H-Matrix approximation for the operator exponential with applications

H-Matrix approximation for the operator exponential with applications 87
2.2 Examples
As the first example let us consider the one-dimensional operator A :
L 1 (0, 1) → L 1 (0, 1) withthe domain D(A) = H0 2 (0, 1) = {u : u ∈
H 2 (0, 1), u(0)=0,u(1)=0} in the Sobolev space H 2 (0, 1) defined by
Au = −u ′′
for all u ∈ D(A).
Here we set X = L 1 (0, 1) (see Definition 2.1). The eigenvalues λ k =
k 2 π 2 (k =1, 2,...) of A lie on the real axis inside of the domain Ω Γ0
enveloped by the path Γ 0 = {z = η 2 +1± iη}. The Green function for the
problem
(zI − A)u ≡ u ′′ (x)+zu(x) =−f(x), x ∈ (0, 1); u(0) = u(1)=0
is given by
G(x, ξ; z) =
i.e., we have
1
√ z sin
√ z
{
sin
√ zxsin
√ z(1 − ξ) if x ≤ ξ,
sin √ zξ sin √ z(1 − x) if x ≥ ξ,
u(x) =(zI − A) −1 f =
∫ 1
0
G(x, ξ; z)f(ξ)dξ.
Estimating the absolute value of the Green function on the parabola z =
η 2 +1± iη for |z| large enoughwe get that ‖u‖ L1 = ‖(zI − A) −1 f‖ L1 ≤
M
1+ √ |z| ‖f‖ L 1
(f ∈ L 1 (0, 1), z∈ C \Ω Γ0 ), i.e., the operator A : L 1 → L 1
is strongly P-positive in the sense of Definition 2.1. Similar estimates for
the Green function imply the strong positiveness of A also in L ∞ (0, 1) (see
[5] for details).
As the second example of a strongly P-positive operator one can consider
the strongly elliptic differential operator
(2.3) L := −
d∑
j,k=1
∂ j a jk ∂ k +
d∑
j=1
b j ∂ j + c 0
(
∂ j :=
∂ )
∂x j
withsmooth(in general complex) coefficients a jk ,b j and c 0 in a domain
Ω witha smoothboundary. For the ease of presentation, we consider the
case of Dirichlet boundary conditions. We suppose that a pq = a qp and the
following ellipticity condition holds
d∑
a ij y i y j ≥ C 1
i,j=1
d∑
yi 2 .
i=1