 
Summary: SIAM J. MATH. ANAL. c 2010 Society for Industrial and Applied Mathematics
Vol. 0, No. 0, pp. 000000
RESULTS ON THE DIFFUSION EQUATION WITH ROUGH
COEFFICIENTS¦
BURAK AKSOYLU AND HORST R. BEYER
Abstract. We study the behavior of the solutions of the stationary diffusion equation as a
function of a possibly rough (L ) diffusivity. This includes the boundary behavior of the solution
maps, associating to each diffusivity the solution corresponding to some fixed source function, when
the diffusivity approaches infinite values in parts of the medium. In ndimensions, n 1, by assuming
a weak notion of convergence on the set of diffusivities, we prove the strong sequential continuity
of the solution maps. In one dimension, we prove a stronger result, i.e., the unique extendability of
the map of solution operators, associating to each diffusivity the corresponding solution operator,
to a sequentially continuous map in the operator norm on a set containing "diffusivities" assuming
infinite values in parts of the medium. In this case, we also give explicit estimates on the convergence
behavior of the map. In the end, we provide connections to preconditioning.
Key words. diffusion equation, diffusion operator, rough coefficients, preconditioning, first
order formulation, mixed formulation, dependence on diffusivity
AMS subject classifications. 35J25, 47F05, 65J10, 65N99
DOI. 10.1137/080738520
1. Motivation. Numerical methods for the diffusion equation with rough coef
