Galerkin Time Stepping Methods for Nonlinear IVP

Continuous and Discontinuous Galerkin Time Stepping Methods for Nonlinear Initial Value Problems with Application to Finite Time Blow-Up

Bärbel Holm Department for Computational Science and Technology, School of Computer Science and Communication, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden  and  Thomas P. Wihler Mathematisches Institut, Universität Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland
Abstract.

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

Key words and phrases:
Initial value problems in Hilbert spaces, Galerkin time stepping schemes, high-order methods, blow-up singularities, existence and uniqueness of Galerkin solutions.
2010 Mathematics Subject Classification:
65J08, 65L05, 65L60
The authors acknowledge the support of the Swiss National Science Foundation (SNF), Grant No. 200021-162990.

1. Introduction

In this paper we focus on continuous Galerkin (cG) as well as on discontinuous Galerkin (dG) time stepping discretizations (of any order) as applied to abstract initial value problems of the form

(1.1)

Here, , for some , is an unknown solution, with values in a real Hilbert space  (with inner product denoted by  and induced norm ). The initial value prescribes the solution  at the start, , and  is a possibly nonlinear, continuous operator. We emphasize that we include, for instance, the case of  being (continuous and nonlinear and) unbounded in the sense that

(1.2)

In the sequel, we will usually omit to explicitly write the dependence on the first argument .

For  and continuous nonlinearities , the well-known Peano Theorem (see, e.g., [17]) guarantees the existence of -solutions  of (1.1) within some limited time range, , for some . Generalizations to problems in Banach spaces are available as well; see, e.g., [9]. Notice that the existence interval for solutions may be arbitrarily small even for smooth : For instance, solutions of (1.1) may become unbounded in finite time, i.e.,

This effect is commonly termed (finite-time) blow-up.

Galerkin Time Stepping

Galerkin-type time stepping methods for initial-value problems are based on weak formulations. For both the cG and the dG time stepping schemes, the test spaces consist of polynomials that are discontinuous at the time nodes. In this way, the discrete Galerkin formulations decouple into local problems on each time step, and the discretizations can therefore be understood as implicit one-step schemes. Galerkin time stepping methods have been analyzed for ordinary differential equations (ODEs), e.g., in [3, 6, 5, 7, 8, 10].

A key feature of Galerkin time stepping methods is their great flexibility with respect to the size of the time steps and the local approximation orders, thereby naturally leading to an -version Galerkin framework. The -versions of the cG and dG time stepping schemes were introduced and analyzed in the works [12, 13, 15, 19]. In particular, in the articles [12, 19], which focus on ordinary initial value problems with uniform Lipschitz nonlinearities, the use of the contraction mapping theorem made it possible to prove existence and uniqueness results for discrete Galerkin solutions, which are independent of the local approximation orders. We emphasize that the -approach is well-known for its ability to approximate smooth solutions with possible local singularities at high algebraic or even exponential rates of convergence; see, e.g., [4, 13, 14, 18] for the numerical approximation of problems with start-up singularities.

Results

The goal of the current paper is to extend the existence results on -type Galerkin time stepping schemes for initial value problems with Lipschitz-type nonlinearities in [12, 19] to problems with nonlinearities which are merely continuous. We emphasize that this generalization is substantial; indeed, it covers, for example, the case of unbounded nonlinearities as in (1.2). We will develop a new technique which is based on writing the weak Galerkin formulations in strong form along the lines of [1, 15]. Subsequently, suitable fixed-point forms will be derived. In the context of the cG method, this is accomplished within an integral equation framework. For the dG scheme, matters are more sophisticated, and a careful investigation of the discrete time derivative operator, which involves a lifting operator from [15], is required on the local polynomial approximation space; this operator turns out to be an isomorphism on the underlying polynomial spaces (with a continuity constant of the inverse operator that is independent of the local polynomial degrees) and allows to transform the strong dG form into a fixed point equation. For both the cG and the dG schemes the application of Brower’s fixed point theorem yields the existence of discrete solutions; see Theorem 1. In particular, as in the case of Lipschitz continuous nonlinearities [1, 15], the existence results do not depend on the local polynomial degrees, and only require the local time steps to be sufficiently small. In this sense, our theory constitutes a discrete version of Peano’s Theorem. Furthermore, employing a contraction argument along the lines of the approach presented in [2], we show that the local Galerkin formulations are uniquely solvable (within a certain range); cf. Theorem 2.

