Adaptive stabilized finite elements via residual minimization onto bubble enrichments

The Adaptive Stabilized Finite Element method (AS-FEM) developed in [16] combines the idea of the residual minimization method with the inf-sup stability offered by the discontinuous Galerkin (DG) frameworks. As a result, the discretizations deliver stabilized conforming approximations and residual representatives in a DG space that can drive automatic adaptivity. In this work, we extend AS-FEM by considering continuous enriched test spaces. Thus, we propose a residual minimization method on a stable Continuous Interior Penalty (CIP) formulation, which considers a C⁰-conforming trial FEM space and a test space based on the bubble enrichment of that trial space. In our numerical experiments, the test space choice significantly reduces the total degrees of freedom compared to the DG test spaces of [16], while recovering the expected convergence rate for the error in the corresponding trial space norm.