# Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors

## TL;DR

This paper develops quasi-optimal adaptive mixed finite element methods for controlling natural norm errors in generalized Hodge Laplace equations, improving error control in both variables for several fundamental PDEs.

## Contribution

It introduces new adaptive mixed finite element methods that achieve quasi-optimal convergence rates and control errors in the natural mixed norm, extending to Hodge Laplace, Poisson, and Stokes equations.

## Key findings

- Proves quasi-optimal convergence for adaptive methods
- Controls errors in both variables in mixed formulations
- Applies to Hodge Laplace, Poisson, and Stokes equations

## Abstract

For a generalized Hodge Laplace equation, we prove the quasi-optimal convergence rate of an adaptive mixed finite element method. This adaptive method can control the error in the natural mixed variational norm when the space of harmonic forms is trivial. In particular, we obtain new quasi-optimal adaptive mixed methods for the Hodge Laplace, Poisson, and Stokes equations. Comparing to existing adaptive mixed methods, the new methods control errors in both variables.