Some convergence and optimality results of adaptive mixed methods in finite element exterior calculus

In this paper, we present several new a posteriori error estimators and two adaptive mixed finite element methods $\mathsf{\text{AMFEM1}}$ and $\mathsf{\text{AMFEM2}}$ for the Hodge Laplacian problem in finite element exterior calculus. We prove that $\mathsf{\text{AMFEM1}}$ and $\mathsf{\text{AMFEM2}}$ are both convergent starting from any initial coarse mesh. A suitably defined quasi error is crucial to the convergence analysis. In addition, we prove the optimality of $\mathsf{\text{AMFEM2}}$. The main technical contribution is a localized discrete upper bound. As opposed to existing literature, our results work on Lipschitz domains with nontrivial cohomology and provide the first norm convergence and optimality results.