Fully automatic hp-adaptivity in three dimensions

Abstract

The paper presents a three-dimensional implementation of the fully automatic hp-adaptive mesh refinement strategy proposed in [L. Demkowicz, W. Rachowicz, Ph. Devloo, A fully automatic hp-adaptivity, J. Sci. Comput. 17 (1–3) (2002) 127–155]. The methodology is based on a minimization of the projection-based interpolation error with a fixed number of degrees-of-freedom. A (coarse) grid is first uniformly refined in both h and p to yield a corresponding fine grid. A two-grid solver is used to solve the problem on the fine grid or iterate towards the fine grid solution. The fine grid solution (possibly partially converged) serves then as a reference solution to be interpolated on the original coarse mesh, and the next, optimal coarse grid to be determined. The difference between the coarse and fine grid solutions provides an excellent error estimate for the coarse grid error. The procedure is repeated, with the coarse and fine meshes generated during the adaptive process simultaneously, until a prescribed error tolerance for the coarse grid error is met. The solution obtained on the last fine grid is considered to be the final product delivered by the adaptive procedure.