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Instance-Optimal Goal-Oriented Adaptivity

  • Michael Innerberger EMAIL logo and Dirk Praetorius

Abstract

We consider an adaptive finite element method with arbitrary but fixed polynomial degree p 1 , where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 2016, 1, 33–68], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. More precisely, the goal error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Numerical experiments underline our theoretical findings.

Funding source: Austrian Science Fund

Award Identifier / Grant number: W1245

Award Identifier / Grant number: SFB F65

Award Identifier / Grant number: P27005

Funding statement: The authors acknowledge support through the Austrian Science Fund (FWF) through the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245), the special research program Taming complexity in PDE systems (grant SFB F65), and the stand-alone project Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005).

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Received: 2019-07-31
Revised: 2019-12-17
Accepted: 2020-02-02
Published Online: 2020-02-18
Published in Print: 2021-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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