FINITE ELEMENT APPROXIMATION OF ... - Semantic Scholar

SIAM J. NUMER. ANAL. Vol. 46, No. 1, pp. 437–471

c 2008 Society for Industrial and Applied Mathematics

FINITE ELEMENT APPROXIMATION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY POISSON RANDOM MEASURES OF JUMP TYPE∗ ERIKA HAUSENBLAS† Abstract. The paper deals with stochastic partial differential equations driven by Poisson random measures of jump type and their numerical approximation. We investigate the accuracy of space and time approximation. As space approximation we consider finite elements and as time approximation the implicit Euler scheme. Key words. stochastic evolution equations, stochastic partial differential equations, numerical approximation, time discretization, space discretization, Poisson random measure AMS subject classifications. 60H15, 60H35, 65C30, 35R30 DOI. 10.1137/050654141

1. Introduction. Our point of interest is the numerical approximation solutions of stochastic partial differential equations (SPDEs) of the following type: ⎧ ∂2 ⎪ du(t, ξ) = ∂ξ ⎪ 2 u(t−, ξ) dt + f (u(t−, ξ), ξ) dt ⎪ ⎨ + ζ∈R g(u(t−, ξ), ξ; ζ) η˜(dζ, ξ, dt), ξ ∈ O, 0 < t < ∞, (1.1) ⎪ ⎪ ⎪ ⎩ u(0, ξ) = u0 (ξ), where f : R×O → R and g : R×R → R are global Lipschitz continuous functions and η is a space time Poisson random measure of impulsive type (see, e.g., Saint Loubert Bié [SLB98] or Peszat and Zabczyk [PZ07]). Non-Gaussian random processes play an increasing rˆ ole in modeling stochastic dynamical systems. Typical examples of non-Gaussian stochastic processes are Lévy processes and processes arising by Poisson random measures. In neurophysiology the driving noise of the cable equation is basically impulsive, e.g., of a Poisson type (see Walsh [Wal86, Chapter 3])—or, on the other hand, Woyczy´ nski describes in [Woy01] a number of phenomena from fluid mechanics, solid state physics, polymer chemistry, economic science, etc., for which non-Gaussian Lévy processes can be used as their mathematical model in describing the related stochastic behavior. Thus, from the point of view of applications one might feel that the restriction to Gaussian noise is unsatisfactory; to handle such cases one can replace the Gaussian noise by a Poisson random measure. Concerning the numerical approximation of SPDEs driven by Wiener noise, several works already exist. Just to mention a few (see, e.g., Gy¨ ongy [Gy¨ o99, Gy¨ o98]), Hausenblas [Hau03, Hau02], Kloeden and Shott [KS01], Printems [Pri01a, Pri01b], Shardlow [Sha99], and Yoo [Yoo00]. Finite element approximation of SPDEs driven by Wiener space time white noise were considered by Allen, Novosel, and Zhang ∗ Received

by the editors March 13, 2006; accepted for publication (in revised form) June 14, 2007; published electronically January 30, 2008. This research was supported by Austrian Science Foundation grant P17273. http://www.siam.org/journals/sinum/46-1/65414.html † Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, 5020 Salzburg, Austria ([email protected]). 437

438

ERIKA HAUSENBLAS

[ANZ98], Cardon-Weber and Millet [CWM04], Germani and Piccioni [GP88], Gy¨ ongy [Gy¨ o99, Gy¨ o98], Gy¨ ongy and Millet [GM05], Millet and Morien [MM05], Walsh [Wal05], and Yan [Yan05]. The numerical approximation of finite-dimensional SDEs driven by Poisson random measures is the topic of, e.g., the work of Gardo´ n [Gar04], Kuznetsov [Kuz01], Protter and Talay [PT97], or Rubenthaler [Rub03]. But the numerical approximation of SPDEs driven by Poisson random measures is barely investigated, and, in the cases known to the author, processes of finite variation are considered, e.g., by Kouritzin, Long, and Sun [KLS03] and Kouritzin and Long [KL02]. If the driving process is of infinite variation, there exists only some works about existence and uniqueness, e.g., Albeverio, Wu, and Zhang [AWZ98], Hausenblas [Hau05, Hau06], and Kallianpur and Xiong [KX87, KX96]. Applebaum and Wu [AW00] and Saint Loubert Bié [SLB98] considered SPDEs with space time Lévy noise. First, we discretize the SPDE spatially by the finite element method, obtaining a system of high-dimensional SDEs that can be solved by the implicit Euler scheme. The noise will be approximated by using the projection onto finite elements. 2. Preliminaries. Let O ⊂ Rd be a convex polygon, 1 < p ≤ 2, and A be a uniformly elliptic, dissipative, second order partial differential operator with either Dirichlet or Neumann boundary conditions. In particular, let A : D(A) ⊂ Lp (O) → Lp (O) be the realization of L = L(x, D) =−

(2.1)

2 2 ∂ ∂ ∂ ai,j (x) + bi (x) + c(x), ∂x ∂x ∂x i j i i,j=1 i=1

with either u = 0 on ∂O × (0, ∞) or ∂u + σu = 0, ∂nL ∂u where ∂n denotes the differentiation along the outer normal unit vector n on O. L We assume that aij ∈ C 2 (O), 1 ≤ i, j ≤ d, strictly bounded away from zero, and σ ∈ C 2 (O). + ˆ ˆ Let (Ω, F, P) be a complete probability space. Let η : B(R)×B(O) ×B(R ) → R+ 1 ˆ ν ∈ L(R) := be a Poisson random measure over (Ω, F, P) with intensity ι = λ×ν, {σ : B(R) → R+ , σ is a Lévy measure}, γ be the compensator, and η˜ = η − γ. Let (Ft )t≥0 be the natural filtration induced by η. Our point of interest is solutions of SPDEs of the following form: ⎧ ⎪ ξ) dt + f (u(t−, ξ), ξ) dt ⎪ ⎨ du(t, ξ) = Au(t, + ζ∈R g(u(t−, ξ), ξ; ζ) η˜(dζ, ξ, dt), ξ ∈ O, t > 0, (2.2) ⎪ ⎪ ⎩ u(0, ξ) = u0 (ξ), ξ ∈ O,

where f : R → R and g : R × O × R → R are globally Lipschitz continuous functions. By a mild solution of (1.1) we understand a process u such that 1λ

denotes the Lebesgue measure.

FINITE ELEMENTS FOR SPDEs OF POISSONIAN TYPE

• u satisfies the integral equation Gt (ζ, ξ)u0 (ζ) dζ + u(t, ξ) = O

t

0+

O

Gt−s (ζ, ξ)f (u(s−, ζ, ζ)) dζ ds

Gt−s (ζ, ξ)g(u(s−, ζ), ζ; z) η(dz, dζ, ds), a.s., t ≥ 0,

+ 0+

t

439

ζ∈O z∈R

∂ where G = {Gt (·, ), 0 < t < ∞} is the Green kernel associated with ∂t +L and the boundary conditions specified above; • for all φ ∈ Cb∞ (O), the R-valued process t → u(t), φ is adapted and c´ adl´ ag. It is shown in Saint Loubert Bié [SLB98] or Hausenblas [Hau05, Example 2.3] that if p < d2 + 1, then there exists a unique solution of problem (1.1) such that u ∈ Lp (Ω; Lp (O)). Our point of interest is the numerical approximation of the solution of (1.1). First, we introduce the space discretization by finite elements, and second we introduce the time discretization, i.e., the Euler scheme.

2.1. Space discretization. The space discretization is done by finite elements. For the definition for a finite element, we follow the book of Brenner and Scott [BS02]. Definition 2.1. Let • K ⊂ Rd be a bounded closed set with nonempty interior and piecewise smooth boundary (the element domain); • P be a finite-dimensional space of functions on K (the space of shape functions); and • N = {N1 , . . . , Nk } be a basis for P (the set of nodal variables). Then (K, P, N ) is called a finite element. Moreover, we denote by diam(K) = maxx,y∈K |x − y|Rd the diameter of K. Definition 2.2. A set K is star-shaped with respect to B, if for all x ∈ K the closed convex hull of {x} ∪ B is a subset of K. Definition 2.3. Suppose K has diameter d and is star-shaped with respect to a ball. Let ρmax = sup{ρ : K is star-shaped with respect to a ball of radius ρ}. Then the chunkiness parameter of K is defined by γ :=

d . ρmax

Moreover, we denote by Pm the set of all polynomials in d variables of degree less than or equal to m. Definition 2.4 (compare the assumptions of Theorem 4.4.4 in [BS02]). We assume that a finite element satisfies the standard conditions for some l and m, iff • K is star-shaped with respect to some ball, m • Pm−1 ⊂ P ⊂ W∞ (O), l ¯ • N ⊂ (C (K)) . Definition 2.5. Let (K, P, N ) be a finite element. The basis {φ1 , . . . , φk } of P dual to N is called the nodal basis of P. Definition 2.6. Given a finite element (K, P, N ), let the set {φ1 , . . . , φk } ⊂ P be the basis dual to N . If v is a function for which all Ni ∈ N , i = 1, . . . , k, are defined, then we define the local interpolant by IK v :=

k i=1

v(Ni )φi .

