3.4 Convergence analysis
3.4.4 Order two error estimate for the five parts modified Strang splitting 40
2qk 1+ Z ⌧
0
f( fs(vk 1
⌧
2qk 1))ds.
We then apply the midpoint quadrature formula to the integral and obtain an error of size O(⌧2) since f is twice continuously di↵erentiable:
f⌧
2(uk) = vk 1+⌧f( f⌧
2(vk 1
⌧
2qk 1)) ⌧
2qk 1+O(⌧2).
We observe that f⌧
2(vk 1 ⌧
2qk 1) =uk. Since by Proposition 3.4.2, uk =f(u(tk)) +O(⌧) and since f(u(tk)) =f(u(tk 1)) +O(⌧), we obtain
f⌧
2(uk) =vk 1 +⌧
2f(u(tk)) + ⌧
2f(u(tk 1)) ⌧
2qk 1+O(⌧).
Since Bvk 1 =bk, and since by hypothesis f(u(tk 1)) qk 1 = k 1 +O(⌧) with k 1 2 D(A) andA k 1 =O(1), we obtain
Bqk=2
⌧(Bvk 1 bk) +Bf(u(tk)) +Bf(u(tk 1)) Bqk 1+BO(⌧) =Bf(u(tk)) +BO(⌧).
We can therefore decompose qk as qk = ˜qk+rk, where Bq˜k = Bf(u(tk)) and rk = O(⌧) and choose k = ˜qk f(u(tk)). ThenB k = 0 and a simple calculation yields kA kk C.
This concludes the proof.
3.4.4 Order two error estimate for the five parts modified Strang splitting
The following lemma is an estimation of the local error for the modified Stang splitting, which states that the five parts modified Strang splitting (3.16) is locally a method of second order.
3.4. CONVERGENCE ANALYSIS 41 Lemma 3.4.14. Under the assumption of section 3.2, the five parts modified Strang split-ting (3.16) satisfies
n+1 =O(⌧2).
Proof. In the exact formula of n+1 (3.28), we use the left rectangle quadrature formula for the first and third integral and a right quadrature formula for the second integral. By Lemma 3.4.4, the quadrature error isO(⌧2). We get
n+1 = e⌧A⌧
2(f(u(tn)) qn) +⌧(qn f(u(tn+⌧))) + ⌧
2(f(˜vn) qn) +O(⌧2).
Expending e⌧Aandu(tn) around⌧, that is, e⌧A⌧2(f(u(tn)) qn) = ⌧2(f(u(tn+1)) qn)+O(⌧2), we obtain the following result:
n+1 = ⌧
2(f(˜vn) f(u(tn+1)) +O(⌧2).
We use the exact formula for ˜vn and u(tn+1) and the Lipschitz continuity of f: k n+1k C⌧
2 e⌧A Z ⌧/2
0
(f( fs(u(tn))) qn)ds+ Z ⌧
0
e(⌧ s)A(qn f(u(tn+s)))ds +O(⌧2).
Since the integrands are uniformly bounded on [0,⌧], we get n+1 =O(⌧2), which concludes the proof.
The following lemma gives the second local error estimates that we need in the proof for the global convergence of the method.
Lemma 3.4.15. Under the assumption of section 3.2, the five parts modified Strang split-ting (3.16) satisfies
n+1 =AO(⌧3) +O(⌧3).
Proof. We start as in the proof of Lemma 3.4.8, by using trapezoidal quadrature formulas to approximate the integrals in formula (3.28) of the local error. By Lemma 3.4.5, the quadrature error made to approximateR⌧
0 e(⌧ s)A(qn f(u(tn+s)))ds is equal toAO(⌧3) + O(⌧3). We get
n+1 = e⌧A⌧ 4
⇣f(u(tn)) qn+f( f⌧/2(u(tn))) qn⌘ +⌧
2 e⌧A(qn f(u(tn))) +qn f(u(tn+⌧)) +⌧
4
⇣f(˜vn) qn+f( f⌧/2(˜vn)) qn⌘
+AO(⌧3) +O(⌧3) By Lemma 3.4.14, we have the following equality:
f( f⌧/2(˜vn)) =f(S⌧(u(tn))) =f(u(tn+⌧)) +O(⌧2). (3.30) We obtain
n+1 = e⌧A⌧
4(f( f⌧/2(u(tn))) qn) ⌧
4e⌧A(f(u(tn)) qn) + ⌧
4(f(˜vn) qn) ⌧
4(f(u(tn+⌧)) qn) +AO(⌧3) +O(⌧3).
