Notation

We begin with a quick recap of frequently used concepts, most of which can be found in [112].

Let O ⊂ R be an open and bounded interval. Fork ≥ 0, let C^k(O) (Cc^k(O)) be the space of k times differentiable real-valued functions (with compact support). Let L²:=L²(O) be the Lebesgue space of square integrable functions, endowed with the normk·kL². Let H₀¹ :=H₀¹(O) be the Sobolev space of weakly differentiable functions with zero trace, endowed with the norm kukH₀¹ =k∇ukL², and letH⁻¹ be its topological dual space. Recall the canonical continuous embeddingI:H₀¹→L² provided by the Poincar´e inequality, and define its dual mapI⁰:L²→H⁻¹ by

hI⁰u, viH⁻¹×H₀¹ =hu, IviL².

foru∈H⁻¹,v∈H₀¹. Its dual mapI⁰⁰:H⁻¹→(L²)⁰ is defined analogously. We will also use the adjoint operator (I⁰)^∗:H⁻¹→L² ofI⁰, defined by

h(I⁰)^∗u, viL² =hu, I⁰viH⁻¹

for u ∈ H⁻¹, v ∈ L². If there is no risk of misunderstanding, we will not mention the use of the embeddingsI,I⁰ andI⁰⁰.

Recall that the negative Laplace operator−∆ :H₀¹→H⁻¹ is defined by h−∆u, viH⁻¹×H¹₀ =h∇u,∇viL².

We note that I⁰⁰◦(−∆) : H₀¹ → (L²)⁰ is continuous with respect to the L² norm. Hence, it can be extended linearly and continuously to the whole of L². The resulting operator L² → (L²)⁰ will also be denoted by −∆. As pointed out in [112, Remark 4.1.14], we note that this operator is a surjective isometry, and we stress that it does not coincide with the Riesz isomorphism for the classical dualization ofL² with itself.

For two separable Hilbert spaces H1, H2, we write L2(H1, H2) for the space of all Hilbert-Schmidt operators fromH1 toH2.

For a Banach spaceBandk≥0, letC^k([0, T];V) be the space ofktimes continuously differentiable curves in V parametrized byt∈[0, T]. For a measurable space (S,A), we denote the Lebesgue-Bochner space of measurable, square integrableB-valued functions byL²(S;B), which is defined e. g. in [86, Definition 1.2.15]. If S is the product two Banach spaces S₁×S₂, we will use L²(S;B) and L²(S₁;L²(S₂;B)) interchangeably, see Lemma 2.B.3 for a justification. Letf ∈L²(Ω×[0, T];B) be a Banach-space valued random function. Then, f is called progressively measurable with respect to a filtration (F^t)_t∈[0,T], if f|^[0,t] is measurable with respect toF^t⊗ B([0, t])− B(B) for allt∈[0, T].

For the productV =V₁× · · · ×V_n of topological spaces, wheren∈N, we define thei-th projection Π_i by Π_i(v) =v_i fori∈ {1, . . . , n}, which is a continuous map by the definition of the product space. We will use Π_i for any such projection, regardless of the respective underlying spaces.

LetT >0 and consider a probability triple (Ω,F,P) and a filtration (F^t)_t∈[0,T], whereF^t ⊂ F for all t∈[0, T]. Expected values with respect toPwill be denoted by E. The filtration is called normal, if it is complete, i. e.F^tcontains allA∈ F withP(A) = 0 for allt∈[0, T], and right-continuous, i. e.

F^t= \

s>t

F^s for allt∈[0, T].

Each filtration (F^t)_t∈[0,T] can be augmented to a normal filtration (Ft^∗)_t∈[0,T] by defining FT^∗ =σ(F^T ∪ N),

Ft^∗= \

s>t

σ(F^s∪ N) for allt∈[0, T),

whereN denotes the collection of allP-zero sets. We refer to [114, p. 45] for details.

We now turn to the finite-dimensional structures which we will use to formulate numerical convergence results. From now on, we fix

O= [0,1]⊂R.

Consider an equidistant grid on the unit interval with grid points (x_i)^Z_i=0withh= _Z¹, Z∈Nandx_i=ih.

Fori= 0, . . . , Z−1 lety_i= i+¹₂

h. Consider the sets of intervals (K_i)_i=0,...,Zand (J_i)i=0,...,Z−1given by

K₀= [x₀, y₀), K_Z = [y_Z−1, x_Z], K_i= [y_i−1, y_i) fori= 1, . . . , Z−1,

Ji= [xi, xi+1) fori= 0, . . . , Z−1. (2.1.7)

We consider the space of grid functions on (xi)^Z_i=0 with zero boundary conditions, which is isomorphic toR^Z−1, and we define the following prolongations (see Figure 2.1).

Definition 2.1.1. Letuh∈R^Z−1 andvh∈R^Z. We then define the piecewise linear prolongation with respect to the grid (xi)i=0,...,Z with zero-boundary conditions by

I_h^plx :R^Z−1,→H₀¹, uh7→u^plx_h :=

Z−1

X

i=0

uh,i+uh,i+1−uh,i

h (· −xi)

1J_i, and the piecewise constant prolongation by

I_h^pcx :R^Z−1,→L², uh7→u^pcx:=

Z−1

X

i=1

uh,i1K_i,

0 h 2h · · · 1 0

u u^pcx u^plx

Figure 2.1: Different prolongations of a spatial grid function

with the conventionuh,0 =uh,Z= 0. The image ofI_h^pcx, i. e. the space of piecewise constant functions on the partition (Ki)^Z_i=0 with zero Dirichlet boundary conditions, will be denoted by S^pcx_h . The L² -orthogonal projection to this space will be denoted by Π^pcx_h . Note thatI_h^pcx :R^Z−1→S_h^pcxis bijective.

