An averaging scheme for macroscopic numerical simulation of nonconvex minimization problems

作者：Li Zhi Ping^*; Carstensen Carsten

来源：BIT Numerical Mathematics, 2007, 47(3): 601-611.

DOI：10.1007/s10543-007-0140-1

• averaging scheme
• nonconvex minimization
• convexification
• macroscopic numerical simulation
• adaptive mesh refinement

Averaging or gradient recovery techniques, which are a popular tool for improved convergence or superconvergence of finite element methods in elliptic partial differential equations, have not been recommended for nonconvex minimization problems as the energy minimization process enforces finer and finer oscillations and hence at the first glance, a smoothing step appears even counterproductive. For macroscopic quantities such as the stress field, however, this counterargument is no longer true. In fact, this paper advertises an averaging technique for a surprisingly improved convergence behavior for nonconvex minimization problems. Similar to a finite volume scheme, numerical experiments on a double-well benchmark example provide empirical evidence of superconvergence phenomena in macroscopic numerical simulations of oscillating microstructures.

• 出版日期2007-9
• 单位北京大学