An a posteriori-driven adaptive Mixed High-Order method with application to electrostatics

Abstract

In this work we propose an adaptive version of the recently introduced Mixed High-Order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism. The latter include, in particular, the numerical modeling of comb-drive and MEMS devices. Mesh adaptation is driven by newly derived, residual-based error estimators. The resulting method has several advantageous features: It supports fairly general meshes, it enables arbitrary approximation orders, and has a moderate computational cost thanks to hybridization and static condensation. The a posteriori-driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions, allowing to fully exploit the high-order of approximation.

A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits

“…From the rightmost columns of Tables 1-3, it can be noticed that the assembly time becomes negligible with respect to the resolution time as finer and finer meshes are considered. This behaviour had already been observed in other HHO implementations (see, e.g., the numerical results in [27]).…”

“…with d Ω denoting the diameter of Ω. Hence, denoting by I k RTN,h : H 1 (Ω) d → RTN k (T h ) the global Raviart-Thomas-Nédélec interpolator whose restriction to each mesh element T ∈ T h coincides with I k RTN,T (see (27)), we arrive at…”

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

“…4 Aside from grid adaptation strategies based on elements subdivision, see, eg, Hartmann and Leicht,5,6 or conforming grid modification, see, eg, Dolejší and Felcman, 7 an alternative approach, based on the adaptive coarsening of a fine mesh, has been proposed by Bassi et al 8 in the context of DG methods and applied by Di Pietro and Specogna to hybrid high-order formulations of electrostatics. 9 Recently, Collis and Houston also developed an agglomeration-based adaptive mesh refinement algorithm coupled with a goal-oriented error estimator applied to the DG approximation of the linear elasticity equations for a homogeneous isotropic material. 10 The adaptive coarsening strategy is attractive from the implementation viewpoint as the background mesh does not change and no nodes nor elements are dynamically added, moved, or removed.…”

An Arbitrary-Order Discontinuous Skeletal Method for Solving Electrostatics on General Polyhedral Meshes

“…This flexibility, in turn, should yield to easier techniques for adaptive mesh refinement, derefinement and non-overlapping domain decomposition with non-matching grids. In particular, the non-conforming-like refinement-as the subgridding proposed in [2]-and the adaptive coarsening strategy [3] are particularly appealing.…”