hp Optimization of finite element approximations: Analysis of the optimal mesh sequences in one dimension

This paper addresses the problem of the construction of the optimal hp meshes for the singular solution x, >1/2, in the one-dimensional setting. First, a review of the asymptotic analysis of Gui and Babuka [1] is given and is used to characterize the optimal hp meshes which are geometrically graded toward the singular point. The asymptotically optimal hp meshes are compared with the hp meshes which are constructed by employing the hp strategy of Rachowicz et al. [2], which employs the principle of maximum change of error per change in the number of degrees of freedom, and the true optimal hp meshes which minimize the error for fixed number of degrees of freedom and which are determined by searching through the entire sequence of the admissible hp meshes. It is shown that, for the singular solution, the meshes constructed by the hp strategy of Rachowicz coincide with the true optimal meshes; nevertheless, there exist various solutions for which the algorithm which employs the principle of maximum change of error may not converge. For this reason the hp strategy of Rachowicz is combined with the classical feedback algorithm to obtain a robust hp strategy which always converges. Further, it is shown that the asymptotically optimal hp meshes are very close to the true optimal hp meshes in the entire range of practical computation. The effect of the accuracy of the error estimator on the optimality of the constructed hp meshes is also analyzed. It is shown that unless the estimator is sufficiently accurate in the element adjacent to the singular point, the meshes constructed by the adaptive hp strategy can differ substantially from the true optimal meshes.