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A posteriori error estimator for obstacle problems

Project manager: Prof. Dr.  rer. nat. B.I. Wohlmuth
Projektbearbeiter: Dipl.. Math. A. Weiß

Project description

In this project, we consider error estimators for obstacle problems. The focus is on error estimators which are defined in terms of H(div) conforming stress approximations and equilibrated fluxes. It can be shown that the error is bounded by the estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order. Moreover bounds for the Lagrange multiplier and efficiency of the estimator are derived.

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Problem with exact solution, error decay and ration between estimated and exact error
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Problem with exact solution: grid at level 2,4,6, active nodes and exact contact set (red)

For the case of non-smooth or discontinuous obstacles, suitable modifications in the estimator are considered. Here, the H(div) conformity must be weakend as the exact solution is no longer H(div) conform. The estimator has been implemented and applied to various examples including two-membrane contact problem problem with non-matching meshes and low regularity problems. Optimal convergence is achieved using adaptive mesh refinement.

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Low regularity problem: Setting and solution for alpha=pi/6 and alpha=pi/2

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Low regularity problem: Optimal convergence by adaptive refinement

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Two-membrane + obstacle problem: Setting and solution
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Two-membrane + obstacle problem: Grid and active sets at level 3,6,9.