440

ERIKA HAUSENBLAS

Definition 2.7. A subdivision of a domain O is a finite collection of element domains{Ki , i = 1, . . . k} such that • Ki ∩ Kj = ∅ if i = j and • ∪Ki = O. Remark 2.1. The nodal basis of a subdivision {Ki , i = 1, . . . k} induces a Voronoi decomposition {Ji ; i = 1, . . . , k} of O by

Ji := ξ ∈ O, |ξ − Ni | ≤ min {|ξ − Nj |} . 1≤i≤k

Definition 2.8. Suppose that O is a polyhedral domain with subdivision T . Assume that each element domain K in the subdivision is equipped with some type of shape functions P and nodal variables N such that (K, P, N ) forms a finite element. Let m be the order of the highest partial derivatives involved in the nodal variables. ¯ the global interpolant is defined by For f ∈ C m (O), IT f = IKi f Ki

for all Ki ∈ T . Example 2.1. Let H = L2 ([0, 1]). Let {xi := Ni , i = 1, . . . , N − 1} be the set of grid points. For 0 ≤ i ≤ N let φi be a pyramid function, i.e., a function, which takes value 1 at the grid point Ni , vanishes at the other grid points, and is linear between the grid points. For i = 1, . . . , N − 1, a finite element (Ki , Ni , Pi ) consists of i 0 the interval [ i−1 N , N ], the nodal variables Ni : C ([0, 1]) v → v(xi ) ∈ R, the shape i−1 1 1 ]. functions φi |Ki and φi−1 |Ki , and Ji = [ N + 2N , Ni + 2N We can now define a family of subdivisions Fh , h = Nh−1 . For each subdivision Nh −1 . For Th , let Vh the linear hull of the shape functions, i.e., the linear hull of {φi }i=1 any v ∈ C([0, 1]), the global interpolant is given by Ih (v) =

Nh

Nih (v)φi .

i=1

Definition 2.9. We call a family of subdivisions {Th , 0 < h ≤ 1} nondegenerate if the chunkiness parameter is uniformly bounded for all T ∈ {Th } and all h ∈ (0, 1]. Let {Th , 0 < h ≤ 1} be a nondegenerate family of subdivisions of a polyhedral domain O in Rd , consisting of kh finite elements of reference type (K, P, N ). We assume that the parameter h corresponds to the maximal diameter of the finite elements, i.e., h = max{1 ≤ i ≤ kh | diam(Ki )}. Let {φh1 , . . . , φhkh } be the set of shape functions and {N1h , . . . , Nkhh } be the set of nodal variables. We denote by Vh the linear hull of the shape functions and Vh its dual. The semidiscrete problem corresponding to (2.2) is to find the process uh = {uh (t) ∈ Rkh , 0 ≤ t < ∞} ⎧ h (t−)) dt ⎪ ⎨ duh (t) = Ahuh (t) dtfh (u + i∈{1,...,kh } ζ∈R gh (uh (t−), i, ζ) η˜h (dζ, di, dt), t > 0, (2.3) ⎪ ⎩ u(0) = ( u0 , N1 , . . . , u0 , Nkh ) ∈ Rkh where Ah : Vh → Vh is the discrete analogue of A defined by (2.4)

Ah φ, χ = Aφ, χ

∀φ, χ ∈ Vh ,


441

+ ˆ ˆ . . . , kh })×B(R ) → R+ is the compensated Poisson random and η˜h : B(R)×P({1, measure with intensity ιh given by

B(R) × B({1, . . . , kh }) (A, B) → ιh ((A, B)) := ∪i∈B λ(Jih ) ν(A) ∈ R+ . Moreover, fh : Vh → Vh and gh : Vh × {1, . . . kh } × R → Vh are given by Vh v → fh (v) :=

kh

f (v(Ni ), Ni )φhi ∈ Vh

i=1

and Vh × {1, . . . kh } × R (v, i, ζ) → gh (v, i, ζ) = g(v(Ni ), Ni ; ζ) chi φhi ∈ Vh ,

−1 . where chi := O φhi (ξ) dξ Remark 2.2. The Poisson random measure η˜h can also be written as a sum of kh independent Poisson random measures (see, e.g., [Lin86, Proposition 5.3.2]). + ˆ To be precise, let η˜hi : B(R)×B(R ) → R+ be pairwise independent compensated Poisson random measures with intensity ν i = λ(Jih ) ν. Then we have for some A ∈ B({1, . . . , kh }) and B ∈ B(R), ν(B) < ∞,

d

ιh (di, dξ) = i∈A

ξ∈B

kh

1A (i)λ(Jih )

i=1

ν i (dξ). B

In particular, we have for all Vh -valued Ft -adapted and c´ adl´ ag processes u = {u(t), 0 ≤ t < ∞} t 0

{1,...,kh }

d

gh (u(s−), di, ζ) η˜h (di, dζ, ds) = ζ∈R

kh t i=1

0

ζ∈R

gh (u(s−), i, ζ)˜ ηhi (dζ, ds).

2.2. Time discretization. One usually prefers to discretize simultaneously in time and space. Thus, let τh be the time step size corresponding to the subdivision Th . One popular way for time discretization is to use the implicit Euler scheme, i.e., uh (t + τh ) − uh (t) + Ah uh (t + τh ) ≈ fh (uh (t)) τh t+τh gh (uh (t); z)˜ ηh (dz, dt). + t

z∈Rkh

So, if vhm denotes the approximation of uh (mτh ), then (2.5)

vhm+1 = (1 − τh Ah )−1 [vhm + τh fh (mτh , vhm ) + ξhm (uh (mτh ))] , vh0 = Ph u0 ,

where we denote by ξhm (uh (mτh )) the kh dimensional random variable

(m+1)τh

(2.6) mτh

ζ∈Rkh

gh (uh (mτh ); ζ) η˜h (dz; ds).

442

ERIKA HAUSENBLAS

The approximation at time t = τh m, m ∈ N, is given by u ˆh (mτ ) := Ih vhm .

(2.7)

The solution is defined by linear interpolation; let t ∈ (mτh , (m + 1)τh ); we define vh (t) by vh (t) = vhm and u ˆh (t) := Ih vh (t),

mτh < t < (m + 1)τh .

Since the implicit Euler scheme is unconditionally stable, no stability conditions arise. Now we can state our main result. Theorem 2.1. Let O be a polyhedral domain and A be a uniformly elliptic, second order differential operator given in (2.1) with Dirichlet or Neumann boundary conditions. For a fixed p, 1 ≤ p ≤ 2, let (Ω, F, (Ft )t≥0 , P ) be the usual filtration and η : O × R × B(R+ ) → R+ be a Poisson random measure of jump type with intensity ι := λ × ν, where ν belongs to the set of p-integrable Lévy measures on R. ¯ × R → R are globally We assume that the functions f : R → R and g : R × O Lipschitz and g linear in the third variable. Moreover, we assume that the initial condition u0 belongs to Lp (O). The space is approximated by a nondegenerate family of subdivisions {Th , 0 < h ≤ 1} of finite elements, satisfying the conditions described in section 2.1. Moreover, the reference element is supposed to be a standard element for l ≥ 0 and m = 2. Let vnk , k ∈ N, be the approximation of (2.2) given by (2.7). Then for all N ∈ N, for all α such that α
0 when p > 1 or m − l − d ≥ 0 when p = 1. Then there exists a positive constant C depending on the reference element, m, l, d, p, and the chunkiness parameter such that for 0 ≤ s ≤ m

T ∈Th

p1 p

v − Ih vW s (T ) p

≤ C hm−s vW m (O) p

for all v ∈ Wpm (O), where the left-hand side should be interpreted, in the case where p = ∞, as maxT ∈Th v − Ih vW s (O) . For 0 ≤ s ≤ l, ∞

d

m−s− p vW m (O) max v − Ih vW∞ s (O) ≤ C h p

∀v ∈ Wpm (O).

