An integrated adaptive finite element method
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We present an integrated adaptive finite element method for numerical approximation of partial differential equations. Traditional approaches for adaptivity usually rely on a well-established iterative sequence of steps: solve, estimate, mark, refine. In the proposed method, we formulate a new paradigm where the error estimator is also used to update the numerical approximation, by correcting the current numerical solution either in the same space or in a refined space. In each linear algebraic step, the update is performed using a p-robust geometric multigrid; and in each refinement step, a new approximation space is defined simultaneously with the new iterate. The resulting approach thus delivers: a sequence of adapted meshes/spaces, and a sequence of numerical approximations along with associated error estimates. Furthermore, the total error is contracted at each step of the integrated methodology. Numerical experiments will illustrate the performance of the approach on selected examples.
