Communications in Mathematical Sciences
Volume 7 (2009)
Numerical averaging of non-divergence structure elliptic operators
Pages: 785 – 804
Many important equations in science and engineering contain rapidly varying operators that cannot be practically sufficiently resolved for accurate solutions. In some cases it is possible to obtain approximate solutions by replacing the rapidly varying operator with an appropri- ately averaged operator. In this paper we use formal asymptotic techniques to recover a formula for the averaged form of a second order, non-divergence structure, linear elliptic operator. For several special cases the averaged operator is obtained analytically. For genuinely multi-dimensional cases, the averaged operator is also obtained numerically using finite difference method, which also has a probabilistic interpretation.
Homogenization; partial differential equations; elliptic partial differential equations; diffusions; finite difference methods
2010 Mathematics Subject Classification
35B27, 35J15, 65L12