T ∈Th

Since the space Vh is finite-dimensional, all norms which can be defined in Vh are equivalent. The purpose of the inverse inequalities is to specify how the equivalent constants depend on the parameter h. Theorem 3.2 (Brenner and Scott [BS02, Theorem 4.5.11]). Assume that the conditions above are satisfied, and let p, q, and l be given constants such that 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, 0 ≤ m ≤ l, and φhi ∈ Wpl (O) ∩ Wqm (O) for i = 1, . . . , kh . Then there exists a positive constant C depending on the reference element, d, m, p, and q such that

T ∈Th

p1 p vW l (T ) p

d m−l+min(0, d p−q)

≤Ch

T ∈Th

q1 vqW m (T ) q

1 p for all v ∈ Vh . When p = ∞, then [ T ∈Th vW s (T ) ] p is interpreted as p maxT ∈Th vqW l (T ) . ∞ Corollary 3.1. Under the conditions above, there exists a positive constant C depending on the reference element such that we have for all 1 > δ ≥ 0

T ∈Th

for all v ∈ Vh .

p1 p

vLp (T )

d −δ+min(0, d p−q)

≤Ch

T ∈Th

q1 vqW −δ (T ) q


445

Proof. The proof is done similarly to the proof of Theorem 3.2 (see Brenner and ˆ P, ˆ N ˆ ) be a finite element of reference type Scott [BS02, Theorem 4.5.11]). Let (K, ˆ = 1 and (K, P, N ) be a satisfying the standard conditions for m and l with diam(K) finite element of the family of subdivisions with diam(K) = hK . Since the subdivision is nondegenerate, a transformation exists, having the form ˆ x ˆ → JK x ˆ + bK ∈ K, TK : K where JK ∈ Rd × Rd being invertible and bK ∈ Rd (see Proposition 4.4.11 in [BS02]). ˆ → R be an arbitrary function and v : K → R be defined by Let vˆ : K −1 x) ∈ R. K x → v(x) := vˆ(JK

Then substitution gives d

1

p v |Lp (K) v |Lp (K) |v|Lp (K) = det(JK ) p |ˆ ˆ = C hK |ˆ ˆ ,

0 < hK ≤ 1,

and v |L∞ (K) |v|L∞ (K) = |ˆ ˆ ,

0 < hK ≤ 1.

Similarly there exists a constant C > 0 such that we have for all 0 < hK ≤ 1 d

1

−j

p v |Wpj (K) v |Wpj (K) |v|Wpj (K) = C det(JK ) p h−j ˆ = C hK |ˆ ˆ K |ˆ

and −j −j j j j v |W∞ v |W∞ |v|W∞ ˆ = C hK |ˆ ˆ . (K) = C hK |ˆ (K) (K)

Duality arguments give for q conjugate to p and for all 0 < hK ≤ 1 |v|Wp−j (K) =

sup w∈Wqj (K)

v, w

≤C |w|Wqj (K)

d(1− q1 )+j

(3.1)

= C hK

sup j ˆ w∈W ˆ q (K)

det(JK ) ˆ v , w

ˆ d

−j

q hK |w| ˆ Wqj (K) ˆ

d ˆ v , w

ˆ p +j ≤ C hK |ˆ v |Wp−j (K) ˆ . |w| ˆ Wqj (K) j w∈W ˆ q (K)

sup

Since Pˆ is finite-dimensional, all norms are equivalent, and there exists a constant Cˆ such that ˆ v | −γ ˆ |ˆ v |Lp (K) ˆ ≤ C |ˆ Wq (K)

∀v ∈ P.

Therefore (3.1) and interpolation arguments imply for γ > 0 that −d

− d −γ

hK p |v|Lp (K) ≤ C hK q

|v|Wq−γ (K)

∀v ∈ P, 0 < hK ≤ 1.

446

ERIKA HAUSENBLAS

Let {Th , 0 < h ≤ 1} be a nondegenerate family of subdivisions. Then, by taking the sum, one can show by the same arguments as in Theorem 4.5.11 in [BS02], that, if the family of subdivisions is nondegenerate, the assertion of Corollary 3.1 holds. 3.1. Approximation of the operator. We can associate with the operator (A, D(A)) the Gelfand triple (V, H, V ), where V → H = H → V densely. In 1 fact, V = D(−A) 2 and V its dual. In our case, where (A, D(A)) is a second order differential operator with boundary conditions, we have V = H 1 (O) if the boundary condition involves first order derivatives and V = H01 (O) if the homogeneous Dirichlet condition is specified on the whole boundary. Now the operator A : V → V defines a bilinear form on H by a(u, v) := Au, v ,

u, v ∈ V.

The corresponding symmetric variational problem is posed as follows. Suppose that the following three conditions are valid: ⎧ (1) (H, ·, · ) is a Hilbert space. ⎪ ⎪ ⎨ (2) V is a (closed) subspace of H. (3.2) (3) a(·, ·) is a bounded, symmetric bilinear form ⎪ ⎪ ⎩ that is coercive on V. Then the symmetric variational problem is the following: For a given F ∈ V , find u ∈ V such that (3.3) a(u, v) = F (v) ∀v ∈ V. Given the family of nondegenerate subdivisions Th of a polyhedral domain O, the Ritz–Galerkin approximation of problem (3.3) is the following: For a given F ∈ Vh , find uh ∈ Vh such that (3.4) a(uh , v) = F (v) ∀v ∈ Vh . Assume that u solves problem (3.3) and uh solves problem (3.4). Then we can find the following estimates on u − uh depending on F . By means of the inequality above and duality arguments, one can verify bounds of the difference u − uh in H 1 (O) and L2 (O). In particular, one can show that, if the family of subdivisions {Th , 0 < h ≤ 1} satisfies the assumptions of Theorem 3.1 with m = 2, l ≥ 0 and p = 2, then (3.5)

inf u − vH 1 (O) ≤ C h uH 2 (O)

v∈Vh

∀u ∈ H 2 (O),

0 < h ≤ 1,

is valid, and therefore (for more details, see Theorem 5.4.8 in [BS02]) (3.6)

u − uh L2 (O) ≤ Ch2 uH 2 (O) ≤ Ch2 F L2 (O) ,

0 < h ≤ 1.

Finally, by using duality arguments, one can show that there exists a h0 > 0 such that (see Theorem 5.8.3 in [BS02]) (3.7) u − uh H 2,−γ (O) ≤ Chγ+1 u − uh H 1 (O) ≤ Ch2+γ F L2 (O) ,

0 < h ≤ 1.

Since our problem (2.2) is well-posed in Lp (O), the error bounds of the difference p (O), γ > d − dp , p ∈ (1, 2]. have to be transferred to Lp (O) or W−γ


447

Corollary 3.2. Assume that (3.5) holds. Then, under the conditions above, there exist positive constants C and h0 such that for all h ≤ h0 there is a unique solution to problem (3.4) satisfying d

d

d

d

u − uh Lp (O) ≤ Ch2− p + 2 F Lp (O) ,

0 < h ≤ 1,

and for γ ∈ (0, 1) u − uh W p

−γ (O)

p ≤ Ch2− p + 2 F W−γ (O) ,

0 < h ≤ 1.

Proof. Since p ≤ 2 and O is bounded we can infer that u − uh Lp (O) ≤ C u − uh L2 (O) . The estimate (3.6) gives u − uh Lp (O) ≤ C h2 F L2 (O) . Theorem 3.2 leads to u − uh Lp (O) ≤ C h2+ 2 − p F Lp (O) . d

d

Similarly, we get u − uh W p

−γ (O)

≤ C u − uh W 2

−γ (O)

.

Estimate (3.7) gives u − uh W p

−γ (O)

≤ C h2+γ F L2 (O) .

Next, we apply negative norm estimates. In particular, Theorem 3.2 leads to u − uh W p

−γ (O)

≤ C h2+ 2 − p F W p d

d

−γ (O)

.

3.2. Approximation of the semigroup. The approximation Ah : Vh → Vh of the operator A is given by the induced bilinear form on Vh . In particular, for any u ∈ Vh we define Ah u by (3.8)

Ah u, v := a(u, v),

v ∈ Vh .

The semidiscrete approximation of the solution u to the abstract Cauchy problem d t > 0, dt u(t) + Au(t) = 0, (3.9) u(0) = u0 is the function uh : [0, ∞) → Vh , determined by the following condition: For all v ∈ Vh we have d dt uh (t), v + a(uh (t), v) = 0, t > 0, (3.10) uh (0), v = u0 , v . In use of the operator Ah , relation (3.10) is written as duh (t) + Ah uh (t) = 0, dt

t > 0.

448

ERIKA HAUSENBLAS

Let Hh be the space Vh equipped with the topology induced from H. Let Rh : H → Hh be the Ritz projection. Recall that the linear operator Ah : Hh → Hh is defined by (3.8). Let Rh : V → Vh be the Ritz operator defined through a(Rh u, vh ) = a(u, v) ∀u ∈ Vh and u ∈ V. By the Lax–Milgram theorem, it can be shown that the Ritz operator is well-defined, i.e., that the element Rh u exists and is unique. Theorem 1.1 of [FSS01] shows that the operator Ah : Hh → Hh generates an analytic semigroup on Hh independently of h. To be precise, there exist some θ ∈ (0, π) and some M ≥ 1 such that C \ Σθ ⊂ ρ(Ah ), where ρ(Ah ) denotes the resolvent set of Ah and Σθ := {z ∈ C, 0 ≤ | arg(z)| ≤ θ} . The semidiscrete approximation admits the error estimate of optimal order for t > 0. Theorem 3.3 ([FSS01, Theorem 2.1]). Given u0 ∈ H, let u and uh be the solutions of (3.9) and (3.10), respectively. Then we have u(t) − uh (t)L2 (O) + h u(t) − uh (t)H 1 (O) ≤ C h2 t−1 u0 L2 (O) ,

t > 0, 0 < h ≤ 1.

