Figure 7.

Examples illustrating how the measure of a primal simplex σ^k (black) and its dual ⋆σ^k (red) need not be the same order of magnitude. (a) In this 2D example, the ratio |⋆ σ¹|/|σ¹| can be made arbitrarily small by increasing the length of σ¹. (b) The ratio |⋆ σ¹|/|σ¹| can be made arbitrarily large by decreasing the length of σ¹. (c) The ratio |⋆ σ²|/|σ²| can be made arbitrarily large by decreasing the area of σ². Thus, a discrete Hodge star involving terms of the form |⋆ σ^k|/|σ^k| may have a bad condition number unless primal and dual mesh quality is controlled.