Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

TL;DR

This paper introduces a goal-oriented adaptive mesh refinement algorithm for a stabilized finite element method based on residual minimization in dual discontinuous Galerkin norms, improving solution accuracy for PDEs.

Contribution

It develops and analyzes a novel goal-oriented adaptive algorithm using saddle-point formulations for residual minimization in dual DG norms, with efficient error estimation for PDEs.

Findings

Effective mesh refinement guided by goal-oriented error estimates.

Stable residual minimization provides accurate solutions and error control.

Performance demonstrated on advection-diffusion-reaction problems.

Abstract

We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal-oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection-diffusion-reaction problems.