Asymptotically optimal mesh gradings are derived for interpolation using \(H^ p\) seminorms. An algorithm for computing an interpolating mesh that approximately minimizes the \(H^ p\) seminorm of the error is obtained. An example illustrates some aspects of the procedure. The idea is extended naturally to the mesh optimization problem for the finite difference or finite element solution of ordinary boundary value problems. Second order, linear equations, discretized by linear elements are considered in detail, and an adaptive, asymptotic equidistribution algorithm is described. Several numerical examples show the relative effectiveness of various choices of norm.