Subdivision $k$-Form Spaces within the Finite Element Exterior Calculus Framework

TL;DR

This paper develops subdivision-based finite element spaces for differential forms on 2D meshes, enhancing smoothness and accuracy while preserving the de Rham complex, leading to improved eigenvalue computations and efficiency.

Contribution

It introduces a framework for subdivision $k$-form spaces that improve regularity and accuracy, and maintains the de Rham complex structure in finite element discretizations.

Findings

Subdivision $k$-form spaces are up to 1.5 times more accurate than conventional FE spaces.

The method preserves the de Rham complex, confirmed by Maxwell eigenvalue tests.

Achieves up to 6 times speed-up for target accuracy with fewer degrees of freedom.

Abstract

This paper introduces discrete differential form spaces over two-dimensional manifold meshes that feature enhanced subdivision-induced inter-element regularity compared to conventional finite element (FE) spaces. This increase in smoothness is achieved by pulling back refined subdivision basis functions along a hierarchy of increasingly fine meshes that are generated by a subdivision algorithm. We introduce a framework that casts several known instances of $k$-form subdivision schemes in the language of FE and derive conditions under which the resulting subdivision-induced hierarchy of FE function spaces satisfies a discrete de Rham complex. The paper further illustrates the enforcing of zero boundary conditions by discarding basis functions close to the mesh boundary and shows that this does not compromise the de Rham complex. To analyse our novel subdivision $k$-form spaces we solve the Maxwell eigenvalue problem to confirm the absence of spurious modes and to study the accuracy of the computed eigenvalues. Recovering accurately the expected analytic eigenvalue spectrum shows that our novel subdivision $k$-form spaces indeed preserve the de Rham complex, since this test case is known to be challenging for methods not preserving this structure. Further, we numerically investigate the approximation errors of these subdivision spaces for given analytic functions. The presented study shows that our method can be employed in two ways. Upon a suitable choice of parameters, the subdivision $k$-form spaces are up to $1.5$ orders of magnitude more accurate in the $L^2$ norm than conventional lowest-order FE spaces with the same number of degrees of freedom. Alternatively, for a given target accuracy, the number of required degrees of freedom can be significantly reduced, resulting in a speed-up by a factor of up to 6 for the discussed test cases.