Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations ^{†}^{†}thanks: This work was funded by CRC 901 Control of selforganizing nonlinear systems: Theoretical methods and concepts of application.
Abstract
Domain decomposition based time integrators allow the usage of parallel and distributed hardware, making them wellsuited for the temporal discretization of parabolic systems, in general, and degenerate parabolic problems, in particular. The latter is due to the degenerate equations’ finite speed of propagation. In this study, a rigours convergence analysis is given for such integrators without assuming any restrictive regularity on the solutions or the domains. The analysis is conducted by first deriving a new variational framework for the domain decomposition, which is applicable to the two standard degenerate examples. That is, the Laplace and the porous medium type vector fields. Secondly, the decomposed vector fields are restricted to the underlying pivot space and the time integration of the parabolic problem can then be interpreted as an operators splitting applied to a dissipative evolution equation. The convergence results then follow by employing elements of the approximation theory for nonlinear semigroups.
Keywords:
Domain decomposition Time integration Operator splitting Convergence analysis Degenerate parabolic equationsMsc:
65M55 65M12 35K65 65J08∎
1 Introduction
Nonlinear parabolic equations of the form
(1) 
equipped with suitable boundary and initial conditions, are frequently encountered in applications. If the diffusion constant vanishes for some values of and , i.e., the equation is degenerate, one obtains a quite different dynamics compared to the linear case. The two main nonlinear features are finite speed of propagation and the absence of parabolic smoothening of the solution. Concrete applications can, e.g., be found when modelling gas flow through porous media, phase transitions and population dynamics. A survey of such applications is given in (Vasquez.2007, , Section 1.3 and Chapter 2). In order to keep the presentation as clearcut as possible, we will mostly ignore the presence of lowerorder advection and reactions terms.
Approximating the solution of a partial differential equation typically results in largescale computations, which require the usage of parallel and distributed hardware. One possibility to design numerical schemes that make use of such hardware is to decompose the equation’s domain into a family of subdomains. The domain decomposition method then consists of an iterative procedure where, in every step, the equation is solved independently on each subdomain and the resulting solutions are thereafter communicated to the adjacent subdomains. This independence of the decomposed equations and the absence of global communication enables the parallel and distributed implementation of domain decomposition methods. For linear parabolic equations the common procedure is to first discretize the equation in time by a standard implicit integrator. Then an elliptic equation on is obtained in every time step, which is iteratively solved by a domain decomposition based discretization. We refer to the monographs Mathew.2008 (); QuarteroniValli.1999 (); ToselliWidlund.2005 () for an indepth treatment of this approach. Another possibility is to apply the domain decomposition method to the full spacetime domain , which leads to an iterative procedure over parabolic problems that can be parallelized both in space and time; see, e.g., Gander.1999 (); GanderHalpern.2007 (); GiladiKeller.2002 ().
When considering nonlinear parabolic problems one finds that there are hardly any results concerning the analysis of domain decomposition based schemes. Two exceptions are the papers KimEtal.2000 (); Lapin.1991 (), where domain decomposition schemes are analyzed for nondegenerate quasilinear parabolic equations and the degenerate twophase Stefan problem, respectively. The lack of results in the context of degenerate equations is rather surprising from a practical point of view, as the equations’ finite speed of propagation is ideal for applying domain decomposition strategies. For example, a solution that is initially zero in parts of the domain will in each time step only propagate to a small number of neighboring subdomains, which limits the computational work considerably. However, from a theoretical perspective the lack of convergence results is less surprising. The issue is that the standard domain decomposition schemes all link together the equations on the subdomains via boundary conditions. As the solutions of degenerate parabolic equations typically lack higherorder regularity, making sense of such boundary linking is, at the very least, challenging.
In order to remedy this, we propose to directly introduce the domain decomposition in the time integrator via an operator splitting procedure. More precisely, let be an overlapping decomposition of the spatial domain , as exemplified in Figure 1. On these subdomains we introduce the partition of unity and the operator decomposition, or splitting,
(2) 
Two possible (formally) firstorder integrators are then the sum splitting
(3) 
which represents a “quick and dirty” scheme that is straightforward to parallelize, and the Lie splitting
(4) 
which is usually more accurate but requires a further partitioning of the subdomains in order to enable parallelization, as illustrated in Figure 1. In contrast to the earlier domain decomposition based schemes, where an iterative procedure is required with possibly many instances of boundary communications, one time step of either splitting scheme only needs the solution of elliptic equations together with the communication of the data related to the overlaps. Similar splitting schemes have, e.g., been considered in the papers Arraras.2015 (); Hansen.2016 (); Mathew.1998 (); Vabishchevich.2013 () when applied to linear, and to some extent semilinear, parabolic problems. However, there does not seem to be any analysis applicable to degenerate, or even quasilinear, parabolic equations in the literature.
