On grid refinement at point singularities for <i>h</i>‐<i>p</i> methods

The h ‐ p method for solving partial differential equations uses discretizations where the grid size h and the polynomial degree p are varied to obtain the most efficiency in the computing of approximate solutions. It has been known for many years that increasing p can dramatically improve efficiency and that the current common practice of keeping p = 1 is a particularly poor choice. For problems with singularities, good efficiency requires that the grid be refined near the points with singularities so the h ‐ p method also requires a grid refinement scheme. We compare two such schemes here which we call E‐Refinement (proposed by Babuska and colleagues recently) and Q‐Refinement (proposed by Rice in the 1960's). Both of these schemes have been shown to have asymptotically optimal order of convergence. The actual behaviours of these two schemes are quite different in practice and the mechanisms for choosing good values of h, p and the grid refinement parameters are also quite different. This paper presents the results of a systematic experimental study of these two methods and discusses the difficulty of choosing highly efficient h, p and grid refinement parameters. Our study confirms that these two refinements are of approximately equal efficiency given such good choices. The data suggest that the Q‐Refinement is more efficient but the advantage is not dramatic given the uncertainties inherent in choosing numerical methods for practical applications. We conclude that it is substantially simpler to make good choices for the Q‐Refinement than for the E‐Refinement.