Residual minimization for goal-oriented adaptivity
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In [12], 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.
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