Time-dependent SteklovPoincar e operators and space-time RobinRobin decomposition for the heat equation Emil Engstr omEskil Hansen

2025-05-06 0 0 335.34KB 25 页 10玖币

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Time-dependent Steklov–Poincar´

e operators and space-time

Robin–Robin decomposition for the heat equation

Emil Engstr¨

om ·Eskil Hansen

October 26, 2022

Abstract Domain decomposition methods are a set of widely used tools for par-

allelization of partial differential equation solvers. Convergence is well studied for

elliptic equations, but in the case of parabolic equations there are hardly any results

for general Lipschitz domains in two or more dimensions. The aim of this work is

therefore to construct a new framework for analyzing nonoverlapping domain de-

composition methods for the heat equation in a space-time Lipschitz cylinder. The

framework is based on a variational formulation, inspired by recent studies of space-

time ﬁnite elements using Sobolev spaces with fractional time regularity. In this

framework, the time-dependent Steklov–Poincar´

e operators are introduced and their

essential properties are proven. We then derive the interface interpretations of the

Dirichlet–Neumann, Neumann–Neumann and Robin–Robin methods and show that

these methods are well deﬁned. Finally, we prove convergence of the Robin–Robin

method and introduce a modiﬁed method with stronger convergence properties.

Keywords Steklov–Poincar´

e operator ·Robin–Robin method ·Nonoverlapping

domain decomposition ·Space-time ·Convergence

Mathematics Subject Classiﬁcation (2000) 65M55 ·65J08 ·35K20

This work was supported by the Swedish Research Council under the grant 2019–05396.

Emil Engstr¨

Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00 Lund, Sweden

E-mail: emil.engstrom@math.lth.se

Eskil Hansen

Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00 Lund, Sweden

E-mail: eskil.hansen@math.lth.se

arXiv:2210.13868v1 [math.NA] 25 Oct 2022

2 Emil Engstr¨

om, Eskil Hansen

1 Introduction

Domain decomposition methods enable the usage of parallel and distributed hardware

and are commonly employed when approximating the solutions to elliptic equations.

The basic idea is to ﬁrst decompose the equation’s spatial domain into subdomains.

The numerical method then consists of iteratively solving the elliptic equation on

each subdomain and thereafter communicating the results via the boundaries to the

neighboring subdomains. An in-depth survey of the topic can be found in the mono-

graphs [25,29].

A recent development in the ﬁeld is to apply this approach to parabolic equations.

The decomposition into spatial subdomains is then replaced by the decomposition

into space-time cylinders. In general, space-time decomposition schemes enable ad-

ditional parallelization and less storage requirements when combined with a standard

numerical method for parabolic problems. The methods have especially gained at-

tention in the contexts of parallel time integrators; surveyed in [8], space-time ﬁnite

elements; surveyed in [26], and parabolic problems with a spatial domain given by a

union of domains with very different material properties [1,13].

There have been several studies concerning the convergence and other theoretical

aspects of space-time decomposition schemes applied to parabolic equations on one-

dimensional or rectangular spatial domains; see [6,9,10,11,19,20]. However, there

are hardly any convergence results for more general settings, e.g., with spatial Lips-

chitz domains in higher dimensions. The one exception is the convergence analysis

considered in [12] for an optimized Schwarz waveform relaxation, where a method-

speciﬁc variational approach is proposed. The latter is similar to the analysis used by

Lions for the Robin–Robin method applied to elliptic equations [23].

This lack of convergence results is in stark contrast to the elliptic setting, where

the analysis of nonoverlapping domain decomposition schemes for Lipschitz domains

in higher dimensions is standard. The basic elliptic framework is to reformulate the

method as an iteration scheme posed on the intersections of the subdomains in terms

of Steklov–Poincar´

e operators. This is the basis for the analysis of the Dirichlet–

Neumann and Neumann–Neumann methods [25]. The Steklov–Poincar´

e framework

even yields the convergence of the Robin–Robin method applied to nonlinear elliptic

equations [4]. Note that the time-dependent Steklov–Poincar´

e operators were intro-

duced without any analysis in [12].

Hence, the aim of this study is twofold. First, we strive to introduce a new vari-

ational framework for time-dependent Steklov–Poincar´

e operators and to derive the

essential properties of these operators, e.g., coercivity and bijectivity. The latter prop-

erty implies that the space-time generalizations of the Dirichlet–Neumann, Neumann–

Neumann, and Robin–Robin methods are all well deﬁned in the parabolic setting.