Similarly to the Proof of Theorem 3.3 one can find error estimates in Lp spaces. Theorem 3.4. Given u0 ∈ H, let u and uh be the solutions of (3.9) and (3.10), respectively. Then we have for 1 ≤ p ≤ 2 and 0 ≤ γ < 1 u(t) − uh (t)W p,−γ (O) ≤ C h2− p + 2 t−1 u0 W p,−γ (O) , d

d

t > 0, 0 < h ≤ 1.

Proof. The proof of Theorem 3.4 is similar to the proof of Theorem 3.3, i.e., Theorem 2.1 in [FSS01]. Assume that u solves problem (3.3) with F = u0 and uh solves problem (3.4) with F = Ih u0 . In Corollary 3.2 we have shown that there exists estimates of the error in Lp spaces. In particular, we have d

d

u − uh Lp (O) ≤ Ch2− p + 2 F Lp (O) . Since u = A−1 u0 and A−1 h Ih u0 , we can write −1 d d A u0 − A−1 Ih u0 p ≤ Ch2− p + 2 u0 Lp (O) . h L (O) Next, since for all λ0 , λ ∈ ρ(A) ∩ ρ(Ah ) we have −1 −1 −1 (λIh − Ah ) − (λ − A) = 1 + (λ − λ0 ) (λIh − Ah ) Ih −1 −1 −1 × (λ0 Ih − Ah ) Ih − (λ0 − A) × (λ0 − A) (λ0 − A) , (3.11) we can conclude that the following estimates: −1 −1 (3.12) (λI − A) u0 − (λIh − Ah ) Ih u0

d

Lp (O)

and for γ ∈ (0, 1) −1 −1 (3.13) (λI − A) u0 − (λIh − Ah ) Ih u0

d

≤ Ch2− p + 2 u0 Lp (O)

p W−γ (O)

d

d

≤ Ch2− p + 2 u0 W p

−γ (O)


449

are valid for all λ ∈ ρ(A) ∩ ρ(Ah ). The proof of Theorem 3.3 relies on the fact that we can represent the semigroup in terms of the resolvent. By comparing to [FSS01, page 58], or to [Paz83, Theorem 1.7.7], we have −tz

1 e · (λI − A)−1 − (λIh − Ah )−1 Ih u0 dz, (3.14) u(t) − uh (t)Ih ≤ 2π Γ where Γ is the boundary of Σθ1 = {z, 0 ≤ | arg(z)| ≤ θ1 }, for θ1 ∈ (θ, π2 ) coming from ∞eiθ1 , passing through the origin and going out to ∞e−iθ1 . Using estimate (3.6) and taking into account the calculations in (3.11) gives

(λI − A)−1 − (λIh − Ah )−1 Ih u0 2 L (O) ≤ C hm u0 L2 (O) ,

(3.15)

u0 ∈ L2 (O).

By substituting estimate (3.15) into (3.14), Theorem 3.3 can be shown. Similarly, by substituting estimate (3.12) into (3.14) one gets C 2− pd + d2 h u(t) − uh (t)Ih Lp (O) ≤ u0 Lp (O) e−tz dz 2π Γ ∞ d d d d ≤ C h2− p + 2 u0 Lp (O) e−tr cos(θ1 ) dr ≤ C h2− p + 2 t−1 u0 Lp (O) . 0

The case γ = 0 of Theorem 3.4 follows. To show Theorem 3.4 for γ > 0 one can use the same computations, but one has to substitute estimate (3.13) instead of (3.12) into (3.14). 3.3. Mesh-dependent norms. For x ∈ Vh , one can define the mesh-dependent inner product kh

v, w h := hd

Nih (v)Nih (w)

i=1

and the norm v2L2 (O) := v, v = hd h

kh

v(Nih )2 .

i=1

It can be shown that there exist some constants c and C such that we have for all h ∈ (0, 1) (see Lemma 6.2.7 of [BS02]) cvL2 (O) ≤ vL2h (O) ≤ CvL2 (O)

∀v ∈ Vh .

Since the natural spaces for SPDEs driven by Poisson random measure are Lp spaces, we will also define mesh-dependent Lp spaces; i.e., for 0 < h ≤ 1 and v ∈ Vh we define vLph (O) :=

kh h p Ni (v) hd

p1

i=1

and Lph (O) := {v ∈ Vh , vLph (O) < ∞}.

450

ERIKA HAUSENBLAS

Similarly to the case where p = 2, one can show that there exist some constants c and C such that we have for all h with 0 < h ≤ 1 (3.16)

cvLp (O) ≤ vLph (O) ≤ CvLp (O)

∀v ∈ Vh ∩ Lp (O).

Remark 3.1. It follows from the nondegeneracy assumptions of the subdivisions that there exist some constants c and C such that c hd ≤ kh ≤ Chd ,

0 < h < 1.

Theorem 1.1 of [FSS01] shows that the operator Ah : Hh → Hh generates an analytic semigroup on Hh with parameter M and ω independent of h. To be precise, there exist some θ ∈ (0, π) and some M ≥ 1 such that C \ Σθ ⊂ ρ(Ah ), where ρ(Ah ) denotes the resolvent set of operator Ah and Σθ := {z ∈ C, 0 ≤ | arg(z)| ≤ θ} . Let X be a Banach space and A : X → X be a operator generating an analytic semigroup on X. Since the constants M , ω, and C in −tA e uX ≤ M eωt uX and −tA C ωt Ae e uX uX ≤ t depend on the resolvent set, in particular, on the angle θ which includes the numerical range, we can conclude that there exist some constants C1 and C2 such that −tA h e (3.17) u Lp (O) ≤ C1 uLp (O) ∀v ∈ Lph (O), 0 < h ≤ 1, h

h

and (3.18)

−tA C2 h Ae u Lp (O) ≤ uLp (O) h h t

∀v ∈ Lph (O),

0 < h ≤ 1.

For more details, see [Paz83, Theorem 2.5.2] and [FSS01]. 3.4. The mass and stiffness matrix. In this section we will investigate properties of the mass and stiffness matrix. Let O be a polyhedron in Rd and {Th , 0 < h ≤ 1} be a nondegenerate family of subdivisions, where the reference finite element ˆ P, ˆ N ˆ ) satisfies the standard conditions for m = l = 1. (K, h The mass matrix Mh of a subdivision is given by Mh = ((mhi,j ))ki,j=1 , where mhi,j := φhi , φhj ,

1 ≤ i, j ≤ kh .

Since the shape functions are locally defined, there exists a number m such that for all h ∈ (0, 1) there exists a set Ch ⊂ Nkh × Nkh such that supp(φhi ) ∩ supp(φhj ) = ∅

∀(i, j) ⊂ Ch ,

0 < h ≤ 1,

and |(i, Nkh ) ∩ Ch | ≤ m,

0 < h ≤ 1.


451

Remark 3.2. The mass matrix is given by Theorem 9.8, page 387 in Ern and Guermond [EG04]. The left-hand side is given by Ah φhl , φhj = ∇φhi , ∇φhj . Since {Th , 0 < h ≤ 1} is a uniformly bounded family of subdivisions, we have c hd−2 , if (i, j) ⊂ Ch , h h ∇φl , ∇φj = 0, elsewhere. −1 Fix some i, 1 ≤ i ≤ kh . Our point of interest is A−1 h φi . Let ui = Ah φi . Then

Ah ui , φhj = φhi , φhj ,

j = 1, . . . , kh .

Since ui ∈ Vh , there exists ui1 , . . . , uikh ∈ R such that ui =

kh

uil φhl .

l=1

Therefore, Ah ui , φhj =

kh

uil Ah φhl , φhj = φhi , φhj

l=1

and by Remark 3.2 c h2 , if (i, j) ⊂ C, uil = 0, elsewhere.

