An hp-hierarchical framework for the finite element exterior calculus

TL;DR

This paper introduces an hp-adaptive finite element exterior calculus framework for solving PDEs on manifolds, featuring a spectral error indicator and demonstrating practical engineering applications.

Contribution

It presents a novel, efficient hp-adaptive framework for FEEC on manifolds, including a spectral error indicator and polynomial order adaptivity.

Findings

Spectral error indicator effectively estimates error and polynomial decay.

The framework demonstrates practical utility through computational examples.

Adaptive hp-method improves solution accuracy on complex manifolds.

Abstract

The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well as the corresponding covariant forms of the constitutive closing relations are naturally expressed using the bundle-valued exterior calculus of differential forms or related algebraic concepts. It can be argued that the appropriate solution method to such PDE problems is given by the finite element exterior calculus (FEEC). The aim of this essay is the exposition of a simple, efficiently-implementable framework for general hp-adaptivity applicable to the FEEC on higher-dimensional manifolds. A problem-independent spectral error-indicator is developed which estimates the error and the spectral decay of polynomial coefficients. The spectral decay rate is taken as an admissibility indicator on the polynomial order distribution. Finally, by elementary computational examples, it is attempted to demonstrate the power of the method as an engineering tool.