Second, we will employ the new framework to give a ﬁrst rigorous proof of the

Robin–Robin method’s convergence when applied to parabolic equations.

For sake of simplicity, we will restrict our attention to the heat equation with

homogeneous Dirichlet conditions. However, our analysis does not depend on any

symmetry of the underlying bilinear form, and a similar framework can be used for

more general parabolic problems and boundary conditions. To limit the technicalities

further, we consider the equation for all times t∈R, which allows a variational for-

Space-time Robin–Robin method 3

Fig. 1: The nonoverlapping decomposition of the space-time cylinder.

mulation with the same test and trial spaces. Our model problem is therefore the heat

equation on the space-time cylinder Ω×R, i.e.,

((∂t−∆)u=fin Ω×R,

u=0 on ∂ Ω ×R,(1)

where the spatial domain Ω⊂Rd,d=2,3, is bounded with boundary ∂ Ω . Note

that a homogeneous initial condition at time t0can be incorporated by prescribing a

source term fthat is zero for times t<t0.

Next, we decompose the spatial domain Ωinto nonoverlapping subdomains Ωi,

i=1,2, with boundaries ∂ Ωi, and denote the interface separating the subdomains Ωi

by Γ. That is,

Ω=Ω1∪Ω2,Ω1∩Ω2=/0,and Γ= (∂ Ω1∩∂ Ω2)\∂ Ω .(2)

The space-time cylinder Ω×Ris thereby decomposed into Ωi×R,i=1,2, as illus-

trated in Figure 1. The current setting is also valid for spatial subdomains Ωigiven as

unions of nonadjacent subdomains, i.e.,

Ωi=∪s

`=1Ωi`and Ωi`∩Ωi j =/0 for `6=j.

With a ﬁxed domain decomposition we can reformulate the heat equation as equa-

tions on Ωi×R,i=1,2, connected via transmission conditions on Γ×R. More

precisely, we have the parabolic transmission problem











(∂t−∆)ui=fin Ωi×Rfor i=1,2,

ui=0 on (∂ Ωi\Γ)×Rfor i=1,2,

u1=u2on Γ×R,

∇u1·ν1=−∇u2·ν2on Γ×R,

(3)

where νidenotes the unit outward normal vector of ∂ Ωi. Alternating between the de-

composed space-time cylinders and the transmission conditions generates the space-

time generalizations of the Dirichlet–Neumann and Neumann–Neumann methods.

4 Emil Engstr¨

om, Eskil Hansen

As ν1=−ν2on Γ, the transmission conditions are also equivalent to the Robin con-

ditions

∇u1·νi+su1=∇u2·νi+su2on Γ×Rfor i=1,2,

where sis a non-zero parameter. The latter reformulation gives rise to the space-time

Robin–Robin method, for which we will prove convergence. The method is given by

ﬁnding (un

1,un

2)for n=1,2,... such that











(∂t−∆)un+1

1=fin Ω1×R,

un+1

1=0 on (∂ Ω1\Γ)×R,

∇un+1

1·ν1+sun+1

1=∇un

2·ν1+sun

2on Γ×R,

(∂t−∆)un+1

2=fin Ω2×R,

un+1

2=0 on (∂ Ω2\Γ)×R,

∇un+1

2·ν2+sun+1

2=∇un+1

1·ν2+sun+1

1on Γ×R,

(4)

where u0

2is a given initial guess, s>0 is a ﬁxed method parameter, and un

iapproxi-

mates u|Ωi×R. Note that the method per se is sequential, but the computation of each

term un

ican be implemented in parallel if Ωiis a union of nonadjacent subdomains.

The standard variational framework for parabolic problems, based on solutions in

the space

H1R,H−1(Ωi)∩L2R,H1(Ωi),

is unfortunately not well suited for our domain decompositions. The issue is that two

functions ui,i=1,2, in the above solution space, which coincide on Γ×Rin the

sense of trace, can not be “pasted” together into a new function in H1R,H−1(Ω);

compare with [3, Example 2.14]. Hence, the transmission problem (3) does not nec-

essarily yield a solution to the heat equation (1) in this context.