(3.19)

3.5. Approximation of the noise. The space time Poissonian noise of jump type is given by a Poisson random measure η : B(O) × B(R) × B([0, ∞)) → R+ with intensity λ × ν, where ν is a Lévy measure over R, such that |z|p ν(dz) < ∞. R

The function g :R×O×R→R is Lipschitz continuous with Nemytskij operator (3.20)

Lp (O) × R (u, z) → G(u; z) = g(u(·); ·; z) ∈ Lp (O),

where g(u(·); ·; z)(ξ) := g(u(ξ); ξ; z). In [Hau05] we have seen that the space time Poissonian noise can be described by a Banach-valued Poisson random measure. In d

−d

p particular, let Z = Bp,∞ (O) and U be the unit sphere in Z. Let χ be defined by

O ξ → χ(ξ) := c δξ ∈ Z,

452

ERIKA HAUSENBLAS

where c = |δξ |−1 Z . The intensity ι of the Z-valued Poisson random measure is given by 1B (ζx)σ(dx)ν(dζ) ∈ R+ , B(Z) B → ι(B) := ζ∈R

x∈U

where σ(A) := λ(χ−1 (A ∩ χ(O))), A ∈ B(Z). By inserting the definition above into the definition of the Besov space, one gets by Proposition A.1 p |G(u, z) − G(v, z)|pZ ι(dz) = |g(u(ξ), ξ; ζ) − g(v(ξ), ξ; ζ)| ν(dζ) dξ z∈Z

ξ∈O

ζ∈R

≤ C |u − v|pLp (O) ,

u, v ∈ Cb0 (O).

Extension to Lp (O) leads to the Lipschitz property (2.10) of G. The approximation can be similarly interpreted. First, the Lebesgue measure is approximated by λh defined by B ({1, 2, . . . , kh }) A → λh (A) :=

kh

λ(Jih )1A (i) ∈ R+ .

i=1

Now the approximation ιh : B({1, . . . , kh }) × B(R) → R+ of the intensity ι is defined by {1, . . . , kh } × B(R) (A, B) → ιh (A, B) :=

kh

ν(dζ) ∈ R+ .

1A (i)λh ({i})

i=1

ζ∈R

The approximated Nemytiskij operator Gh is defined by Vh × {1, . . . , kh } × B(R) (u, i, ζ) → Gh (u, i, ζ) := g(u(Nih ), Nih ; ζ) chi φhi . Similarly to the above we can show the following proposition. Proposition 3.1. There exists some C > 0 such that for any h > 0 and for any γ > 12 (d − dp ) and δ < 2γ − d − dp the following inequality holds for all uh , vh ∈ Vh : p (−Ah )−γ [Gh (uh , z) − Gh (vh , z)]p p ι (dz) ≤ C hdδp uh − vh Lp (O) . L (O) h z∈Zh

Proof. Let u, v ∈ Vh . Then we have (−Ah )−γ [Gh (uh , z) − Gh (vh , z)]p p ι (dz) L (O) h z∈Z

k p h −γ h = E (−Ah ) [Gh (uh , ζ) − Gh (vh , ζ)] μi (dζ, ds) p R i=1

,

L (O)

where μhi , i = 1, . . . , kh , are independent Poisson random measures with intensity λh (i)ν. The generalized Burkholder inequality gives (−Ah )−γ [Gh (uh , z) − Gh (vh , z)]p p ι (dz) L (O) h z∈Z kh

≤

j=1

R

(−Ah )−γ g(uh (Njh ), Njh ; ζ) − g(vh (Njh ), Njh ; ζ) chj φhj p p λ(Jjh )ν(dζ). L (O)


453

Since g(uh (Njh ), Njh ; ζ) − g(vh (Njh ), Njh ; ζ) ∈ R and Ah is linear, we can continue z∈Zh

kh (−Ah )−γ [Gh (uh , z) − Gh (vh , z)]p p ι (dz) ≤ L (O) h

j=1

R

g(uh (Njh ), Njh ; ζ) − g(vh (Njh ), Njh ; ζ) (−Ah )−γ chj φhj p p λ(Jjh )ν(dζ). L (O) p d Since the family of subdivisions is nondegenerate, we have φhj Lp (O) ∼ h p , chj ∼ h−d , and λ(Jjh ) ∼ hd . Therefore, by using also Remark 3.2, we get

z∈Zh

(−Ah )−γ [Gh (uh , z) − Gh (vh , z)]p p ι (dz) L (O) h

kh g(uh (Njh ), Njh ; ζ) − g(vh (Njh ), Njh ; ζ)p (−Ah )−γ chj φhj p p λ(Jjh )ν(dζ) ≤C L (O) R

j=1

≤Ch

γ2

kh g(uh (Njh ), Njh ; ζ) − g(vh (Njh ), Njh ; ζ)p chj φhj p p j=1

L (O)

R

≤ C hγ2−dp+d

λ(Jjh )ν(dζ)

kh g(uh (Njh ), Njh ; ζ) − g(vh (Njh ), Njh ; ζ)p λ(Jjh )ν(dζ). j=1

R

According to the Lipschitz condition of g we have z∈Z

(−Ah )−γ [Gh (uh , z) − Gh (vh , z)]p p ι (dz) L (O) h

≤ C hγ2−dp+d

kh

p hd uh (Njh ) − vh (Njh ) .

j=1

The equivalence between Lp (O) and Lph (O) gives the assertion. In order to calculate the error induced by the approximation of the Poisson random measure, we have to express the measure ιh in terms of ι. Proposition 3.2. There exists some C > 0 such that for any h > 0 and for any γ > 12 (d − dp ) and δ < 2γ − d − dp the following inequality holds for all uh ∈ Vh : Z

p dδp (−Ah )−γ Ih G(u, z)p p 1 + u (ι − ι )(dz) ≤ C h p h h L (O) . L (O)

Remark 3.3. In our calculation we use the fact that for γ > d− dp in W p,−γ (O) the noise vanishes for h → 0. But, by plugging the difference ι−ιh into the definition of Z, one can show directly that the approximated noise converges to the functional-valued representation of the space time Poissonian noise in Z.

454

ERIKA HAUSENBLAS

Proof. Embedding Zh into Z by i → chi φhi leads to (−Ah )−γ Ih G(u, z)p p (ι − ι )(dz) ≤ h L (O) Z

ξ∈O

ζ∈R

⎡ ⎤p kh h h h h ⎦ (−Ah )−γ ⎣Ih G(u, z)δξ − 1Jjh (ξ)g(u(Nj ), Nj , ζ)cj φj dζ dξ p j=1 L (O) ⎡ kh kh (−Ah )−γ ⎣ ≤ g(u(Nlh ), Nlh , ζ)φhl δξ h ξ∈J ζ∈R i i=1 l=1 ⎤p kh h h h h ⎦ − 1Jjh (ξ)g(u(Nj ), Nj , ζ)cj φj dζ dξ p j=1 L (O) k p k h h ≤ g(u(Nlh ), Nlh , ζ)φhl δξ dζ dξ (−Ah )−γ ξ∈Jjh ζ∈R j=1

+

l=1

kh j=1

ξ∈Jjh

ζ∈R

Lp (O)

(−Ah )−γ g(u(Njh ), Njh , ζ)chj φhj p p dζ dξ L (O)

=: S1 + S2 . By Corollary 3.1 we obtain S1 ≤ h−δp

k p h g(u(Nlh ), Nlh , ζ)φhl δξ dζ dξ. (−Ah )−γ ζ∈R

kh j=1

ξ∈Jjh

l=1

Z

Inequality (A.1) leads to −δp

S1 ≤ C h

k p h −γ h h h g(u(Nl ), Nl , ζ)φl (−Ah ) ζ∈R

kh j=1

l=1

dζ.

Lp (Jjh )

Since there exists a constant m such that |{i | sup φhi ∩ Jjh = ∅}| ≤ m for all i, j = kh h 1, . . . , kh and h > 0, and i=1 φi (ξ) = 1, we can infer by Remark 3.2 that S1 ≤ C h(2γ−δ)p

p h h h g(u(N ), N , ζ)φ l l l ζ∈R l,sup φh ∩J h =∅

kh j=1

≤ C h(2γ−δ)p m

l

kh j=1

ζ∈R

j

dζ

Lp (Jjh )

g(u(Njh ), Njh , ζ)φhj p p dζ. L (O)

By the linearity of g and the equivalence of Lph (O) and Lp (O) we get ⎛ S1 ≤ C h(2γ−δ)p m ⎝1 +

kh j=1

⎞ p p hd u(Njh ) ⎠ ≤ h(2γ−δ)p m 1 + |u|Lp (O) .

455


By applying Remark 3.19 we get for S2 kh (−Ah )−γ g(u(Njh ), Njh , ζ)chj φhj p p S2 ≤ λ(Jjh ) dζ L (O) ζ∈R

j=1

≤ C h2γp

kh

λ(Jjh )

j=1

≤ C h2γp−dp

kh

ζ∈R

λ(Jjh )hd

j=1 kh

≤ C h2γp−dp+d

g(u(Njh ), Njh , ζ)chj φhj p p dζ L (O)

ζ∈R

g(u(Njh ), Njh , ζ) φhj p p

λ(Jjh )hd

j=1

ζ∈R

L (O)

dζ

g(u(Njh ), Njh , ζ) dζ .

