# Eigenvalue analysis of preconditioned partial differential operator

today, I encountered a confused problem by accident, 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 operator with variable coefficients, $\mathcal{L} = d_1(x,y)\frac{\partial^2 u(x,y)}{\partial x^2} + d_2(x,y)\frac{\partial^2 u(x,y)}{\partial y^2} + v(x,y)u(x,y)$, where the coefficients $d_1(x,y),d_2(x,y),v(x,y)\geq 0$. For simplicity, we use the following notations to explain the discretization by FDM: $$d_1(x,y),\ d_2(x,y),\ v(x,y) \rightharpoonup D_1, D_2, V\ \ (\mathrm{three\ diagonal\ matrices,\ whose\ entries\ are\ all\ nonnegative})\\ \frac{\partial^2 u(x,y)}{\partial x^2}\rightharpoonup L_1 (\mathrm{discretized\ matrix})\\ \frac{\partial^2 u(x,y)}{\partial y^2}\rightharpoonup L_2 (\mathrm{discretized\ matrix})\\$$ so we can obtain the discretized matrix of $\mathcal{L}$ as follows, $A:= D_1L_1 + D_2L_2 + VI$ ($I$ is the identity matrix). Maybe, we want to construct a preconditioner as $P:=d_1L_1 + d_2L_2+\hat{v}I$, where $d_1,d_2,\hat{v}$ are all the mean value of diagonal entries (for example, if $D_1 = diag(s_1,s_2,\ldots,s_n)$, then $d_1 = \frac{s_1 + s_2+\ldots + s_n}{n}$). in fact, this preconditioner can be regard as the discretized matrix of $\mathcal{L} = d_1\frac{\partial^2 u(x,y)}{\partial x^2} + d_2\frac{\partial^2 u(x,y)}{\partial y^2} + \hat{v}u(x,y)$, so I want to know that:

(1) Is this idea reasonable?

(2) if so, how to analyze the spectrum of preconditioned matrix $P^{-1}A$, for the sake of simplicity, take $v(x,y) = 0,$, I can write $$(d_1L_1 + d_2L_2)^{-1}(D_1L_1 + D_2 L_2) = I + (d_1L_1 + d_2L_2)^{-1}[(D_1 - d_1I)L_1 + (D_2-d_2 I) L_2],$$ 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.

I have do a small test problem: the spectral distributions of original and preconditioned matrix, please see the following figures:

• Have you tried to do this with a small test problem? – Kirill Nov 2 '14 at 16:19
• @Kirill, thank you for your kind reminder, I have add the spectral distributions of a couple of small matrices. Also I use the GMRES(20) method to solve the resulted linear systems needed 1222 iteration steps, however, if I add the preconditioner $P$, it only needs 22 iteration steps. – Hsien-Ming Ku Nov 2 '14 at 17:37
• This is close to Jacobi preconditioner, so it seems reasonable, but is it better than Jacobi? How are you solving linear systems $P^{-1}x=b$? If you can demonstrate by experiment that it performs well, that alone might be sufficient, because analysing preconditioners mathematically can be difficult. – Kirill Nov 3 '14 at 1:19