In addition, we apply our general theory to initial value problems with nonlinearities of algebraic growth, i.e., , with , , and for a given range of ; in this case, the initial value problem (1.1) features a solution that blows up in a finite time . We will show that a careful selection of locally varying time steps in the cG and dG time stepping schemes results in discrete solutions that blow up as well; in this context, we mention the paper [16] which illustrates the importance of variable step size selection. More precisely, following some ideas from [11], we derive an analysis which allows to choose the local time steps a posteriori as the time marching process is moving forward. We develop a time step selection algorithm which guarantees the existence and uniqueness of local solutions, and provides a numerical approximation of the exact blow-up time. Moreover, we prove a convergence result which shows that the blow-up time can be approximately arbitrarily well if the time steps are scaled sufficiently small.

The concepts and technical tools developed in our current work constitute an important stepping stone with regard to the numerical treatment of finite time blow-up problems in the context of nonlinear parabolic partial differential equations.

Outline

Our article is organized as follows: Section 2 presents the cG and dG time stepping schemes. Furthermore, Section 3 centres on the development of existence proofs for discrete solutions. The question of uniqueness is addressed in Section 4. Moreover, the application of our results to algebraically growing nonlinearities causing finite time blow-ups will be worked out in Section 5. Finally, the article closes with a few concluding remarks in Section 6.

Notation

Throughout the paper, Bochner spaces will be used: For an interval and a real Hilbert space  as before, the space consists of all functions that are continuous on  with values in . Moreover, introducing, for , the norm

we write to signify the space of measurable functions so that the corresponding norm is bounded. We notice that  is a Hilbert space with inner product and induced norm

respectively.

2. Galerkin Time Discretizations

In this section we present the -cG and -dG time stepping methods as applied to (1.1).

2.1. -cG Time Stepping

On an interval , , consider time nodes which introduce a time partition of  into  open time intervals , . The (possibly varying) length of a time interval is called the time step. Furthermore, to each interval we associate a polynomial degree  which takes the role of a local approximation order. Moreover, given a (real) Hilbert space , an integer , and an interval , the set

signifies the space of all polynomials of degree at most  on  with values in .

In practical computations, the Hilbert space , on which (1.1) is based, will typically be replaced by a finite-dimensional subspace , , on each interval , . The -orthogonal projection from  to  is defined by

(2.1)

With these definitions, the (fully discrete) -cG time marching scheme is iteratively given as follows: For given initial value  (with , where  is the initial value from (1.1)), we find  through the weak formulation

(2.2)

for any . Notice that, in order to enforce the initial condition on each individual time step, the local trial space has one degree of freedom more than the local test space. Furthermore, if , we remark that the continuous Galerkin solution  is globally continuous on .

2.2. -dG Time Stepping

In order to define the discontinuous Galerkin scheme, some additional notation is required: We define the one-sided limits of a piecewise continuous function at each time node by

Then, the discontinuity jump of at , , is defined by , where we let , with  being the initial condition from (1.1). Then, the (fully discrete) -dG time stepping method for (1.1) reads: Find such that

(2.3)

for any . We emphasize that, in contrast to the continuous Galerkin formulation, the trial and test spaces are the same for the discontinuous Galerkin scheme. This is due to the fact that the initial values are weakly imposed (by means of an upwind flux) on each time interval.

3. Existence of Discrete Solutions

In this Section our goal is to show existence of solutions to the discrete local problems (2.2) and (2.3):

Theorem 1.

Let . Then, if the local time step  is chosen sufficiently small (independent of the local polynomial degree ), then the continuous Galerkin method (2.2) and the discontinuous Galerkin method (2.3) on  both possess at least one solution  and , respectively.

Our general strategy of proof is to represent the Galerkin formulations in terms of strong equations, and then to derive suitable fixed-point formulations. Subsequently, the existence of discrete solutions will follow from the application of Brower’s fixed point theorem.

3.1. Existence of cG Solutions

We begin by rewriting (2.2) as finding  such that

Here, denotes the -projection onto the space , which is uniquely defined by

(3.1)

Thence, noticing that , we obtain the strong form

Integration results in

(3.2)

We see that the operator

(3.3)

maps  into itself, and hence, the integral equation (3.2) is a fixed point formulation,

(3.4)

on . In particular, any solution of (3.2) will solve (2.2).

We are now ready to prove Theorem 1 for the continuous Galerkin method (2.2): For some  (with ) let us define the set

where

(3.5)

Since  is continuous, its maximum on the compact set ,

(3.6)

exists. We let

Then, we introduce

(3.7)

where , with .

