hopefully instructive for the readers to
understand the final formulation and on the other makes it easier to follow the error estimates
in the following sections.
We start with the discretized macro problem. For the spatial discretization we use linear finite
elements based on a triangulation THand for the time discretization we use the implicit Euler
method. Precisely, let VH⊂H1
0(Ω) be the space of all piecewise linear functions which are zero
on ∂Ωand let τ=T/N be the time step size. For 1≤n≤Nwe set tn=nτ. Further, we define
U0
H:= QHu0, where QH:L2(Ω) →VHis the L2-projection.
Let Un
Hthen be the solution of the discretized equation
D∂Un
H
∂t ,ΦHE0+B[tn, Un
H,ΦH] = fn,ΦH0for all ΦH∈VH,(3.1)
where fn(x) = f(tn, x)and ¯
∂Un
H
∂t = (Un
H−Un−1
H)/τ. Here, the discrete bilinear form B[tn,·,·]
is defined for any ΦH,ΨH∈VHvia
B[tn,ΦH,ΨH] := ˆΩ∇ΨH(x)·A0(tn, x)∇ΦH(x)dx
=X
K∈TH
ˆK∇ΨH(x)·A0(tn, x)∇ΦH(x)dx
≈X
K∈TH|K|∇ΨH(xK)·A0(tn, xK)∇ΦH(xK).(3.2)
In the last step we approximated the integral with a quadrature formula, where xKdenotes
hte barycenter of K∈ TH. If we now consider the individual summands, we could calculate
A0(tn, xK)starting from equation (2.7). However, this would have several disadvantages. First,
we would have to compute A0(tn, xK)for all time points tn, which would require a lot of memory
depending on the time step size. Furthermore, we want to change the boundary conditions
later, which is not possible with this approach. Therefore, the idea is to compute ∇ΨH(xK)·
A0(tn, xK)∇ΦH(xK)directly. For this, set I,K := {xK}+Iand use reformulation (2.7) to give
∇ΨH(xK)·A0(tn, xK)∇ΦH(xK)
=1
2|I|ˆtn+2
tnˆI,K ∇ΨH|I,K (xK)·
:=a
n,K (t,x)
z }| {
a(tn, xK,t
2,x
)
∇(ΦH|I,K (xK) + ∇ΦH|I,K (xK)χ(tn, xK
t
2,x
)
| {z }
=:˜
Φ
)dxdt
=1
2|I|ˆtn+2
tnˆI,K ∇ΨH|I,K (xK)·a
n,K (t, x)∇φ
#(tn, xK,t
2,x
)dxdt, (3.3)
where
φ
#= ΦH|I,K +˜
Φ∈VH+X((tn, tn+2), I,K ),(3.4)
5
摘要：

FullydiscreteHeterogeneousMultiscaleMethodforparabolicproblemswithmultiplespatialandtemporalscales*DanielEckhardt„BarbaraVerfürth…Abstract.Theaimofthisworkisthenumericalhomogenizationofaparabolicproblemwithseveraltimeandspatialscalesusingtheheterogeneousmultiscalemethod.Wereplacetheactualcellproblem...

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