An Anisotropic Error Indicator Based on Zienkiewicz--Zhu Error Estimator: Application to Elliptic and Parabolic Problems

Adaptive finite elements with high aspect ratio for the computation of coalescence using a phase-field model

“…The goal of the adaptive algorithm is then to equidistribute the error indicator in the directions of maximum and minimum stretching and to align the directions of maximum and minimum stretching with the directions of maximum and minimum error. We refer to [26] for a theoretical justification in the framework of the heat equation and to [24] for phase field problems.…”

“…In other words r 1;K and r 2;K are the directions of maximum and minimum stretching, while k 1;K and k 2;K measure the amplitude of stretching. Proceeding as in [24,26], we introduce c hs the continuous, piecewise linear approximation in time defined by c hs ðx; tÞ ¼ t À t nÀ1 s c n h ðxÞ þ t n À t s c nÀ1 h ðxÞ; t nÀ1 6 t 6 t n ; x 2 X: ð12Þ …”

“…We then proceed as in [25,26] to build a mesh having elements with high aspect ratio, using the BL2D mesh generator [30]. At each vertex, the estimated error in the directions of maximum and minimum stretching is equidistributed, which yields new desired values of maximum and minimum stretching.…”

“…The model uses refinement and coarsening criteria based on an error indicator which was justified theoretically for simpler problems in [25,26].…”

“…Third, adaptive finite elements with high aspect ratio are used in order to reduce the number of mesh vertices, therefore, a suitable error indicator must be introduced to define the refinement and coarsening criteria. For details about the adaptive procedure, we refer to papers published elsewhere for simpler problems [24][25][26]. The use of adaptive finite elements with high aspect ratio is of great interest for coalescence problems, due to the fact that a realistic value of interface thickness (about 1 nm) has to be taken for a quantitative description of the problem.…”