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

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 goaloriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection-diffusionreaction problems.

Automatically adaptive stabilized finite elements and continuation analysis for compaction banding in geomaterials

“…Previously, this technique has been successfully applied to the resolution of scalar advection-reaction 35 and advection-dominated diffusion 36,37 problems as well as incompressible flows 38,39 and goal-oriented adaptivity. 40…”

“…Nevertheless, the total number of DOFs is a small fraction of the DOFs associated with a fully‐resolved mesh for continuous or discontinuous finite elements. Previously, this technique has been successfully applied to the resolution of scalar advection‐reaction 35 and advection‐dominated diffusion 36,37 problems as well as incompressible flows 38,39 and goal‐oriented adaptivity 40 …”

Error representation of the time-marching DPG scheme

“…Conversely, authors in [22,23,37] apply and analyze the DPG method in space together with different time-stepping schemes for parabolic problems. In other works like [24,[38][39][40], the authors employ DPG-related ideas for solving both transient and frequency-domain problems, employing minimum residual methods or the corresponding mixed problems.…”

Incompressible flow modeling using an adaptive stabilized finite element method based on residual minimization

“…Therefore, we can design adaptive refinement strategies that simultaneously consider both pressure and velocity solutions. Other applications for adaptive stabilized FEM include goal‐oriented adaptivity in advection‐diffusion‐reaction problems, 22,23 direct and nonlinear problems such as weak constraint enforcement for advection‐dominated diffusion problems 24‐26 …”