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)

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Abstract

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 08:23
Last Modified:09 Jun 2011 08:23

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