The Mathematical Institute, University of Oxford, Eprints Archive

Preconditioning the Advection-Diffusion Equation: the Green's Function Approach

Loghin, Daniel and Wathen, A. J. (1997) Preconditioning the Advection-Diffusion Equation: the Green's Function Approach. Technical Report. Unspecified. (Submitted)



We look at the relationship between efficient preconditioners (i.e., good approximations to the discrete inverse operator) and the generalized inverse for the (continuous) advection-diffusion operator -- the Green's function. We find that the continuous Green's function exhibits two important properties -- directionality and rapid downwind decay -- which are preserved by the discrete (grid) Green's functions, if and only if the discretization used produces non-oscillatory solutions. In particular, the downwind decay ensures the locality of the grid Green's functions. Hence, a finite element formulation which produces a good solution will typically use a coefficient matrix with almost lower triangular structure under a "with-the-flow" numbering of the variables. It follows that the block Gauss-Seidel matrix is a first candidate for a preconditioner to use with an iterative solver of Krylov subspace type.

Item Type:Technical Report (Technical Report)
Subjects:D - G > Functional analysis
H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1312
Deposited By: Lotti Ekert
Deposited On:09 Jun 2011 07:23
Last Modified:29 May 2015 19:00

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