Multi-level hp-adaptivity and explicit error estimation

Abstract

Recently, a multi-level hp-version of the finite element method (FEM) was proposed to ease the difficulties of treating hanging nodes, while providing full hp-approximation capabilities. In the original paper, the refinement procedure made use of a-priori knowledge of the solution. However, adaptive procedures can produce discretizations which are more effective than an intuitive choice of element sizes h and polynomial degree distributions p. This is particularly prominent when a-priori knowledge of the solution is only vague or unavailable. The present contribution demonstrates that multi-level hp-adaptive schemes can be efficiently driven by an explicit a-posteriori error estimator. To this end, we adopt the classical residual-based error estimator. The main insight here is that its extension to multi-level hp-FEM is possible by considering the refined-most overlay elements as integration domains. We demonstrate on several two-and three-dimensional examples that exponential convergence rates can be obtained.

A painless automatic hp-adaptive strategy for elliptic problems

“…Inheriting some of the existing hp-adaptive algorithms is simply unfeasible with the new data structures employed by Zander et. al [31]. For example, the local projections considered by Demkowicz [3] in the context of hierarchical h-basis functions considered in our work would lead to an inefficient and overly complex implementation.…”

“…Step strategy [15] that performs first an h-adaptive step followed by a p-adaptive one that leads to non-optimal results; (c) the work of Demkowicz et al presented in [3,4,16] and applied in several contexts, e.g. [17][18][19][20][21][22][23][24][25][26][27][28], that produces almost optimal meshes but needs from solving the problem over a globally refined ( h 2 , p + 1)-grid, which is often prohibitively expensive, and also requires a sophisticated implementation; (d) the work of Houston et al [29], which estimates the regularity of the solution with the Legendre coefficients [30] and, (e) the contribution of Zander et al [31] and applications [32][33][34][35], which combine their multi-level data structure [9, 10] with a classic residual-based estimator [36]. We refer to [30] for a recent (Oct. 2014) review and comparison of some of the existing methods in terms of computational time versus the number of degrees of freedom (dofs).…”

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

“…For singular problems, the uniform multi‐level hp ‐refinement shows exponential convergence in the preasymptotic range, which can be extended by increasing the refinement depth . For boundary conforming problems, an a posteriori error estimator for the multi‐level hp ‐FEM coupled with a smoothness indicator was developed, where automatically generated hp ‐FEM discretizations with nonuniform p ‐distributions further improved the efficiency.…”

Parallelization of the multi-level hp-adaptive finite cell method

“…Furthermore, a priori refinement allows mesh partitioning to be directly performed on the refined grid. Multi-level hp-refinement can, however, be extended to perform refinement driven by error indicators and estimators, as shown in [43], and is a topic of ongoing research [44]. Such automatic hp-refinement would, however, require more involved fully parallel refinement strategies and data structures such as those described in [7,3], which perform a distribution of the initial elements, followed by parallel refinement on a subset of elements and a subsequent re-balancing of the refined grid on the granularity of either initial elements or refined elements in order to guarantee scalability.…”