Direct methods in the calculus of variations
Direct methods in the calculus of variations
Strong convergence of numerical solutions to degenerate variational problems
Mathematics of Computation
Numerical solution of the scalar double-well problem allowing microstructure
Mathematics of Computation
Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods
SIAM Journal on Numerical Analysis
Numerical Analysis of Compatible Phase Transitions in Elastic Solids
SIAM Journal on Numerical Analysis
A Posteriori Finite Element Error Control for the P-Laplace Problem
SIAM Journal on Scientific Computing
Multiscale resolution in the computation of crystalline microstructure
Numerische Mathematik
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Macroscopic simulations of non-convex minimisation problems with enforced microstructures encounter oscillations on finest length scales - too fine to be fully resolved. The numerical analysis must rely on an essentially equivalent relaxed mathematical model. The paper addresses a prototype example, the scalar 2-well minimisation problem and its convexification and introduces a benchmark problem with a known (generalised) solution. For this benchmark, the stress error is studied empirically to asses the performance of adaptive finite element methods for the relaxed and the original minimisation problem. Despite the theoretical reliability-efficiency gap for the relaxed problem, numerical evidence supports that adaptive mesh-refining algorithms generate efficient triangulations and improve the experimental convergence rates optimally. Moreover, the averaging error estimators perform surprisingly accurate.