The linearity of g and the equivalence of Lph (O) and Lp (O) give p S2 ≤ C h2γp−dp+d 1 + uLp (O) . 4. Stability of the finite element approximation. In this section we shall investigate the stability of the approximation by finite elements. Since the semidiscrete problem (2.3) is finite-dimensional, for all 0 < h ≤ 1 there exists a unique solution uh = {uh (t) ∈ Rkh , 0 ≤ t < ∞} (see, e.g., Theorem 8 of [Pro04]). In this section we show that the family of unique solution {uh , 0 < h ≤ 1} is uniformly bounded. Theorem 4.1. Let uh = {uh (t), 0 ≤ t < ∞} be a solution to problem (2.3). Then, under the conditions of Theorem 2.1, for all p ∈ (1, 2], with p < d2 + 1, and for all T > 0 there exists some constant such that p

sup E uh (t)Lp (O) ≤ C,

0 < h ≤ 1.

0≤t≤T

Proof. First, we show that there exist some q < ∞ and some constants C1 and C2 such that t p q p q (E uh (s) ) ds, t > 0 and 0 < h ≤ 1. (Euh (t) ) ≤ C1 + C2 0

The Gronwall lemma yields the assertion. The variation of constant formula gives t uh (t) = e−tAh uh (0) + e−(t−s)Ah fh (uh (s)) ds +

t kh ζi ∈R

0 i=1

0

e−(t−s)Ah gh (uh (s); i, ζi ) η˜hi (dζi ; ds),

t > 0.

Taking the expectation leads to p p E uh (t)Lp (O ≤ Cp e−tAh uh (0)Lp (O) t p −(t−s)Ah + Cp E e fh (uh (s)) ds 0

Lp (O)

k p h t + Cp E e−(t−s)Ah gh (uh (s); i, ζi ) η˜hi (dζi ; ds) 0 p ζi ∈R i=1

=: S1 + S2 + S3 .

L

456

ERIKA HAUSENBLAS

If u0 ∈ Lp (O), then the first entity S1 is finite. The second entity can be calculated directly. The Minkowski inequality gives t p −(t−s)Ah e fh (uh (s)) ds S2 = E 0

t −(t−s)Ah ≤E fh (uh (s)) e

Lp (O)

Lp (O)

0

p

ds

.

In section 3.2 we have seen that there exist ω ∈ R, M ≥ 1, and θ ∈ (0, π) such that for all h ∈ (0, 1) the operator Ah generates an analytic semigroup of (ω, M, θ) on Lph (O) and the inequality (3.17) holds. In section 3.3 we have seen that there exist some constants c and C such that for all 0 < h ≤ 1 we have (see (3.16)) (4.1)

c wLp (O) ≤ wh Lp (O) ≤ C wLp (O) ,

w ∈ Vh

h

where w

Lp h (O)

:= h

kh

d

|wh (Ni )|p ,

w ∈ Vh .

i=1

Therefore, a constant C < ∞ exists such that for all 0 < h ≤ 1 −(t−s)Ah (4.2) Ah v ≤ C vLp (O) , v ∈ Lp (O) ∩ Vh . e p L (O)

Thus,

t

S2 ≤ C E 0

p fh (uh (s))Lp (O) ds

.

Again by (4.1) we can infer that S2 ≤ C E

t

0

p f (uh (s))Lp (O) ds h

.

By the globally Lipschitz property of f it follows that t p 1 + uh (s)Lp (O) ds . S2 ≤ C E h

0

Again by (4.1) we get S2 ≤ C E

t 0

1 + uh (s)Lp (O)

p

ds

.

The Hölder inequality leads to p S2 ≤ C tp 1 + sup E uh (s)Lp (O) . 0≤s≤t

Next, we estimate the term k p h t S3 = E e−(t−s)Ah gh (uh (s); i, ζi ) η˜hi (dζi ; ds) 0 p ζi ∈R i=1

L (O)

.


457

Since Lp (O) is of M -type p, we can infer by Proposition 3.3 in [Hau05] S3 ≤ E

t

0

kh p 1 E e−(t−s)Ah gh (uh (s); i; ζi ) ν(dζi ) ds. kh i=1 ζi ∈R Lp (O)

In section 3.2 we have seen that there exist ω ∈ R, M ≥ 1, and θ ∈ (0, π) such that for all h ∈ (0, 1) the operator Ah generates an analytic semigroup of (ω, M, θ) on Lp (O). In particular (3.18) holds; i.e., there exists a constant C < ∞ such that for all h > 0 −(t−s)Ah Ah v e

(4.3)

Lp h (O)

≤

C vLp (O) , h t

v ∈ Lph (O).

Because of inequality (4.3), and since the norms in Lph (O) and Lp (O) are equivalent, we have for all γ < p1

t

S3 ≤ 0

kh p 1 −γp (t − s) E A−γ h gh (uh (s); i, ζi ) Lp (O) ν(dζi ) ds. kh i=1 ζi ∈R

Since gh (uh (s); i, ζ) = Ni (G(uh (s); ζi ))φi and since Ah is linear, we get

kh p 1 (t − s)−γp E Ni (G(uh (s); ζi ))A−γ h φi Lp (O) ν(dζi ) ds. kh i=1 ζi ∈R

t

S3 ≤ 0

The Hölder inequality gives S3 ≤ 0

(4.4)

kh

t

1 kh −γp

(t − s)

i=1

ζi ∈R

p p E Ni (G(uh (s); ζi ))Lp (O) A−γ h φi L∞ (O) ν(dζi ) ds.

Moreover, by (3.19) we know that for any i = 1, . . . , kh we have A−γ h φi =

kh

uil φhl ,

l=1

where c h2γ , if (i, l) ⊂ Ch , uil = 0, elsewhere, and there exists some constant m0 such that for any i and 0 < h < 1 {(i, Nkh ) ∩ Ch ≤ m0 . (4.5) Therefore, we have (4.6)

−γ A φi ∞ ≤ m0 h2γ . h L (O)

458

ERIKA HAUSENBLAS

By substituting (4.6) in (4.4) we obtain S3 ≤

t kh 0 i=1 −γp

(t − s)

(4.7)

λ(Jih )

m0 h

2γp ζ∈R

p E g(uh (s, Nih ), Nih ; ζ)Lp (O) ν(dζi ) ds.

The inverse inequality given in Corollary 3.1 gives for all δ > d −

kh

t

S3 ≤

C 0

λ(Jih )

i=1

(t − s)−γp h2γp−δp ≤C

t kh 0 i=1

d p

ζ∈R

p E g(uh (s, Nih ), Nih ; ζi )

λ(Jih )(t − s)−γp h(2γ−δ)p

ζi ∈R

d −d

p (O) Bp,∞

ν(dζ) ds

p E g(uh (s, Nih ), Nih ; ζ)

d −d

p (O) Bp,∞

ν(dζ) ds,

which gives S3 ≤ C

t kh 0 i=1

λ(Jih )(t

−γp

− s)

h

(2γ−δ)p ζi ∈R

p E g(uh (s, Nih ), Nih ; ζ) ν(dζ) ds.

The mapping g is linear in the last variable and globally Lipschitz in the first variable. Thus we can infer that S3 ≤ C

t kh 0 i=1

p λ(Jih )(t − s)−γp h(2γ−δ)p 1 + E uh (s, Nih ) ds.

By the equivalence of the Lph (O) and Lp (O) norms we infer that S3 ≤ C 0

t

p (t − s)−γp h(2γ−δ)p 1 + E |uh (s)|Lp (O) ds.

Let q be so large that γp < 1 − 1q . Since there exists a δ > d − dp such that 2γ − δ > 0, the H¨ older inequality gives t q q p 1 + E |uh (s)|Lp (O) ds. (E |S3 |) ≤ c(t) 0

Summing up leads to the assertion. 5. Proof of the main result. The error of the time discretization follows from Theorem 3.1 of [HM06a]. By the Lipschitz continuity of f and g we infer that G(u; z)p d −d ι(dz) ≤ CupLp (O) , u ∈ Lp (O), z∈Z

p (O) Bp,∞

and F (u)Lp (O) ≤ CuLp (O) ,

u ∈ Lp (O).