Lemma 2.1.2. Letη∈L². Then,Π^pcx_h η →η inL² forh→0.

Proof. The proof is a simpler version of the proof of Lemma 2.5.6 below, which is why it is omitted here.

Leth·,·i:=h·,·il² denote the inner product arising from the Euclidean normk·k:=k·kl² onR^Z−1. For a matrixA∈R(Z−1)×(Z−1), kAk denotes the matrix norm induced byk·k, i. e.

kAk:= sup

x∈R^Z−1\{0}

kAxk

kxk . (2.1.8)

Let ∆h∈R(Z−1)×(Z−1)be the matrix corresponding to the finite difference Laplacian on grid functions on (xi)^Z_i=0 with zero Dirichlet boundary conditions, i. e.

∆_h=−1 h²





















2 −1

−1 2 −1

0

−1 . .. . .. . .. . .. −1

0

⁻¹ ₋²₁ ⁻₂¹





















. (2.1.9)

Recall that −∆_h is symmetric and positive definite (for a formal argument, see Lemma 2.4.1 below).

Hence, the following definition is admissible.

Definition 2.1.3. OnR^Z−1, we define the inner products h·,·i0,h·,·i1andh·,·i−1 by hu, vi0=hhu, vi

hu, vi1=h−∆hu, vi0

hu, vi−1=

(−∆h)⁻¹u, v

0

foru, v∈R^Z−1.

Remark 2.1.4. The inner producth·,·i0in Definition 2.1.3 corresponds to the L²norm onOby the fact that

I_h^pcx : R^Z−1,k·k0

→(S_h^pcx,k·kL²) is an isometry, i. e.

hu, vi0=hI_h^pcxu, I_h^pcxviL² foru, v∈R^Z−1.

0 τ 2τ · · · T v0

v v^pct−

v^pct+

v^plt

Figure 2.2: Different prolongations of a time grid function

Furthermore, Definition 2.1.3 suggests to viewhu, vi1 andhu, vi−1 as discrete analogues of theH₀¹ and H⁻¹ norm on O, respectively. These connections are more subtle and will be made more precise in Lemma 2.4.7, Lemma 2.4.9 and Proposition 2.4.29 below.

Next, we consider a lattice for the time interval [0, T], T > 0. For τ > 0 such that T = N τ, N ∈ N, consider the equidistant grid (0, τ,2τ, . . . , N τ). We then define the following prolongations of grid functions (see Figure 2.2).

Definition 2.1.5. Let (vk)^N_k=0 ⊆ R be a grid function on the previously described grid of length τ.

Then we define the piecewise linear prolongation v^plt : [0, T] → R, the left-sided piecewise constant prolongationv^pct-: [0, T]→Rand the right-sided piecewise constant prolongationv^pct+: [0, T]→Rby

v^plt(t) = t−tτ

τ v_bt/τc+1+tτ+τ−t τ v_bt/τc, v^pct-(t) =v_bt/τc,

v^pct+(t) =v_bt/τc+1.

Definition 2.1.6. Let N, Z ∈ N and (uk,l)k=0,...,N;l=1,...,Z−1 ⊂ R be a function on the space-time grid covering [0, T]×[0,1], with time grid length τ and space grid length h, such that τ N = T and Zh = 1. Committing a slight abuse of notation, we define the componentwise time prolongations u^plt, u^pct-, u^pct+: [0, T]→R^Z−1by

u^plt(t) := (u·,l)^plt(t)^Z−1

l=1 :=

(uk,l)^N_k=0^plt

(t)Z−1 l=1 , u^pct-(t) := (u·,l)^pct-(t)Z−1

l=1 :=

(uk,l)^N_k=0 pct-(t)Z−1

l=1 , u^pct+(t) := (u·,l)^pct+(t)Z−1

l=1 :=

(uk,l)^N_k=0^pct+

(t)Z−1 l=1 , and the componentwise spatial piecewise constant prolongationu^pcx: [0,1]→R^N+1by

u^pcx(x) := (u^pcx_k (x))^N_k=0:=

(uk,l)^Z−1_l=1 ^pcx (x)N

k=0

,

where we used the extensions from Definition 2.1.1 and 2.1.5. Finally, we define the full prolongations u^plt,pcx, u^pct-pcx, u^pct+pcx: [0, T]×[0,1]→R

by

u^plt,pcx(t, x) = u^plt(t)^pcx

(x) = (u^pcx(x))^plt(t), u^pct-pcx(t, x) = u^pct-(t)^pcx

(x) = (u^pcx(x))^pct-(t), and u^pct+pcx(t, x) = u^pct+(t)pcx

(x) = (u^pcx(x))^pct+(t).

Im Dokument Stochastic partial differential equations arising in self-organized criticality (Seite 15-19)