42 CHAPTER 3. FIVE PARTS MODIFIED STRANG SPLITTING We observe that ⌧2e⌧A(f(u(tn)) qn) + ⌧2(f(u(tn+⌧)) qn) is the trapezoidal quadrature formula forR⌧
0 e(⌧ s)A(f(u(tn+s))) qn)dsand that e⌧A⌧2(f( f⌧/2(u(tn))) qn) +⌧2(f(˜vn) qn) is the trapezoidal quadrature formula for R⌧
0 e(⌧ s)A(f( f(⌧ s)/2 s/2qn D+qs n s/2qn
f
s/2(u(tn))) qn)ds. Since f(⌧ s)/2 s/2qn D+qs n s/2qn fs/2(u(tn)) =u(tn) +O(⌧), the second integrand satisfies the hypotheses of Lemma 3.4.5, by Lemmas 3.4.12 and 3.4.13.
Applying Lemma 3.4.5, we therefore have a quadrature error of the formAO(⌧3) +O(⌧3),
n+1 = 1 2
Z ⌧ 0
e(⌧ s)A(f( f⌧ s 2
qn s 2
D+qn
s
qn s 2
fs
2(u(tn))) qn)ds 1
2 Z ⌧
0
e(⌧ s)A(f(u(tn+s)) qn)ds+AO(⌧3) +O(⌧3).
Applying the midpoint quadrature method to both integrals, we obtain
n+1 = ⌧
4e⌧2A(f(S⌧2(u(tn))) qn) ⌧
4e⌧2A(f(u(tn+ ⌧
2)) qn) +AO(⌧3) +O(⌧3).
Using Lemma 3.4.14, the Lipschitz continuity of f, and the boundedness of e⌧2A we have the desired result,
n+1 =AO(⌧3) +O(⌧3), which concludes the proof.
We now improve Lemma 3.4.15 as follows using Lemma 3.4.6 instead of Lemma 3.4.5 for the quadrature error.
Lemma 3.4.16. Under the assumption of section 3.2, the five parts modified Strang split-ting (3.16) satisfies the following. There exists ↵>0 such that
n+1 = ( A)1 ↵O(⌧3) +O(⌧3).
Using the previous results for the local error, we can prove Proposition 3.4.3. It is the main global error estimate that we obtain without using Theorem 3.2.1.
Proof of Proposition 3.4.3. The proof is similar to the proof of the Proposition 3.4.10, except for the bounds of the local errors n, since if (3.17) is satisfied, k =AO(⌧3) +O(⌧3) and n =O(⌧2). Hence
kenk C⌧2(1 +|log(⌧)|)e(n 1)C⌧ C⌧2(1 +|log(⌧)|)eCT =C⌧2(1 +|log(⌧)|), which concludes the proof.
Proof of Theorem 3.4.1. We follow the proof of Proposition 3.4.2, using Lemma 3.4.16 to remove the log(⌧) term in the global error estimate of Proposition 3.4.3.
3.5. NUMERICAL EXPERIMENTS 43
3.5 Numerical experiments
In this section we perform several numerical experiments in dimensiond= 1,2 to illustrate the performance of the five parts modified Strang splitting (3.16) when applied to di↵usion problems with various nonlinearities. The norm we use to compute the numerical error is
kukL1([0,T],L2(⌦)) = sup
t2[0,T]ku(t)kL2(⌦),
where we use the trapezoidal rule to approximate theL2(⌦) norm. Considering the domain (0,1)d equipped with an uniform spatial grid with size x, we use the usual second order finite di↵erence approximation of the Laplacian. We compute the di↵usion flows in time exactly using the Matlab package described in [53] for approximating matrix exponentials and the '1 matrix function (3.12).