Hence, the goal of this paper is twofold. First, we aim to derive a new energetic, or variational, framework that allows a proper interpretation of the operator decomposition (2) for two commonly occurring families of degenerate parabolic equations. These are the Laplace type evolutions, where the prototypical example is given by , and the porous medium type equations, where in the simplest case. For the porous medium application we will use the strategic reformulation
of the decomposition (2), in order to enable an energetic interpretation.
Secondly, we will strive to obtain a general convergence analysis for the domain decomposition based time integrators, including the sum and Lie splitting schemes. The main idea of the convergence analysis is to introduce the nonlinear Friedrich extensions of the operators and , via our new abstract energetic framework, and then to employ a Laxtype result from the nonlinear semigroup theory BrezisPazy.1972 ().
2 Function spaces
Throughout the analysis , , will be an open, connected and bounded set and the parameter is fixed. Next, let be a family of overlapping subsets of such that holds. Here, each is either an open connected set, or a union of pairwise disjoint open, connected sets such that . On we introduce the partition of unity such that
For details on the construction of explicit domain decompositions and partitions of unity we refer to (Arraras.2015, , Section 3.2) and (Mathew.1998, , Section 4.1).
The related weighted Lebesgue space can now be defined as the set of all measurable functions on such that the norm
is finite. The space is a reflexive Banach space, which follows by observing that the map is an isometric isomorphism (DrabekEtAl.1997, , Chapter 1). We will also make frequent use of the product space , equipped with the norm
which is again a reflexive Banach space (AdamsFournier.2003, , Theorem 1.23).
Next, let be a real Hilbert space and denote the space of distributions on by . For a given we introduce the linear operator
which is assumed to be continuous in the following fashion.
Assumption 1
If in then, for ,
As the regularity of the weights implies that for all , we can define the product by
With this in place we can introduce our energetic spaces and as subspaces of given by
and
respectively. On the energetic spaces we consider the operators
where maps to the corresponding functions that can be represented by, and maps to the corresponding functions that can be represented by, respectively.
Lemma 1
.
Proof
For an arbitrary it follows, for , that
for every and . As , we have a representation of in , i.e., for every . Hence, .
Next, assume that . Then we can write
for every and . Let be the zero extension of to the whole of . We can then define the measurable function on as , which satisfies
Furthermore, the norm of can be bounded by
This yields that for , i.e., and we thereby have the identification . ∎
Lemma 2
If Assumption 1 holds, then the operators and , , are linear and closed.
Proof
The linearity of the operators is clear, since is a linear operator. Let the sequence satisfy
Assumption 1 then yields that
for every and . Hence, can be represented by the function , i.e., holds and the operator is therefore closed. The closedness of follows by the same line of reasoning. ∎
On the energetic spaces and , , we define the norms
respectively.
Lemma 3
If Assumption 1 holds, then the spaces and , , are reflexive Banach spaces.
Proof
Consider the reflexive Banach space , equipped with the norm , and introduce the linear and isometric operator
The graph of the closed operator coincides with the image , which makes a closed linear subset of . Here, is a reflexive Banach space (AdamsFournier.2003, , Theorem 1.22) and, as is isometric, it is isometrically isomorphic to . Hence, the latter is also a reflexive Banach space. The same line of argumentation yields that is a reflexive Banach space. ∎
Hereafter, we will assume the following.
Assumption 2
The set is dense in .
Under this assumption it also holds that is a dense subsets of . By the construction of the energetic norms, one then obtains that the reflexive Banach spaces and are densely and continuously embedded in and we have the following Gelfand triplets
Here, the density of in and , respectively, follows, e.g., by (GGZ.1974, , Bemerkung I.5.14). For future reference, we denote the dual pairing between a Banach space and its dual by , and the Riesz isomorphism from to by
Here, the Riesz isomorphism satisfies the relations
for all , and .
Remark 1
Throughout the derivation of the energetic framework we have assumed that the partition of unity consists of elements in . This is somewhat restrictive from a numerical point of view, but this regularity is required if nothing else is known about the operator . Fortunately, in concrete examples; see Sections 6 and 7, one commonly has that . If we then choose a partition of unity in , we have the property that for every , and we can once more derive the above energetic setting by testing with functions in , instead of in .
