SBP-FDEC: Summation-by-Parts Finite Difference Exterior Calculus

TL;DR

The paper develops SBP-FDEC, a structure-preserving discretization that extends FEEC ideas to SBP-FD methods to obtain divergence- and curl-free discretizations for Maxwell's equations. It builds a discrete de Rham complex using nodal and integral degrees of freedom linked by histopolation, enabling compatible gradient, curl, and div operators without explicit basis functions. The method extends to 2D and 3D via tensor products, yielding commuting discrete complexes and an energy framework that supports both divergence preservation and energy conservation with appropriate time integrators. Numerical tests on periodic 2D Maxwell problems confirm convergence consistent with SBP-FD orders, exact divergence preservation, and Crank-Nicolson energy conservation, highlighting practical impact for high-order electromagnetics with SBP-FD operators.

Abstract

We demonstrate that we can carry over the strategy of Finite Element Exterior Calculus (FEEC) to Summation-by-Parts (SBP) Finite Difference (FD) methods to achieve divergence- and curl-free discretizations. This is not obvious at first sight, as for SBP-FD no basis functions are known, but only values and derivatives at points. The key is a remarkable analytic relationship that enables us to construct compatible operators using integral and nodal degrees of freedom. Pre-existing SBP-FD matrix operators can then be used to obtain nodal values from the integral degrees of freedom to derive a scheme with the desired properties.

Figures (21)

• Figure 1: Discrete de Rham complex in 1D.
• Figure 2: 2D de Rham complex
• Figure 3: 2D dual de Rham complex
• Figure 4: Storage of the degrees of freedom for 2D functions
• Figure 5: Discrete dual de Rham complex in 2D

Theorems & Definitions (3)

• Remark 1
• Proposition 1
• proof