459

Thus, Theorem 3.1 of [HM06a] gives the order of convergence in time for α < p2 −d+ dp . Here we investigate the accuracy of the space discretization, i.e., the difference Δ(t) = u(t) − uh (t)Ih at time t. To show the assertion, we will show that for a certain q < ∞ there exist constants C1 , C2 , and C3 such that for all 0 < h ≤ 1 q p E Δ(t)Lp (O) t q C1 p E Δ(t)Lp (O) ds (5.1) ≤ q hpqδ1 + C2 hpqδ2 + C3 hpqδ3 + C4 t 0 where δ1 , δ2 , and δ3 are the constants specified in Theorem 2.1. Now the second part of the assertion of Theorem 2.1 follows by an extension of Grownwall’s lemma which is a simple consequence of Exercise 3, page 190 in [Hen81] and the proof of which is omitted. Lemma 5.1 (Gronwall-type lemma). Suppose that α > 0 and c1 , c2 , c3 > 0. Assume that u : R+ → R+ is a measurable function such that the function t → t1−α u(t) is locally integrable on R+ and a.s. on R+ , t u(s) ds. u(t) ≤ c1 tα−1 + c2 + c3 0

Then, a.s. on R , +

∞ $ # u(t) ≤ c1 tα−1 Γ(α) + c2

[c3 t]m . Γ(m + α) m=0

To show the second part of Theorem 2.1, we will show the following: Let q and γ be two real numbers such that (5.2)

d−

d < γ < 2p p

and

γp 1 0 there exists some constant C < ∞ such that C −(t−s)A (5.7) Av −ρ ≤ ρ vLp (O) , v ∈ Lp (O). e Wp (O) t2 Thus, we can infer for any γ > 0 that t γp p p (t − s)− 2 E g(u(s); z) − g(uh (s); z)W −γ (O) ι(dz) ds. E I3 (t)Lp (O) ≤ p

Z

0+

d

−d

p (O) → Wp−γ (O) By Sobolev inequalities for any γ > d − dp we have Z = Bp,∞ continuously. Therefore, t γp p p (t − s)− 2 E (g(u(s); z) − g(uh (s); z))Z ι(dz) ds. E I3 (t)Lp (O) ≤

Z

0+

The Lipschitz property of g leads to t γp p p (t − s)− 2 E u(s) − uh (s)Lp (O) ds. E I3 (t)Lp (O) ≤ 0+

If γ satisfies (5.2), then (5.8)

γp 2

< 1 − 1q , and we can infer by the H¨ older inequality that

p

E I3 (t)Lp (O)

q

≤ c(t)

t 0

p

E u(s) − uh (s)Lp (O)

q

ds.

By arguments similar to those used in Proposition 3.2 in [Hau05] we get for the fourth term t p p E I4 (t)Lp (O) ≤ E e−A(t−s) (I − Ih ) g(uh (s); z) ν(dz) ds. 0+

Lp (O)

Z

d

−d

p (O) → Wp−γ (O) By the smoothing property of A, i.e., (5.7), and since Z = Bp,∞ d 2 d continuously for all γ > d − p , we obtain for any ρ < p − d + p

p

E I4 (t)Lp (O) t p p d ≤ (t − s)− 2 (ρ+d− p ) E A−ρ (I − Ih ) G(uh (s); z)Z ι(dz) ds 0+ t

≤

0+

Z

−p 2

(t − s)

(ρ+d− dp )

ζ∈R

p E A−ρ (I − Ih ) g(uh (s), ·, ζ)Lp (O) ν(dζ) ds,

where G is defined in (3.20). Theorem 3.1 leads to p p pρ E I4 (t)Lp (O) ≤ C(t) h sup E g(uh (s), ·; ζ)Lp (O) ν(dζ) 0≤s≤t

ζ∈R

463


and the Lipschitz property of g to p E I4 (t)Lp (O)

≤ C(t) h

pρ

1 + sup 0≤s≤t

p E uh (s)Lp (O)

.

By Proposition 4.1 there exists some constant C < ∞ such that p

sup E uh (s)Lp (O) ≤ C,

h > 0.

0≤s≤t

Thus, we can infer that there exists some constant C < ∞ such that E I4 (t)Lp (O) ≤ C hpρ , p

(5.9)

0 ≤ t ≤ T.

The Burkholder inequality leads for any 12 (d − dp ) < γ
0 there exists some constant C < ∞ such that −tA C e Av Lp (O) ≤ ρ vWp−ρ (O) , t2

(5.12)

v ∈ Wp−ρ (O), t > 0.

Inequality (5.12) leads for all ρ ≥ 0, with p(γ + ρ) < 2, to I61 ≤ C hpγ(2+ 2 − p ) t kh −p (γ+ρ) h 2 × (t − s) λ(Ji ) d

d

0+

d

d

Wp−ρ (O)

ζ∈R

i=1

≤ C(γ, ρ) hpγ(2+ 2 − p ) sup

p Egh (uh (s); i, ζ)

kh

0≤s≤t i=1

λ(Jih )

ν(dζ) ds

p Egh (uh (s); i, ζ)

ζ∈R

Wp−ρ (O)

ν(dζ),

where C . 2 − p(γ + ρ)

C(γ, ρ) =

By Sobolev inequalities, we know that L1 (O) → Wp−ρ (O) continuously. Thus we obtain kh p d d I61 ≤ C hpγ(2+ 2 − p ) sup λ(Jih ) Egh (uh (s); i, ζ) ν(dζ) 1 0≤s≤t i=1

≤ C hpγ(2+ 2 − p ) sup d

d

kh

0≤s≤t i=1

≤ C hpγ(2+ 2 − p ) sup d

d

kh

0≤s≤t i=1

ζ∈R

λ(Jih )

ζ∈R

λ(Jih )

ζ∈R

L (O)

p p Eg(uh (s, Nih ), Nih ; ζ) chi φhi L1 (O) ν(dζ) p Eg(uh (s, Nih ), Nih ; ζ) ν(dζ).

Since g is Lipschitz continuous and linear in the third variable, we get I61 ≤ C hpγ(2+ 2 − p ) sup d

d

kh

0≤s≤t i=1

p λ(Jih ) 1 + E uh (s, Nih )Lp ds. h

465


In section 3.3 we have seen that there exist some constants c and C such that for all 0 < h < 1 we have (see (3.16)) c wLp (O) ≤ hpγ(2+ 2 − p ) wh Lp ≤ C wLp (O) , d

d

h

w ∈ Vh .

Thus, by taking into account Remark 3.1 we can write d d p I61 ≤ Ct1−p(γ+ρ) hpγ(2+ 2 − p ) sup 1 + E uh (s)Lp (O) . 0≤s≤t

p

Since E uh (s)Lp (O) is bounded, there exists a constant C < ∞ such that I61 ≤ Ct1−p(γ+ρ) hpγ(2+ 2 − p ) . d

(5.13)

d

Next, we have to handle the term I62 . Because of (3.18) and (3.16), there exists a constant C < ∞ such that for all 0 < h ≤ 1 −(t−s)Ah Ah v e

(5.14)

Lp (O)

≤

C vLp (O) , t

Because of inequality (5.14), we have for all 0 < δ < I62 ≤

t

(t − s)−δp

0

(5.15) ζ∈R

kh

v ∈ Lp (O).

1 p

λ(Jih )

i=1

p 1 1 E(−Ah )−δ Ih e− 2 A(t−s) − e− 2 Ah (t−s) gh (uh (s); i, ζ) p

ν(dζ) ds.

L (O)

In Remark 3.1 we have shown that there exist some constants c and C such that hd ≤ ckh ≤ hd . By (3.19) we know that for any i = 1, . . . , kh we have (−Ah )−δ φi =

(5.16)

kh

uil φhl ,

l=1

where c h2δ , if (i, l) ⊂ Ch , uil = 0, elsewhere. Moreover, there exists some constant m0 such that for any i and 0 < h < 1 {(i, Nkh ) ∩ Ch } ≤ m0 . By substituting (5.16) in (5.15) we obtain I62 ≤ h2δp ζ∈R

0

t

(t − s)−δp m0

kh

λ(Jih )

i=1

p 1 1 E Ih e− 2 A(t−s) − e− 2 Ah (t−s) gh (uh (s); i, ζ) p

L (O)

ν i (dζ) ds.

466

ERIKA HAUSENBLAS

By applying Theorem 3.4 we can infer that for any γ ∈ (01), we have I62 ≤ C hpδ+pγ(2+ 2 − p ) d

ζ∈R

d

t

(t − s)−(δ+γ)p m0

0

p Egh (uh (s); i, ζ) p

kh

λ(Jih )

i=1

ν(dζ) ds,

L (O) d

−d

p where ρ = γ/(2 − dp + d2 ). By Theorem 3.1 and since Z = Bp,∞ (O) → Wp−γ (O) continuously, we get for any ρ > d − dp

I62 ≤ C hp(δ−ρ)+pγ(2+ 2 − p ) d

d

t

(t − s)−(δ+ρ)p m0

0

p Egh (uh (s); i, ζ) ν(dζ) ds.

kh

λ(Jih )

i=1

Z

ζ∈R

Provided that δ ≥ ρ we can omit the first term in the exponent of h. The mapping g is linear in the last variable and globally Lipschitz in the first. Thus we can infer that d pγ(2+ d 2−p)

t

I62 ≤ C h

(t − s)−(δ+ρ)p m0

0

kh

p λ(Jih ) 1 + E uh (s, Nih ) ds.

i=1

By Remark 3.1 it follows that I62 ≤ C hpγ(2+ 2 − p ) d

d

t

(t − s)−(δ+ρ)p m0

0

kh

p hd 1 + E uh (s, Nih ) ds,

i=1

and by (3.16) I62 ≤ C hpγ(2+ 2 − p ) d

d

0

t

p (t − s)−(δ+ρ)p 1 + E uh (s)Lp (O) ds.

