# An adaptive stabilized conforming finite element method via residual minimization on dual discontinuous Galerkin norms

## TL;DR

This paper introduces a new adaptive stabilized finite element method that minimizes residuals in dual norms, providing efficient error control and optimal convergence for problems with sharp layers.

## Contribution

It develops a novel residual minimization approach in a dual norm framework, ensuring stability and adaptivity in finite element approximations.

## Key findings

- Achieves competitive error reduction rates on smooth solutions.
- Attains optimal decay rates for adaptive mesh refinement.
- Effective in resolving solutions with sharp layers.

## Abstract

We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf-sup stability. We formulate this residual minimization as a stable saddle-point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement. Numerical results on an advection-reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers.