DEC approximations for Hodge-Laplacian problems are analyzed through FEEC

What the study found: The authors present a framework that interprets Discrete Exterior Calculus (DEC) numerical schemes using Finite Element Exterior Calculus (FEEC). They show that cochains on primal and dual meshes are equivalent to Whitney and generalized Whitney forms, which lets them analyze DEC approximations with FEEC tools.
Why the authors say this matters: The authors conclude that this framework makes it possible to rigorously study DEC approximations, including convergence rates for Hodge-Laplacian problems. They also say it helps explain superconvergence phenomena.
What the researchers tested: They developed a theoretical framework linking DEC and FEEC, then applied it to the Hodge-Laplacian problem in full k-form generality on well-centered meshes. They also included numerical results.
What worked and what didn't: The authors rigorously proved convergence rates for the Hodge-Laplacian problem on well-centered meshes. Their numerical results illustrated optimality of the derived convergence rates, and they also presented numerical results consistent with their explanation of superconvergence phenomena.
What to keep in mind: The abstract specifies well-centered meshes and full k-form generality for the convergence result, so the stated analysis is limited to that setting. Other limitations are not described in the available summary.

Key points

• The paper links Discrete Exterior Calculus schemes with Finite Element Exterior Calculus.
• Cochains on primal and dual meshes are shown to match Whitney and generalized Whitney forms.
• The authors rigorously prove convergence rates for the Hodge-Laplacian problem on well-centered meshes.
• Numerical results illustrate optimality of the derived convergence rates.
• The framework is also used to explain superconvergence phenomena.