Provably convergent anisotropic output-based adaptation for continuous finite element discretizations

Abstract

The expansion of modern computing power has seen a commensurate rise in the reliance on numerical simulations for engineering and scientific purposes. Output error estimation combined with metric-based mesh adaptivity provides a powerful means of quantifiably controlling the error in these simulations, for output quantities of interest to engineers and scientists. The Mesh Optimization via Error Sampling and Synthesis (MOESS) algorithm, developed by Yano for Discontinuous Galerkin (DG) discretization, is a highly effective method of this class. This work begins with the extension of the MOESS algorithm to Continuous Galerkin (CG) discretization which requires fewer Degrees Of Freedom (DOF) on a given mesh compared to DG. The algorithm utilizes a vertex-based local error decomposition, and an edge-based local solve process in contrast to the element-centric construction of the original MOESS algorithm.

Numerical results for linear problems in two and three dimensions demonstrate the improved DOF efficiency for CG compared to DG on adapted meshes. A proof of convergence for the new MOESS extension is then outlined, entailing the description of an abstract metric-conforming mesh generator. The framework of the proof is rooted in optimization, and its construction enables a proof of higher-order asymptotic rate of convergence irrespective of singularities. To the author's knowledge, this is the first such proof for a Metric-based Adaptive Finite Element Method in the literature. A three dimensional Navier Stokes simulation of a delta wing is then used to compare the new formulation to the original MOESS algorithm. The required stabilization of the CG discretization is performed using a new stabilization technique: Variational Multi-Scale with Discontinuous sub-scales (VMSD).

Numerical results confirm that VMSD adapted meshes require significantly fewer DOFs to achieve a given error level when compared to DG adapted meshes; these DOF savings are shown to translate into a reduction in overall CPU time and memory usage for a given accuracy