Question

Example 1.3.16. Find the Eigenvalues of the inverse of the matrix
1 0
3 4
0
[A.U M/J 2014] [A.U A/M 2015 R-08]
0
0
A =
4​

Answer 1

Answer:

today, encountered a confused problem, but I have no ideas to deal with it fully. The question can be described as follows: for example, when we need to use FDM/FEM to discrete the partial differential

the coefficients di(x, y), d₂(x, y), v(x, y) ≥ 0. For simplicity, we use the following notations to explain the discretization by FDM:

operator with variable coefficients, C = d₁(x, y) +d₂(x, y) +v(x,y)u(x, y), where

di(x,y), d2(x, y), v(x, y) D₁, D2,

V (three diagonal matrices, whose entries are all nonnegative) 8²u(x, y)

L1 (discretized matrix)

a²u(x, y) L2(discretized matrix)

Oy2

so we can obtain the discretized matrix of Las follows, A: D₁L₁ + D₂L2 + VI (I is the identity matrix). Maybe, we want to construct a preconditioner as P diLi+d₂L2+ I, where d1, d2, û are all the mean value of diagonal entries (for example, if D₁ diag(81, 82,..., 8n), then

-). in fact, this preconditioner can be regard as the discretized matrix of

L=d₁ +d₂ + vu(x, y), so I want to know that

(1) Is this idea reasonable?

(2) if so, how to analyze the spectrum of preconditioned matrix P¹ A, for the sake of simplicity, take

v(x, y) = 0,, I can write

(dLid₂L₂) (D₁L₁ + D₂L₂) =I+ (d₁ L₁ + d₂L₂) ¹ (D₁-d₁I)L₁ + (D₂ - d₂I) L₂],

it means that there are some eigenvalue closed to 1 and this can translate a rapid convergence of Krylov subspace methods? However, this analysis seems to be very rough. So are there some more detailed spectral analysis of preconditioned matrix ? Maybe, some one can provide some references to explain this idea ?

(3) Maybe, some other boundary condition can be also considered.