  In this paper, a meshless Hermite-HDMR finite difference method is proposed
to solve high-dimensional Dirichlet problems. The approach is based on the
local Hermite-HDMR expansion with an additional smoothing technique. First, we
introduce the HDMR decomposition combined with the multiple Hermite series to
construct a class of Hermite-HDMR approximations, and the relevant error
estimate is theoretically built in a class of Hermite spaces. It can not only
provide high order convergence but also retain good scaling with increasing
dimensions. Then the Hermite-HDMR based finite difference method is
particularly proposed for solving high-dimensional Dirichlet problems. By
applying a smoothing process to the Hermite-HDMR approximations, numerical
stability can be guaranteed even with a small number of nodes. Numerical
experiments in dimensions up to $30$ show that resulting approximations are of
very high quality.
