Hierarchical error estimates for the energy functional in obstacle problems

Abstract

Abstract. We present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems. The main result is that the energy of the exact solution is, up to oscillation terms, equivalent to an appropriate hierarchical estimator. The proof does not invoke any saturation assumption. Moreover, we prove an a posteriori error estimate indicating that the estimator from [12] is asymptotically reliable and we give sufficient conditions for the validity of a saturation assumption. Finally, we corroborate and complement our theoretical results with a numerical example.

Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

“…Examples of linear elliptic problems with inequality constraints are obstacle and contact problems [30,21]. Different types of error estimators for obstacle problems can be found in, e.g., [8,38,42]. By measuring the error in the solutions as well as in the constraining force, the first efficient and reliable residual-type estimator for obstacle problems has been derived in [31].…”

Numerical simulation of coarsening in binary solder alloys

“…Hierarchical error estimates rely on the solution of local defect problems. While originally introduced for linear elliptic problems [14,17,44,65] this technique was successfully extended to nonlinear problems [3], constrained minimization [45,49,50,61,66] and nonsmooth saddle point problems [38,33]. Thermomechanical stress is caused by different thermal expansion coefficients and the mismatch of the different constituents [19].…”

On Hierarchical Error Estimators for Time-Discretized Phase Field Models

“…for θ = 0, the resulting error estimator (8) was proposed in [17] and later analyzed in [20,22,24]. Here we concentrate on θ > 0 and investigate robustness for θ → 0.…”

“…In early papers, upper bounds are often derived from the so-called saturation assumption that the extended space Q provides a more accurate approximation than S. It turned out later that local equivalence to residual estimators provides upper bounds up to data oscillation and, conversely, that small data oscillation implies the saturation assumption [7,12]. For a direct proof based on local L 2 -projections we refer to [24].…”

“…This work was supported by the DFG Research Center MATHEON. 21,22,24]. In this numerical study, we consider hierarchical error estimators for semi-linear elliptic problems as arising from the time discretization of Allen-Cahn and Cahn-Hilliard equations.…”