A class of new implicit compact sixth-order approximations for Poisson equations and the estimates of normal derivatives in multi-dimensions

In this piece of work, a family of compact implicit numerical algorithms for (∂u/∂n) of order of accuracy six are proposed on a 9- and 19-point compact cell for two- and three- dimensional Poisson equations ∆2u=f which are quite often useful in mathematical physics and engineering, where ∆2 is either two or three dimensional Laplacian operator. First, we propose a family of new numerical algorithms of order of accuracy six for the computation of the solution of 2D and 3D Poisson equations on 9- and 27-points compact stencil, respectively. Then with the aid of the numerical solution of u, we propose a new family of compact sixth order implicit numerical algorithms for the estimates of (∂u/∂n). The proposed algorithms are free from derivatives of the source functions, which makes our algorithms more efficient for computation. Suitable iteration techniques are used for computation to demonstrate the sixth order convergence of the proposed algorithms. Numerical results are tabulated, confirming the usefulness of the suggested numerical algorithms.