Let  be arbitrary, and  such that

Then, using Bochner’s Theorem as well as the Cauchy-Schwarz inequality, yields

Taking into account the boundedness of the -projection on  (with constant 1) leads to

Therefore,

Thus, we have , and more generally, it follows . Finally, since  is convex and compact, and  is continuous, Brower’s fixed point theorem implies that there exists at least one solution of (3.4) in , and thus of (2.2).

3.2. Existence of dG Solutions

The situation for the dG method is more involved. We will commence by looking at a discrete dG time operator appearing in the dG formulation.

3.2.1. Discrete dG Time Operator

Following [15, Section 4.1] we define the lifting operator, for ,

by

on a real Hilbert space , with inner product , and norm .

In view of this definition with , we have for the dG solution  from (2.3):

for any . Here, is the -orthogonal projection from (2.1), and is the -projection from (3.1).

Then, since , , all belong to , we arrive at the strong formulation

(3.8)

of (2.3). The term on the left-hand side of this equation is the -dG time discretization of the continuous derivative operator . This motivates the definition of a discrete operator

(3.9)

given by

(3.10)

For the proof of existence of solutions of (3.8) it is important to notice that the linear operator  is invertible.

Proposition 1.

Let  be a real Hilbert space, and . Then, the operator  from (3.9)–(3.10) is an isomorphism on . In addition, there exists a constant  independent of the time step  and the local approximation order  such that, for any , there holds the bound

(3.11)

for any .

In order to establish this estimate, we require two auxiliary results which will be proved first.

Lemma 1.

Let , and  a real Hilbert space. Then, there holds

(3.12)

for any .

Proof.

Let us first consider the lifting operator  on the unit interval , defined by

Referring to [15, Eq. (35) and Lemma 8] there holds the explicit formula

where

with  signifying the family of Legendre polynomials on  (with degrees ), scaled such that ; cf. [15, Eq. (9) and Lemma 1]. Combining the above identities, we obtain

Noticing the telescope sum as well as the fact that  and , we arrive at

Then, employing the fact that

(3.13)

results in

Now we define the affine mapping

(3.14)

A scaling argument implies that

see [15, Lemma 7]. Hence, by a change of variables, , , we conclude that

Noticing that, for , there holds equality in the above bound, completes the proof. ∎

Lemma 2.

Let , and  a real Hilbert space. Then, the bound

(3.15)

holds true for any .

Proof.

Let . We define

where  is the -th Legendre polynomial on , which we scale such that  (cf. the proof of Lemma 1), and  is the affine element mapping from (3.14). Then,

Involving (3.13) shows

(3.16)

Furthermore, is orthogonal to the space  (where ) with respect to the inner product in . In particular, since , we have

So, noticing that , it follows that

Therefore, using Hölder’s inequality and recalling (3.16), we conclude that

Dividing by  shows the desired bound. ∎

We are now ready to show Proposition 1.

Proof of Proposition 1.

Consider  . We choose  such that . It holds that

Applying the triangle inequality as well as Bochner’s Theorem, and recalling (3.12), this implies that

Inserting the bound (3.15) results in

and applying Hölder’s inequality completes the proof with . ∎

Remark 1.

The proof of Proposition 1 reveals the upper bound . We emphasize, in particular, that the estimate (3.11) is uniform with respect to the local polynomial degree  as .

Remark 2.

Upon setting  in (3.11), we obtain

(3.17)

for any .

3.2.2. Fixed Point Formulation and Existence of Discrete dG Solutions

As for the cG method we prove the existence of solutions of (2.3) by means of a fixed point argument. For this purpose, we will derive a suitable fixed point formulation, and return to the case . Noticing the fact that , we observe that, on , there holds

and recalling (3.8), we can write

Applying Proposition 1 we infer that

this is the ‘dG-version’ of the integral equation (3.2) for the cG method. Now, for given  (where as before ) we define the operator

by

(3.18)

Then, solves (3.8) if and only if  satisfies

(3.19)

We will now prove the existence of solutions to the local -dG time stepping scheme (2.3): Consider  (with ), and define the set

where

(3.20)

Due to the continuity of , its maximum on the compact set ,

(3.21)

exists. We choose

(3.22)

where  is the constant from (3.11), and introduce

(3.23)

with , .

Consider any . From the definition of  in (3.18), and from (3.17) with , we conclude that

The boundedness of the -projection on  (with constant 1) implies that

Then, we obtain

since . This implies that