In the numerical experiments that follow, we denote the classical Strang splitting (3.2) by Strang, the modified Strang splitting (3.7) constructed in [24, 25] by StrangM3, the five parts modified Strang splitting (3.16) with correction (3.2) byStrangM5a and the five parts modified Strang splitting (3.16) with correction (3.3) by StrangM5b. In Table 3.1, we recall the number of evaluations of the di↵usion flows (3.5, 3.8) and the number of evaluations of the source term flows (3.4, 3.6) required at each step of the algorithm for the four considered numerical methods. We also collect the total number of flows after n steps, where we consider that the correction steps (3.15) are negligible. We recall that for Strang and StrangM5b, only one evaluation of the di↵usion flow and one evaluation of the source term flow is needed after the first step, due to the semigroup property of the exact flows (see Remark 3.3.2).
Splitting methods Di↵usion flows Source term flows Total number of for one step for one step flows for n steps
Classical Strang (3.2) 1 1 2n+ 1
StrangM3 (3.7) 2 1 3n
StrangM5a (3.16, 3.17) 1 2 3n
StrangM5b (3.16, 3.18) 1 1 2n+ 1
Table 3.1: Number of evaluations of the di↵usion flows (3.5, 3.8) and of the source term flows (3.4, 3.6) needed by the considered splitting methods.
A quadratic nonlinearity(see Figure 3.1) First, we apply the splitting methodsStrang (3.2), StrangM3 (3.7), StrangM5a (3.16, 3.17) and StrangM5b (3.16, 3.18) to a problem given in [25, Example 5.2]. We then change the nonlinearity to see how the splitting methods behave. The non linearities we consider aref(u) =u2 andf(u) = 5u2. The casef(u) = u2 is the one presented in [25, Example 5.2]. We perform the experiment with mixed bound-ary conditions, u(0) = 1, @nu(1) = 1. We choose a smooth initial condition that satisfies the prescribed boundary conditions. We obtain the following equation with m = 1 and
44 CHAPTER 3. FIVE PARTS MODIFIED STRANG SPLITTING
10-3 10-2
Time step 10-8
10-6 10-4 10-2
Error
Strang StrangM3 StrangM5a StrangM5b
f(u) =u2
10-3 10-2
Time step 10-8
10-6 10-4 10-2
Error
Strang StrangM3 StrangM5a StrangM5b
f(u) = 5u2
Figure 3.1: Comparison between the splitting methods Strang (3.2), StrangM3 (3.7), StrangM5a (3.16, 3.17) and StrangM5b (3.16, 3.18) when applied to problem (3.31) for m = 1 and m = 5 on [0,1] with inhomogeneous mixed boundary conditions. The conver-gence curves of StrangM5a and StrangM5b overlap. Reference slopes of order one and two are given in dotted lines.
m= 5.
@tu(x, t) =@xxu(x, t) +mu(x, t)2, u(0, t) = 1, @nu(1, t) = 1, u(x,0) = 1 + 2
⇡ 2
⇡cos(1
2⇡x). (3.31)
The correction we use for StrangM3, given in [25] for m = 1, is qn =m+ 2mxun(1). We use 500 spatial points to discretize the interior of⌦. We compute the solution at final time T = 0.1. The chosen time steps are ⌧ = 0.02·2 k, k = 0, . . . ,6. The reference solution is computed with the classical fourth order explicit Runge-Kutta method and a very small time step ⌧ = 0.02·2 14 ⇡10 6. In the splitting algorithms, we use the analytic formulas for computing the flows f⌧
2(un) and f q⌧ n(un).
We observe in Figure 3.1 that the splittingsStrangM3, StrangM5a and StrangM5b are all of order two compared to the classical splitting Strang which is of order one. The methodsStrangM5a and StrangM5b have a slightly worse error constant compared to the splitting StrangM3 of [25] for f(u) = u2. Recall that StrangM3 costs two di↵usion flows per time step while Strang, StrangM5a and StrangM5b cost only one. For f(u) = 5u2, StrangM5a and StrangM5b become a lot more accurate.
