Differential complexes and numerical stability

TL;DR

The paper argues that numerical stability for PDE discretizations can be achieved by preserving differential complex structure at the discrete level. By constructing discrete de Rham and elasticity complexes with commuting projections, it shows how stability and convergence follow from exactness properties and space mappings, enabling stable Galerkin and mixed methods. It demonstrates, through Poisson, Maxwell, and elasticity examples, that appropriate finite element spaces (e.g., edge/Nedelec, Raviart-Thomas, elasticity elements) eliminate spurious modes and provide quasioptimal error bounds. This geometric, functorial framework unifies diverse discretization strategies and guides the design of stable, topology-aware finite element methods for geometry-rich PDE problems.

Abstract

Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial differential equations often hinges on capturing subtle aspects of the structure of the system in the discretization. In many cases the differential geometric structure captured by a differential complex has proven to be a key element, and a discrete differential complex which is appropriately related to the original complex is essential. This new geometric viewpoint has provided a unifying understanding of a variety of innovative numerical methods developed over recent decades and pointed the way to stable discretizations of problems for which none were previously known, and it appears likely to play an important role in attacking some currently intractable problems in numerical PDE.

Figures (8)

• Figure 1: A mesh marked with the locations of the degrees of freedom for Lagrange finite elements of degree $2$ and a typical such finite element function.
• Figure 2: Element diagrams. First row: discontinuous elements of degrees $0$, $1$, and $2$ in two dimensions. Second row: Lagrange elements of degrees $1$, $2$, and $3$ in two dimensions. Third and fourth rows: the corresponding elements in three dimensions.
• Figure 3: Element diagrams for some finite element approximations to vector fields in two and three dimensions. Multiple dots are used as markers to indicate the evaluation of all components of a vector field. Arrows are used for normal moments on codimension $1$ subsimplices and for tangential components on edges. Left: continuous piecewise linear fields. Middle: face elements of lowest order. Right: edge elements of lowest order.
• Figure 4: The face (left) and edge (right) elements of the second lowest order in 2- and 3-dimensions.
• Figure 5: The point plot shows the first 40 eigenvalues computed with piecewise linear finite elements with respect to the triangulation shown ($\bullet$) versus the exact eigenvalues ($+$). The surface plot shows the computed eigenfunction associated to the fourth eigenvalue. The mesh has 737 vertices, of which 641 are interior, and 1,376 triangles.