H-Matrix approximation for the operator exponential with applications

88 I. P. Gavrilyuk et al.
This operator is associated with the sesquilinear form
⎛
⎞
∫ d∑
d∑
a(u, v) = ⎝ a ij ∂ i u ∂ j v + b j ∂ j u v + c 0 uv⎠ dΩ.
Ω
i,j=1
Our algorithm needs explicit estimates for the parameters of the parabola
which in this example have to be expressed by the coefficients of the differential
operator. Let
C 2 := inf
x∈Ω
1 ∑
∣2
j
∂b j
∂x j
− c 0
∣ ∣∣∣
,
j=1
C 3 := √ d max |b j (x)|,
x,j
|u| 2 1 = ∑ j |∂ ju| 2 be the semi-norm of the Sobolev space H 1 (Ω), ‖·‖ k be the
norm of the Sobolev space H k (Ω) (k =0, 1,...) with H 0 (Ω) =L 2 (Ω),
and C F the constant from the Friedrichs inequality
|u| 2 1 ≥ C F ‖u‖ 2 0
for all u ∈ H 1 0 (Ω).
This constant can be estimated by C F =1/(4B 2 ), where B is the edge
of the cube containing the domain Ω. It is easy to show that in this case
with V = H0 1(Ω),H = L √ 2(Ω) it holds ξ u ≥ C 1 |u| 2 1 − C 2‖u‖ 0 ≥ C 1 C F −
C 2 , |η u |≤C 3 |u| 1 ≤ C 3 (ξu + C 2 )/C 1 , so that the parabola Γ δ is defined
by the parameters δ 0 = C 1 ,δ 1 = C 2 ,κ= C 3 and the lower bound of sp(A)
can be estimated by γ 1 = C 1 C F − C 2 >γ 0 . Now, the desired parabola Γ 0
is constructed as above by putting a = (γ 1−γ 0 )δ 0
(γ 1 +δ 1 )κ
, see Sect. 2.1.
The third example is given by a matrix A ∈ R n×n whose spectrum
satisfies Resp(A) > 0. In this case, the parameters of the parabola Γ 0 can be
determined by means of the Gershgorin circles. Let A = {a ij } n i,j=1 , define
C i = {z : |z − a ii |≤
n ∑
j=1,j≠i
Then by Gershgorin’s theorem,
a ij },D j = {z : |z − a jj |≤
sp(A) ⊂C A := (∪ i C i ) ∩ (∪ j D j ) .
n ∑
i=1,i≠j
a ij }.
The corresponding parabola Γ 0 is obtained as the enveloping one for the set
C A withsimple modifications in the case Re(C A ) ∩ (−∞, 0] ≠ ∅.
2.3 Representation of the operator exponent
Let L be a linear, densely defined, closed, strongly P-positive operator in
a Banachspace X. The operator exponent T (t) ≡ exp(−tL) (operatorvalued
function or a continuous semigroup of bounded linear operators on