A meteorology model with an integral source term(see Figure 3.2) We apply the splitting methods Strang (3.2), StrangM3 (3.7), StrangM5a (3.16, 3.17) and StrangM5b (3.16, 3.18) to a problem presented in [59, Equation 3.1], where we replace the left Dirichlet boundary condition u(0, t) = 2(2 p
t) by u(0, t) = 2(2 t) to have a
3.5. NUMERICAL EXPERIMENTS 45
Figure 3.2: Comparison between the splitting methods Strang (3.2), StrangM3 (3.7), StrangM5a(3.16, 3.17) and StrangM5b(3.16, 3.18) when applied to the integro-di↵erential equation (3.32) with time dependent boundary conditions. Reference slope ones and two are drawn in dotted line. On the right picture, we compare the error with the number of evaluations of the di↵usion flows (3.5, 3.8) and the number of evaluations of the source term flows (3.4, 3.6) computed during the time integration.
time continuously di↵erentiable boundary condition. The considered di↵erential equation is the following
for ⌦ = [0,1]. We choose 500 points to discretize the interior of [0,1]. We then apply the splitting methods with di↵erent time steps ⌧ = 2 ·10 2 ·2 k, k = 0, . . . ,6. The reference solution is computed with the classical fourth order explicit Runge-Kutta method and a very small time step ⌧ = 0.02·2 14 ⇡ 10 6. To solve the integral, we use the trapezoidal quadrature formula with the 502 nodes given by the space discretization. To solve @tu = f(u) and @tu = f(u) qn, we apply five steps of the classical order four explicit Runge-Kutta method with a time step 10⌧ . Note that we compute @tu =f(u) and
@tu=f(u) qn with more precision than is needed in practice to emphasize on the error of the splitting itself. Instead, one can simply use one step of a second order Runge-Kutta method. We compute the solution at final time T = 0.1.
We observe in the left picture of Figure 3.2 that the modified splitting StrangM3 given in [25] has a slightly better error constant than StrangM5a and StrangM5b. In the right picture of Figure 3.2 however, we recover that the splittingStrangM5b requires less evalua-tions of the flows to obtain a given precision (see Table 3.1 and Remark 3.3.2). Since both flows have similar computational costs, the splitting StrangM5b is slightly more accurate for the same computational cost. Moreover the construction of the correction qn for the splitting StrangM5b requires less computational cost and is easier to implement than the
46 CHAPTER 3. FIVE PARTS MODIFIED STRANG SPLITTING correction qn of StrangM3 given by (3.9). Indeed, one needs then to evaluate f on the boundary at each step of the algorithm.
Case of a sti↵ nonlinearity (see Figure 3.3) In the next experiments, we compare Strang (3.2), StrangM3 (3.7), StrangM5a (3.16, 3.17) and StrangM5b (3.16, 3.18) when applied to a sti↵ problem in a two dimensional domain ⌦ = [0,1]⇥[0,1]. We choose the nonlinearity f(u) = (1 Msin(⇡x) sin(⇡y))u2 where M = 1 in the nonsti↵ case and M = 100 in the sti↵ case. We impose Dirichlet boundary conditions on the left boundary and Neumann boundary conditions on the bottom, top and right boundaries.
The boundary conditions are chosen to be consistent with the initial conditionu0 = ex+e2 y. We obtain the following problem
@tu(x, y, t) =@xxu(x, y, t) +@yyu(x, y, t) + (1 Msin(⇡x) sin(⇡y))u(x, y, t)2, u(0, y, t) = 1 + ey
2 , @nu(1, y, t) = e
2, @nu(x,0, t) = 1
2, @nu(x,1, t) = e 2, u(x, y,0) = ex+ ey
2 , (3.33)
where x, y 2 ⌦, t 2[0, T] and with M = 1 or M = 100. The correction functions qn are constructed with a multigrid iterative smoothing algorithm, analogously to [25, Algorithm 1]. We recall that the method StrangM3 in [25] was proposed for nonsti↵ nonlinearities.
Indeed note that qn is of size O(M) and becomes large for the sti↵ case M = 100. For instance in dimension one, on the domain [0,1] withf(u) = (1 Msin(⇡x))u2and boundary conditions u(0, t) = 1, @nu(1, t) = 1, one obtains qn = 1 + (M⇡un(1)2+ 2un(1))x. We use a mesh with size x = 1281 to discretize the interior of ⌦. We compute the solution at final time T = 0.1. The chosen time steps are ⌧ = 0.1·2 k, k = 0, . . . ,8. The reference solution is computed with the classical fourth order explicit Runge-Kutta method and a really small time step⌧ = 0.1·2 14 ⇡10 6. In the splitting algorithms, we use the analytic formulas of f⌧
2(un) and f q⌧ n(un) .
