Random walk for elliptic equations and boundary layer
- Milstein, Grigori N.
2010 Mathematics Subject Classification
- 35J25 60J15 65N99
- Boundary value problem, weak methods of numerical integration of SDE, random walk, boundary layer
We consider the Dirichlet problem for equations of elliptic type in a domain G with a boundary ∂G. A probabilistic representation of solutions to the problem is connected with a system of stochastic differential equations (SDE). Unlike usual approximation of SDE when a time-discretization is exploited, here a space-discretization is recommended. We construct weak approximations for which an estimate of their errors contains derivatives of the required solution to the Dirichlet problem only of lower order. In particular, it is important for problems with a boundary layer. We simulate a Markov chain in G on the basis of a one-step approximation using variable step in the space. The chain should be stopped entering a sufficiently small neighborhood of the boundary ∂G. We estimate the average number of steps before stopping and state some convergence theorems.