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

Robert Piel, Werner Bauer

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.

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

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 -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 -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 -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 -form spaces are up to orders of magnitude more accurate in the 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.

Paper Structure

This paper contains 38 sections, 20 theorems, 116 equations, 15 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background: FEEC and the subdivision algorithm
  3. Finite Element Exterior Calculus
  4. Meshes, simplices and adjacency operators
  5. Subdivision algorithm and mesh subdivision
  6. General description of subdivision
  7. Formal description of a subdivision algorithm
  8. Construction of mesh hierarchies using subdivision
  9. Subdivision $k$-form spaces
  10. Definition of subdivision $k$-forms via subdivision operators
  11. Constructing a basis for the subdivision $k$-form spaces
  12. Brief summary:
  13. Differential complexes of spaces of subdivision $k$-forms
  14. Subdivision k-form spaces with zero boundary conditions
  15. Implementation of spaces of subdivision $k$-forms
  16. ...and 23 more sections

Key Result

Proposition 3.4

Given an intermediate mesh level ${\color{ForestGreen}\ell}$ with ${\color{ForestGreen} 0} \leq {\color{ForestGreen}\ell} \leq {\color{ForestGreen} L}$, the accumulated $k$-form subdivision operator can be decomposed relative to level ${\color{ForestGreen}\ell}$ by $\mathcal{A}^{k}_{{{\color{ForestG

Figures (15)

  • Figure 1: The Loop subdivision algorithm maps a coarse mesh (left) to a fine mesh (right). First, the topology is refined by quadrisecting all faces (centre). The subdivision rule then maps any fine vertex to a set of coarse vertices and assigns a weight to each of them. Finally, the geometric averaging step computes the positions of the fine vertices as a weighted average of the positions of the coarse vertices.
  • Figure 2: The subplots show $0$-, $1$- and $2$-form constrained basis functions according to Definition \ref{['def:alternative_char_for_subdiv_spaces']} on level ${\color{ForestGreen}\ell} =0$ with ${n_s} = 5$. Note that the smoothness increases in the sense of Remark \ref{['remark:subdiv_smoothness']} despite the basis functions remaining in the same Sobolev space. The colorbar indicates the scalar values of the $0$- and $2$-form basis functions. For the $1$-form basis function, the colorbar and the white lines indicate the magnitude and direction of the vector field. All basis functions are plotted on their respective supports according to Proposition \ref{['prop:support']}.
  • Figure 3: Full commutative diagram of the spaces of subdivision $k$-forms including the subdivision, inclusion, and derivative operators. The black lines connect the individual hierarchies of subdivision $k$-form spaces for $k = 0$ (left), $k=1$ (centre) and $k=2$ (right). The derivative operators $\mathbf{d}$ and $\overset{\sim}{\newline{\mathbf{d}}}$ (dashed grey lines) join the spaces of the hierarchies to form de Rham complexes.
  • Figure 4: Approximation of the physical domain $\mathcal{M}$ through Loop subdivision.
  • Figure 5: Approximation error of ${\omega}^{0}_{}$ from Eq. \ref{['eq:differential_forms_for_projection']} when projected into $\mathcal{S}\Lambda^{0}_{{\color{ForestGreen}\ell} \vert {\color{ForestGreen} L}}$ for fixed finest levels ${\color{ForestGreen} L}$. The refinement number ${n_r}$ parametrises the nested spaces (according to Proposition \ref{['prop:uniform_spaces_nested']}) along a hierarchy. As an example, we annotated the hierarchy with ${\color{ForestGreen} L} = 7$.
  • ...and 10 more figures

Theorems & Definitions (77)

  • Definition 2.1: Topological refinement step
  • Definition 2.2: Subdivision rule
  • Definition 2.3: Subdivision scheme
  • Definition 2.4: Mesh subdivision operator
  • Definition 3.1: $k$-form subdivision operator with matrix representation
  • Remark 3.2
  • Definition 3.3: Accumulated $k$-form subdivision operator
  • Proposition 3.4
  • proof
  • Definition 3.5: Subdivision $k$-form spaces
  • ...and 67 more