We observe in the top pictures of Figure 3.3 that StrangM5a and StrangM5b are slightly more accurate than the modified Strang splittingStrangM3 forM = 1 and drastically more accurate for the sti↵ case M = 100. Indeed, as explained above, in this later sti↵ case, the accuracy of the splitting StrangM3 deteriorates due to the large correction involved.
Moreover, in the bottom pictures of Figure 3.3, we observe that for the same number of evaluations of the di↵usion flows, StrangM5b is more accurate than all the other methods also for M = 1. Indeed, here, the di↵usion flows dominates the total cost and we recall that StrangM5b only requires one evaluation of the di↵usion flow and one evaluation of the source term flow at each step (see Table 3.1 and Remark 3.3.2), analogously to the classical Strang splittingStrang.
3.5. NUMERICAL EXPERIMENTS 47
10-3 10-2 10-1
Time step 10-6
10-4 10-2 100
Error
Strang StrangM3 StrangM5a StrangM5b
Nonsti↵caseM = 1.
10-3 10-2 10-1
Time step 10-6
10-4 10-2 100
Error
Strang StrangM3 StrangM5a StrangM5b
Sti↵case M = 100.
101 102
Number of evaluations of the diffusion flows 10-6
10-4 10-2 100
Error
Strang StrangM3 StrangM5a StrangM5b
Nonsti↵caseM = 1.
101 102
Number of evaluations of the diffusion flows 10-6
10-4 10-2 100
Error
Strang StrangM3 StrangM5a StrangM5b
Sti↵case M = 100.
Figure 3.3: Comparison between the splitting methods Strang (3.2), StrangM3 (3.7), StrangM5a (3.16, 3.17) and StrangM5b (3.16, 3.18) when applied to equation (3.5) with M = 1 andM = 100, on [0,1]2 with inhomogeneous mixed boundary conditions. Reference slopes of order one and two are given in dotted lines.
Chapter 4
Superconvergence of the Strang
splitting when using the Crank-Nicolson scheme for parabolic PDEs with
Dirichlet and oblique boundary conditions
This chapter is identical to the paper [11] in collaboration with Christophe Besse and Gilles Vilmart, except for the unpublished Figure 4.6.
4.1 Introduction
We consider a parabolic semilinear di↵erential problem, in a smooth bounded domain ⌦ inRd in dimensiond 1, for t 2[0, T], of the form
@tu(x, t) =Du(x, t) +f(x, u(x, t)) in ⌦⇥(0, T], Bu(x, t) =b(x) on @⌦⇥(0, T],
u(x,0) =u0(x) in ⌦, (4.1)
where, for 1< p < 1,D:W2,p(⌦)!Lp(⌦) is a linear di↵usion operator andf :Lp(⌦)! Lp(⌦) is a possibly nonlinear source term. The operatorB represents boundary conditions of type Dirichlet, Neumann or Robin. When time-discretizing a problem of this form, it can be advantageous to use a splitting method in order to divide the problem (4.1) into two parts: the source equation
@tu(x, t) =f(x, u(x, t)) in ⌦⇥(0, T], (4.2) and the di↵usion equation
@tu(x, t) = Du(x, t) in ⌦⇥(0, T], Bu(x, t) =b(x) on @⌦⇥(0, T]. (4.3) 49
50 CHAPTER 4. STRANG SPLITTING WITH CRANK-NICOLSON We use respectively the notations ft and Dt to denote the exact flows of the subprob-lems (4.2) and (4.3). The advantage of this subdivision is that equations (4.2) and (4.3) can often be solved more efficiently than the main problem (4.1). The classical splitting method used to approximate the main problem (4.1) is the Strang splitting. Starting from an arbitrary initial datum un, one step of the Strang splitting method, with a time step
⌧ >0, applied to equation (4.1) is given by un+1 = f⌧
Interchanging the roles of the di↵usion part and nonlinear part, it is also possible to define one step of the Strang splitting method as
un+1 = D⌧
2
f⌧ D
⌧
2(un). (4.5)
In both cases, we start the procedure with the initial conditionu0 of the parabolic prob-lem (4.1). Both methods (4.4) and (4.5) are formally of order of accuracy two. However, a reduction of order occurs in general, particularly for inhomogeneous boundary conditions, as observed in [39] and [40]. A suitable correction of the splitting algorithm has been proposed in [25], to avoid order reduction phenomenon. In [12], an alternate correction was proposed that depends only on the flow f⌧
2 and facilitates the calculation of the cor-rection. In this paper, we will however not use the techniques developed in [25], [12]. We prove that when the Crank-Nicolson scheme is used to solve the di↵usion equation (4.3) in the splitting (4.4), there is no reduction of order away from a neighbourhood oft = 0.
