# How to calculate electric potential distribution of nodes in subgrid using FDM?

Boundary: the potential of the upper electrode is 7 V, and lower is 0. Potential of two edges is shown in the following figure. It is easy to calculate the potential distribution of nodes using Finite Difference Method with Laplace' equation. But when adding subgrid (the red grid of the following figure), how could I calculate the potential of subnodes? Thank you, guys.

• Do you understand how to do it in the case of the top image? – Kyle Kanos Jun 25 '18 at 1:07
• @Kyle Kanos Oh, yes, when using 2nd order accurate FD approximation, I could calculate the case in the upper figure. So should I use 4th order accurate to sort the case of the second figure? Expecting some details. Many thanks. – Joe ZZ Jun 25 '18 at 6:36
• Each of those 4 smaller squares should average to one of the larger squares. So you kinda have 2 domains to solve: one is a $6\times7$ block the other a $2\times14$ block. Just apply the same discretization & scheme to both domains, but for the larger you'll need to average the values of the smaller domain. – Kyle Kanos Jun 25 '18 at 9:54
• This is a sort of "static mesh refinement" (outline here), which is a looser form of adaptive mesh refinement (which I discuss in this answer of mine). – Kyle Kanos Jun 25 '18 at 9:56
• @Kyle Kanos Right, I got some of it. But I am confused that how I could calculate nodes at the interface, namely the nodes at 3rd row. Thanks for patience. – Joe ZZ Jun 25 '18 at 15:41