In 12%DIFDELCMD < ]%%% , the authors introduce an adaptive, stabilized finite element method (FEM), which solves a stable saddle-point problem . The method delivers a stable continuous approximation and projects the residual onto a broken polynomial space . This projection delivers a reliable error estimate that drives mesh refinement by minimizing a discrete energy norm  . In this work, we extend this framework to goal-oriented adaptivity (GoA) . We solve the primal and adjoint problems using the same initial saddle-point formulation , but with different right-hand sides. Additionally , we obtain two alternative error estimates , which are efficient to guide refinements. Several numerical examples illustrate the framework's performance on diffusion-advection-reaction problems.
 %DIFDELCMD < ]%%% 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.
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alternative error estimates , which are efficient to guide refinements. Several numerical examples illustrate the framework's performance on diffusion-advection-reaction
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