Computerized methods and computer apparatus for determining the values of physical quantities in problems governed by Poisson or Helmholtz equations, and method for fabricating an electronic component

A method of fabricating an electronic component, the method comprising:

• (a) forming a numerical model of physical structure of the electronic component in a three-dimensional space, the numerical model comprising, for each of a plurality of discretization points in the three-dimensional space, a known value of an electrical potential of the electronic component at the discretization point, an unknown electrical charge distribution at the discretization points being related to the electrical potential of the discretization points, by a Poisson equation expressed via an interaction matrix A having elements which correspond to respective pairs of the discretization points,
• (b) repeatedly applying a clustering algorithm, to classify the discretization points into a hierarchical tree structure having a plurality L+1 of levels, labelled by an integer index l=0, . . . ,L,
• where level l=0 of the tree structure is all the discretization points, level l=1 of the tree structure is composed of non-overlapping clusters of the discretization points, and
• for l=2, . . . ,L, level 1 of the tree structure is composed of, for each cluster of the (l−1)-th level, a respective plurality of non-overlapping clusters of the discretization points of that cluster of the (l−1)-th level,
• the clustering algorithm defining the clusters by minimizing distances between discretization points of each cluster to a respective center of that cluster, and maximizing distances between points of each cluster to the centers of other clusters;
• (c) using the hierarchical tree structure to form a compressed interaction matrix which is expressed recursively according to the levels;
• (d) determining the electrical charge distribution of the discretization points from the values of the electrical potential at the discretization points and the compressed interaction matrix;
• (e) determining whether the determined electrical charge distribution meets a criterion;
• and
• (f) determining that the electronic component is to be fabricated according to the numerical model when the electrical charge distribution meets the criterion, or that the electronic component is not to be fabricated according to the numerical model when the electrical charge distribution does not meet the criterion.