Solution of elliptic equations without initial boundary values

A method for the numerical solution of elliptic equations is presented here, which does not require the computation of initial boundary values. This method is particularly advantageous for a multidimensional system with complex source density distribution for which the accurate computation of boundary values can be several orders of magnitude more expensive than the elliptic problem resolution itself. Associated with the newly developed `transmitted-wave´ boundary condition, it can allow a multi-scale resolution of an elliptic equation on a system of grids having specific resolution adjusted to the variations of density, giving capabilities similar to tree codes. This elliptic solver is used in the electromagnetic PIC code BPIC3D for the computation of the initial electric and magnetic fields in simulations of beam transport in Heavy-Ion Inertial Fusion