3 Energetic extensions of the vector fields
With the function spaces in place, we are now able to define the general energetic extensions of our vector fields.
Assumption 3
For a fixed , let fulfill the properties below.

The map fulfills the Carathéodory condition, i.e., is continuous for a.e. and is measurable for every .

The growth condition holds for a.e. and every , where and is nonnegative.

The map is monotone, i.e., for every and a.e. the inequality holds.

The map is coercive, i.e., there exists and such that for every and a.e. the condition holds.
Compare with (Zeidler.1989, , Section 26.3).
We introduce the full energetic operator as
The operator is well defined, as for and by () we obtain that . Furthermore, we define the decomposed energetic operators , , by
These operators are well defined, as
is finite for every , due to (). This family of operators is a decomposition of , as it fulfills
We can now derive the basic properties of the energetic operators.
Lemma 4
Proof
We will only derive the properties for , as the same argumentation holds for . The strict monotonicity of the operator follows using (), as
holds for all with .
Next, we prove that is hemicontinuous, i.e., is continuous on for . Consider a sequence in with limit and introduce
As holds for almost every , due to (), and
where the righthand side is an element, we obtain that
by the dominated convergence theorem. This implies that is hemicontinuous, and the same trivially holds for .
Last, we prove the coercivity of . By assumption (), we have
for every . Hence, we have the limit
as , which implies the coercivity of . ∎
Proof
As and are all, by Lemma 4, strictly monotone, hemicontinuous and coercive, their bijectivity follows by the Browder–Minty theorem; see, e.g., (Zeidler.1989, , Theorem 26.A).∎
4 Friedrich extensions of the vector fields
The energetic setting is too general for the convergence analysis that we have in mind. We therefore introduce the nonlinear Friedrich extensions of our vector fields, i.e., we restrict the domains of the energetic operators such that they become (unbounded) operators on the pivot space . More precisely, we define the Friedrich extension of the full vector field by
Analogously, we introduce the Friedrich extensions , , of the decomposed vector fields by
Proof
Before we continue with our analysis we recapitulate a few properties of a general maximal dissipative operator . The resolvent
is well defined, for every , and nonexpansive, i.e.,
The latter follows directly by the definition of dissipativity. Furthermore, the resolvent and the related Yosida approximation satisfies the following.
Lemma 6
If is maximal dissipative, then
in for every and , respectively.
The proof of Lemma 6 can, e.g., be found in (Barbu.1976, , Proposition II. 3.6) or (Deimling.1985, , Proposition 11.3). Next, we will relate the full vector field with its decomposition .
Proof
Choose a , then and the sum satisfies the relation
for all . Hence, , which yields that and . ∎
Unfortunately, the set is in general not equal to , as does not necessarily imply that for every . This issue is well known and we will encounter it when decomposing the Laplacian; compare with Section 6. We will therefore assume that the mild regularity property below holds.
Assumption 4
for all .
Under this assumption one has the following identification, which is sufficient for our convergence analysis.
Proof
By Lemma 7 and the fact that the maximal dissipative operator is closed (Barbu.1976, , Proposition II.3.4), we obtain that
Next, choose an arbitrary . Since
we can define via
for every . By Lemma 6, we have the limits
Hence, the set is dense in , i.e., its closure in is equal to . ∎
5 Abstract evolution equations and their approximations
With the Friedrich formulation of our full vector field , the parabolic equations all take the form of an abstract evolution equations, i.e.,
(5) 
on . Furthermore, with the decomposition , the splitting schemes (3) and (4) are given by the operators
respectively. Here, and are both approximations of the exact solution at time .
As the resolvent of a maximal dissipative operator is well defined and nonexpansive on , it is a natural starting point for a solution concept. To this end, consider the operator family defined by
where the limit is well defined in for every and ; see (CrandallLiggett.1971, , Theorem I). The operator family is in fact a (nonlinear) semigroup and each is a nonexpansive operator on . The unique mild solution of the evolution equation (5) is then given by the function , which is continuous on bounded time intervals. An extensive exposition of the nonlinear semigroup theory can, e.g., be found in Barbu.1976 ().
There is a discrepancy between the domain of the solution operator, i.e., , and the fact that the operators and are not necessarily invariant over it. In order to avoid several technicalities induced by this, we will assume the following.
Assumption 5
The domain is dense in .
As is the closure of , one has the inclusions
which implies that . Hence, is also dense in when Assumption 5 holds.
We can now formulate the following simplified version of the Laxtype convergence result given in (BrezisPazy.1972, , Corollary 4.3).