This superconvergence property appears specific to the Crank-Nicolson scheme and when another Runge-Kutta method or even the exact flow itself is used to approximate the dif-fusion subproblem, the order reduction of the splitting (4.4) is not avoided. We denote by
D,CN
t the numerical flow of the Crank-Nicolson method for the di↵usion problem (4.3).
We obtain the following splitting method, where D⌧ has been replaced by D,CNt in the splitting (4.4),
We prove that the splitting (4.6) has no order reduction when the nonlinearity f = f(x) only depends on the space variable. More precisely, we prove in this case the following exact representation of the error at timetn=n⌧,
un u(tn) = r(⌧A)n en⌧A A 1(Du0+f), (4.7) where A is the restriction of the operator D to D(A) = {u 2 W2,p(⌦) ; Bu = 0 on @⌦}, the set of functions satisfying the homogeneous boundary condition Bu(x) = 0 on the boundary @⌦, and where r(z) = (1 + z2)/(1 z2) is the stability function of the Crank-Nicolson scheme.
As seen in [44, Theorem 4.2 and Theorem 4.4], for the simplified case wheref and bare both zero in (4.1), that is for@tu(x, t) =Au(x, t), second order convergence results for A-stable methods (see [35, Chapter IV.3]) usually require u0 2 D(A2). In contrast, L-stable methods are second order convergent outside a neighbourhood of the origin even if u0 2 Lp(⌦). The Crank-Nicolson scheme, although it is not L-stable but only A-stable, is second order convergent outside a neighbourhood of the origin for u0 2 D(A) (see [37, Theorem
4.2. ANALYTICAL FRAMEWORK 51 2.1]). Maximal parabolic regularity of A-stable Runge-Kutta methods is studied in [43].
To the best of our knowledge, there exists no result in the literature which proves already that the Crank-Nicolson scheme is second order convergent outside a neighbourhood of the origin when applied to a nonlinear parabolic problem with inhomogeneous boundary conditions and an initial condition u0 2{u2W2,p(⌦) ; Bu=b on @⌦}.
This is a direct consequence of the convergence result of the splitting (4.6) since forf = f(x), the splitting with Crank-Nicolson (4.6) is equal to the Crank-Nicolson scheme (4.13) applied to the whole problem (4.1).
We also provide numerical experiments in the case where f depends on the solution u, in which case the splitting (4.6) is di↵erent from the Crank-Nicolson scheme, and observe that the splitting method (4.6) remains second order convergent in this case. Note also that the superconvergence property does not hold for the other splitting methods given by un+1 = D,CN⌧
2
f⌧
D,CN⌧
2 (un) nor does this splitting preserve stationary states for f =f(x).
This chapter is organized as follows. In Section 4.2, we describe an appropriate analyt-ical framework for the analysis, where D is chosen to be a second order elliptic operator and B a first order di↵erential operator corresponding to Dirichlet or oblique boundary conditions. In Section 4.3, we describe precisely the algorithm corresponding to the split-ting (4.6). In Section 4.4, we restrict ourselves to the case wheref does not depend on the solutionu. We first provide an exact representation of the local error (Proposition 4.4.10).
We then prove formula (4.7) for the global error (Theorem 4.4.3) and additionally conclude that the stationary states are preserved by the splitting method (4.6) (Remark 4.4.6). In Section 4.5, we provide numerical experiments to illustrate the properties of the splitting method (4.6) compared to several natural splitting methods in dimensions one and two for constant and nonlinear terms.
