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Sparsity comparison of polytopal finite element methods

Christoph Lehrenfeld, Paul Stocker, Maximilian Zienecker

TL;DR

The paper analyzes sparsity and coupling patterns of polytopal finite element methods for a Laplace-type model problem, comparing VEM, DG variants (DG, HDG/HHO), and Trefftz DG across diverse periodic polytopal meshes. It shows that skeleton-based reductions (HDG/HHO/VEM) dramatically lower global nonzeros on simple meshes, while Trefftz DG (TDG1/TDG2) achieves superior sparsity on highly faceted polytopal cells, especially at higher order. The study provides explicit ncdof and nnze scaling laws and discusses how mesh topology dictates solver cost, offering practical guidance for method selection in complex geometries. The results underscore the importance of mesh geometry in choosing efficient discretizations for polytopal finite elements and supply data for practitioners to optimize performance.

Abstract

In this work we compare crucial parameters for efficiency of different finite element methods for solving partial differential equations (PDEs) on polytopal meshes. We consider the Virtual Element Method (VEM) and different Discontinuous Galerkin (DG) methods, namely the Hybrid DG and Trefftz DG methods. The VEM is a conforming method, that can be seen as a generalization of the classic finite element method to arbitrary polytopal meshes. DG methods are non-conforming methods that offer high flexibility, but also come with high computational costs. Hybridization reduces these costs by introducing additional facet variables, onto which the computational costs can be transfered to. Trefftz DG methods achieve a similar reduction in complexity by selecting a special and smaller set of basis functions on each element. The association of computational costs to different geometrical entities (elements or facets) leads to differences in the performance of these methods on different grid types. This paper aims to compare the dependency of these approaches across different grid configurations.

Sparsity comparison of polytopal finite element methods

TL;DR

The paper analyzes sparsity and coupling patterns of polytopal finite element methods for a Laplace-type model problem, comparing VEM, DG variants (DG, HDG/HHO), and Trefftz DG across diverse periodic polytopal meshes. It shows that skeleton-based reductions (HDG/HHO/VEM) dramatically lower global nonzeros on simple meshes, while Trefftz DG (TDG1/TDG2) achieves superior sparsity on highly faceted polytopal cells, especially at higher order. The study provides explicit ncdof and nnze scaling laws and discusses how mesh topology dictates solver cost, offering practical guidance for method selection in complex geometries. The results underscore the importance of mesh geometry in choosing efficient discretizations for polytopal finite elements and supply data for practitioners to optimize performance.

Abstract

In this work we compare crucial parameters for efficiency of different finite element methods for solving partial differential equations (PDEs) on polytopal meshes. We consider the Virtual Element Method (VEM) and different Discontinuous Galerkin (DG) methods, namely the Hybrid DG and Trefftz DG methods. The VEM is a conforming method, that can be seen as a generalization of the classic finite element method to arbitrary polytopal meshes. DG methods are non-conforming methods that offer high flexibility, but also come with high computational costs. Hybridization reduces these costs by introducing additional facet variables, onto which the computational costs can be transfered to. Trefftz DG methods achieve a similar reduction in complexity by selecting a special and smaller set of basis functions on each element. The association of computational costs to different geometrical entities (elements or facets) leads to differences in the performance of these methods on different grid types. This paper aims to compare the dependency of these approaches across different grid configurations.
Paper Structure (25 sections, 13 equations, 12 figures, 15 tables)

This paper contains 25 sections, 13 equations, 12 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Polytopal finite element methods for a model problem
  3. Elliptic model problem
  4. DG formulation of the model problem
  5. Hybrid DG (and Hybrid High Order) formulation of the model problem
  6. Static condensation
  7. Hybrid High Order (HHO) methods
  8. Trefftz DG formulation of the model problem
  9. Virtual Element Method formulation of the model problem
  10. Sparsity measures and basic coupling relations
  11. Notation
  12. Geometrical entities with same local topology.
  13. Ratio between entities of a specific type and elements.
  14. Neighborhood relation between mesh entities.
  15. Number of coupling degrees of freedom (ncdof)
  16. ...and 10 more sections

Figures (12)

  • Figure 1: Couplings of a DG degree of freedom with all dofs from one of the neighbouring elements.
  • Figure 2: A Trefftz DG dof couples like standard DG dof, however, the number of dofs is reduced by for TDG2 and further by all dofs marked for TDG1.
  • Figure 3: HDG facet dof couples with all dofs on facets directly adjacent to neighbouring elements.
  • Figure 4: Comparison of Hybrid DG dof of order $3$ with (w) and without (wo) static condensation. An element dof couples with its own elements dof and those of neighbouring facets, but different volume elements no longer interact directly in the resulting matrix. Further, by applying static condensation element dofs can be removed from the global system, remaining only with facet dofs and couplings to their adjacent counterparts.
  • Figure 5: All types of dofs couple with all other dofs that share a common element. Element dofs do not couple with neighboring element dofs and can be condensed out by a Schur complement strategy.
  • ...and 7 more figures