In order to remedy this, we consider a framework with solutions in

H1/2R,L2(Ωi)∩L2R,H1(Ωi),

which originates from [22] and resolves the above issue. This H1/2-setting also en-

ables a trace theory valid for spatial Lipschitz domains [3] and the related bilinear

form becomes coercive-equivalent [5]. This will be the starting point for our analy-

sis. Note that the H1/2-setting has been used by [3] in the context of boundary element

methods and was later employed for space-time ﬁnite elements [5,17,27].

The analysis is organized as follows: In Sections 2 and 3we derive the properties

of the required function spaces and operators. We introduce the variational formula-

tions and prove equivalence between the heat equation and the transmission problem

in Section 4. The time-dependent Steklov–Poincar´

e operators are analysed in Sec-

tion 5 and three standard space-time domain decompositions are proven to be well

deﬁned. In Sections 6 and 7we prove convergence of the space-time Robin–Robin

method in L2R,H1(Ω1)×L2R,H1(Ω2)and the convergence of a modiﬁed ver-

sion of the method in

H1/2R,L2(Ω1)∩L2R,H1(Ω1)×H1/2R,L2(Ω2)∩L2R,H1(Ω2),

respectively.

Space-time Robin–Robin method 5

2 Preliminaries

The spatial domain Ω⊂Rdand its decomposition (2) into Ωi,i=1,2, is assumed to

satisfy the properties below.

Assumption 1 The sets Ω,Ωi,i=1,2, are bounded and Lipschitz. The spatial in-

terface Γand the sets ∂ Ω \∂ Ωi, i =1,2, are all (d−1)-dimensional Lipschitz man-

ifolds.

For a description of Lipschitz domains, see [16, Chapter 6.2]. The assumptions are

made in order ensure the existence of the spatial trace operator, as well as, to allow the

usage of Poincar´

e’s inequality. As a notational convention, operators only depending

on space or time are “hatted” and their extensions to space-time are denoted without

hats, e.g.,

∆:H1

0(Ω)→H−1(Ω)and ∆:L2R,H1

0(Ω)→L2R,H−1(Ω).

Furthermore, cand Cdenote generic positive constants.

Consider the spatial function spaces

V=H1

0(Ω),V0

i=H1

0(Ωi),and Vi={v∈H1(Ωi):(ˆ

T∂ Ωiv)∂ Ωi\Γ=0}.

Here, ˆ

T∂ Ωi:H1(Ωi)→H1/2(∂ Ωi)denotes the spatial trace operator, see [16, Theo-

rem 6.8.13]. The norm on Viand V0

iis given by

kvkVi=k∇vk2

L2(Ωi)d+kvk2

L2(Ωi)1/2.

By Poincar´

e’s inequality and Assumption 1 we have that v7→k∇vkL2(Ωi)dis an equiv-

alent norm on Viand V0

i. The Hilbert spaces V,Vi, and V0

iare all separable. The space

H1/2(∂ Ωi)is deﬁned as

H1/2(∂ Ωi) = {v∈L2(∂ Ωi):kvkH1/2(∂ Ωi)<∞}with

kvkH1/2(∂ Ωi)=Z∂ ΩiZ∂ Ωi

|v(x)−v(y)|2

|x−y|ddxdy+kvk2

L2(∂ Ωi)1/2.(5)

Denoting the extension by zero from Γto ∂ Ωiby ˆ

Eiwe deﬁne the Lions–Magenes

space as

Λ={µ∈L2(Γ):ˆ

Eiµ∈H1/2(∂ Ωi)}with kµkΛ=kˆ

EiµkH1/2(∂ Ωi).

Since H1/2(∂ Ωi)is a separable Hilbert space, so is Λ. On Vithe trace operator takes

the form

Ti:Vi→Λ:v7→ (ˆ

T∂ Ωiv)Γ,

and is bounded; see [4, Lemma 4.4].

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Time-dependentSteklovPoincar´eoperatorsandspace-timeRobinRobindecompositionfortheheatequationEmilEngstr¨omEskilHansenOctober26,2022AbstractDomaindecompositionmethodsareasetofwidelyusedtoolsforpar-allelizationofpartialdifferentialequationsolvers.Convergenceiswellstudiedforellipticequations,butinth...

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Time-dependent SteklovPoincar e operators and space-time RobinRobin decomposition for the heat equation Emil Engstr omEskil Hansen

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