Structure-preserving Discretization of the Hessian Complex based on Spline Spaces

TL;DR

This paper introduces a novel discretization method for the Hessian complex using spline spaces, leveraging isogeometric analysis to achieve high-order convergence and stability, with potential applications in solving Hodge-Laplacian problems.

Contribution

It proposes a structure-preserving discretization approach for the Hessian complex based on spline spaces, extending isogeometric analysis techniques within the FEEC framework.

Findings

Achieves inf-sup stability for the discretized Hessian complex

Enables high-order convergence with smooth test functions

Provides a simple approach for stable discretizations of Hodge-Laplacians

Abstract

We want to propose a new discretization ansatz for the second order Hessian complex exploiting benefits of isogeometric analysis, namely the possibility of high-order convergence and smoothness of test functions. Although our approach is firstly only valid in domains that are obtained by affine linear transformations of a unit cube, we see in the approach a relatively simple way to obtain inf-sup stable and arbitrary fast convergent methods for the underlying Hodge-Laplacians. Background for this is the theory of Finite Element Exterior Calculus (FEEC) which guides us to structure-preserving discrete sub-complexes.