A meshfree Random Walk on Boundary algorithm with iterative refinement

Abstract A hybrid continuous Random Walk on Boundary algorithm and iterative refinement method is constructed. In this method, the density of the double layer boundary integral equation for the Laplace equation is resolved by an isotropic Random Walk on Boundary algorithm and calculated for a set of grid points chosen on the boundary. Then, a residual of the boundary integral equation is calculated deterministically, and the same boundary integral equation is solved where the right-hand side is changed with the residual function. This process is repeated several times until the desired accuracy is achieved. This method is compared against the standard Random Walk on Boundary algorithm in terms of their labor intensity. Simulation experiments have shown that the new method is about 200 times more efficient, and this advantage increases with the increase of the desired accuracy. It is noteworthy that the new hybrid algorithm, unlike the standard Random Walk on Boundary algorithm, solves the Laplace equation efficiently also in non-convex domains.