4.2 Analytical framework
We follow closely the framework and the notations given in the book [50, Chapter 3]. Let
⌦ be an open bounded subset of Rd with C2 boundary @⌦ and dimension d 1. For
where aij and bi are real continuous functions on ⌦. We assume that the matrix [aij(x)]ij
is symmetric and uniformly positive definite on ⌦, i.e for all x 2 ⌦ and for all ⇠ 2 Rd,
⇠T[aij(x)]ij⇠ c⇠T⇠, wherec > 0 is independent ofx. The source termf :Lp(⌦)!Lp(⌦) is assumed continuously di↵erentiable and we assume that the initial condition u0 belongs toW2,p(⌦).
The linear operator B : W2,p(⌦) ! W1,p(⌦) is either defined for all u 2 W2,p(⌦) as Bu=u, which corresponds to Dirichlet boundary conditions or it is a first order di↵erential operator defined for allu2W2,p(⌦) as
52 CHAPTER 4. STRANG SPLITTING WITH CRANK-NICOLSON where i and ↵ are uniformly continuous and di↵erentiable on ⌦, which corresponds to Robin boundary conditions. We assume that↵ is not zero everywhere on @⌦. The degen-erate case, where↵ is the zero function, corresponding to Neumann boundary conditions, is discussed in Remark 4.2.1 below. If B is a first order operator, we assume that the uniform nontangentiality condition is satisfied for all x2@⌦,
Xd
i=1
i(x)~ni(x) c,
wherec >0 is independent of x and where ~n(x) is the outwardly normal unit vector. On the boundary@⌦, Bu|@⌦ is the trace of Bu 2W1,p(⌦) on @⌦and is therefore an element of Lp(@⌦). To avoid heavy notations, we simply writeBu =b, on @⌦. We assume that b is a twice continuously di↵erentiable function on the boundary @⌦. To avoid sti↵ness of the solution or boundary layers at the initial time t = 0, we assume in addition that the initial conditionu0 2W2,p(⌦) satisfies the boundary conditionsBu0(x) =b(x) on @⌦.
The space {u 2 W2,p(⌦) ; Bu = b on @⌦} is difficult to handle since it is not a linear subspace of Lp(⌦) if b is not the zero function on @⌦. Therefore, we provide a reformulation of the problem (4.1) with homogeneous boundary conditions. We choose a function z 2 W2,p(⌦) which satisfies the boundary conditions Bz = b on @⌦. Such a function always exists with the assumptions that we made onB, band @⌦. We define the function ˜u = u z, which satisfies the following di↵erential problem with homogeneous boundary conditions,
@tu(x, t) =˜ D˜u(x, t) +f(x,u(x, t) +˜ z(x)) +Dz(x) in ⌦⇥(0, T], Bu(x, t) = 0 on˜ @⌦⇥(0, T],
˜
u(x,0) =u0(x) z(x) in⌦. (4.8)
We define the operator (A,D(A)) as the restriction of the operator D to the domain D(A) = {u 2 W2,p ; Bu = 0 on@⌦}, i.e. Au = Du for all u 2 D(A). The operator A therefore includes the homogeneous boundary conditions in its domain. Under the above assumptions, the operator A is a closed densely defined linear operator satisfying the two following properties (see [50, Theorem 3.1.13] and [58, page 92]):
1. The resolvent set of A, ⇢(A) = { 2 C ; I A is an isomorphism}, contains the closure of the set ⌃✓ ={z 2C ; z 6= 0, |arg(z)|<⇡ ✓}, where ✓ 2(0,⇡2) is fixed,
⇢(A) ⌃✓. (4.9)
2. For all 2⌃✓, the resolvent ofA,R( , A) = ( I A) 1, satisfies the following bound for the operator norm,
kR( , A)k M
| |, (4.10)
where M 1.
Note that, since 0 2 ⇢(A) by (4.9), the operator A is invertible and A 1 is bounded.
The operatorA is therefore the infinitesimal generator of an analytic uniformly bounded semigroup denoted etA, given by
etA = 1 2⇡i
Z
eztR(z, A)dz,
4.3. THE SPLITTING WITH CRANK-NICOLSON 53