The Hölder inequality leads to d pγ(2+ d 2−p)

I62 ≤ C h

C(t) 1 + sup 0≤s≤t

p E uh (s)Lp (O)

and Theorem 4.1 that there exists a constant C such that (5.17)

I62 ≤ C hpγ(2+ 2 − p ) , d

d

0 < h ≤ 1.

Collecting together (5.4), (5.5), (5.6), (5.8), (5.9), (5.10), (5.13), and (5.17) gives the assertion. 6. Discussions and conclusions. The next step is to confirm the theory described in this paper by numerical experiments. One of the problems which arises here is that, if the intensity of the Poisson random measure is given, the distribution is known only in a few cases. Therefore, it is difficult to find a benchmark where the solution is known analytically and to compare the numerical result with the exact solution.

467


Moreover, because of this problem, the distribution of the random variables ξhm (·), m ∈ N, appearing in (2.6), is not given a priori. Different approaches exist to solve this problem; e.g., Rubenthaler [Rub03] cut off the jumps, which are smaller than a certain threshold, calculated the time of the first jump, and applied the Euler scheme until the time of the first jump. Then he simulated a random variable according to the distribution of size of the jump, with the condition that the jump is larger than the threshold. This step is repeated until the time T is reached. In [HM06a, HM06b] we approximated the intensity by a discrete distribution. In fact, first we cut off the small jumps by setting νh (B) = 0 for a neighborhood B of the origin, and then we approximated νh by a discrete distribution νhn . To be precise, let η˜hn be the real-valued compensated Poisson random measure given by the intensity νhn . Now for any x ∈ R the random variable (m+1)τh g(x; ζ) η˜hn (dz; ds) ζ∈Rkh

mτh

can be simulated directly by the sum of N independent random variables with distribution ν n1(R) νhn , where N is exponentially distributed with parameter νhn (R). h By taking into account that the small jumps are cut off, one may ask if the perturbation of the small jumps can be approximated by, e.g., a Wiener process. This has been done, e.g., by Kallianpur and Xiong [KX95]. Since we are in Banach spaces of M -type p, 1 ≤ p ≤ 2, the convergence of a simple random walk is not necessarily given. One alternative is to choose another scaling, which we also consider in [HM06b]. Appendix A. Besov spaces and their properties. An introduction to Besov spaces is given by Runst and Sickel [RS96]. We are interested is the continuity of the mapping G described in (3.20). To be precise, we prove the following result. Proposition A.1. If f ∈ S(Rd ), then for each a ∈ Rd the Schwartz distribution d

−d

p (Rd ), and moreover there exists a constant C = f δa belongs to the Besov space Bp,∞ C(d, p) independent of f such that p p |f δa | d −d da = |f |Lp (Rd ) .

Rd

p (Rd ) Bp,∞

n

−n

p Thus, there exists a unique bounded linear map Λ : Lp (Rd ) → Lp (Rd , Bp,∞ (Rd )) such that [Λ(f )](a) = f δa , f ∈ S(Rd ), a ∈ Rd . Moreover, for any s ≥ 0 there exists some constant C such that p p |f δa | d −d−s da ≤ C |f |W −s (Rd ) .

Rd

p Bp,∞

(Rd )

p

In the following we denote the value of Λ(f ) at a, where f ∈ Lp (Rd ) by f δa . Let us recall the definition of the Besov spaces as given in [RS96, Definition 2, pages 7–8]. First we choose a function ψ ∈ S(Rd ) such that 0 ≤ ψ(x) ≤ 1, x ∈ Rd , and 1, if |x| ≤ 1, ψ(x) = 0 if |x| ≥ 32 . Then put

⎧ ⎨ φ0 (x) = ψ(x), x ∈ Rd , φ1 (x) = ψ( x2 ) − ψ(x), x ∈ Rd , ⎩ φj (x) = φ1 (2−j+1 x), x ∈ Rd , j = 2, 3, . . . .

468

ERIKA HAUSENBLAS

−1 With the choice of φ = {φj }∞ being the Fourier and the j=0 as above and F and F inverse Fourier transformations, respectively, (acting in the space S (Rd ) of Schwartz distributions), we have the following definition. Definition A.1. Let s ∈ R, 0 < p ≤ ∞, and 0 < q ≤ ∞. If 0 < q < ∞ and f ∈ S (Rd ), we put ⎞ q1 ⎛ ∞ q 2sjq F −1 [φj Ff ]Lp ⎠ , |f |B s = ⎝ p,q

j=0

= sup 2sj F −1 [φj Ff ]Lp .

|f |B s

p,∞

j∈N

s (Rd ) the space of all f ∈ S (Rd ) for which |f |B s is finite. We denote by Bp,q p,q

Lemma A.2. If h ∈ S(Rd ), h ∈ S(Rd ), and a ∈ Rd , then |(hδa ) ∗ g|Lp (Rd ) = |h(a)||g|Lp (Rd ) .

Proof. Let us recall that hδb ∈ S (Rd ) assigns to a function g ∈ S(Rd ) a value δa (hg) = h(a)g(a). Since by the definition of the convolution of a distribution with a test function [(hδa ) ∗ g](x) = (hδa )(τx gˇ) = h(a)(τx gˇ)(a) = h(a)ˇ g (a − x) = h(a)g(x − a), x ∈ Rd , we have

p

p

|(hδa ) ∗ g|Lp (Rd ) =

|g(x − a)h(a)| dx p p = |h(a)|p |g(x)| dx = |h(a)|p |g|Lp (Rd ) . Rd

Rd

Lemma A.3. If ϕ ∈ S(Rd ), λ > 0, and g(x) := ϕ(λx), x ∈ Rd , then 1

|F −1 g|Lp (Rd ) = λd( p −1) |F −1 ϕ|Lp (Rd ) . Proof. The proof follows with simple calculations. Proof of Proposition A.1. Obviously it is enough to prove the first part of the proposition. We will use the definition of the Fourier transform as in [RS96]. In particular, with ·, · being the scalar product in Rd , we put −d/2 (Ff )(ξ) := (2π) e−i x,ξ f (x) dx, ξ ∈ Rd , f ∈ S(Rd ). Rd

Let us fix f ∈ S(R ). Since F infer that, for j ∈ N∗ , d

−1

(ϕu) = (2π)−d/2 (F −1 ϕ) ∗ (F −1 u), ϕ ∈ S, u ∈ S , we

|F −1 [ϕj F(f δa )]|Lp (Rd ) = (2π)−d/2 |(F −1 ϕj ) ∗ (f δa )|Lp (Rd ) = (2π)−d/2 |f (a)|F −1 ϕj |Lp (Rd ) 1

1

= (2π)−d/2 2d( p −1) 2−jd( p −1) |f (a)||F −1 ϕ1 |Lp (Rd ) . d( 1 −1)

p (Rd ) as requested, and the equality Hence, f δa belongs to the Besov space Bp,∞ (A.1) follows immediately. In order to show (A.1) note that γ s+γ f Bp,∞ (Rd ) = I(s)f Bp,∞ (Rd ) ,

469


# $s where I(s) = F −1 1 + ξ 2 2 F. Next, −1 # $s 2 2 F ϕ 1 + ξ F j

δa f da p d Rd L (R ) s −1 # $ −1 2 2 1+ξ ≤ F ϕj F F δa f da Rd

−1 # −1 $s 2 2 1+ξ ≤ F ϕj Lp (Rd ) F F

Rd

Moreover, −1 # $s 2 2 F 1+ξ F

Rd

Lp (Rd )

δa f da

.

L1 (Rd )

δa f da

L1 (Rd )

# $s i y,ξ 2 2 −i a,ξ = 1+ξ e e f (a) dξ da dy Rd Rd Rd # $s i y,ξ 2 2 −i a,ξ = e e f (a) da dξ dy 1+ξ Rd Rd Rd s s i y,ξ −1 2 2 dy = = F e F (I + Δ) f (ξ) dξ F (I + Δ) f (y) dy d d R Rd R s s ≤ F −1 F (I + Δ) 2 f = (I + Δ) 2 f = f W s (Rd ) .

L1 (Rd )

L1 (Rd )

p

In summary, one can show that for any s ∈ R there exists some C such that f δa da ≤ C f W s (Rd ) , f ∈ Cb0 (Rd ). s+ d −d Rd

Bp

p

p

(Rd )

Since the operator I(s) is positive for s ≤ 0, it follows that f δa s+ pd −d da ≤ C f W s (Rd ) , f ∈ Cb0 (Rd ), f ≥ 0. Rd

Bp

(Rd )

p

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