The multiscale finite element method for nonlinear continuum localization problems at full fine-scale fidelity, illustrated through phase-field fracture and plasticity

The residual-driven iterative corrector scheme recently presented by the authors for linear problems has opened a pathway to achieve the best possible fine-mesh accuracy in the multiscale finite element method (MsFEM). In this article, we focus on a series of algorithmic and variational extensions that enable efficient residual-driven correction for nonlinear localization problems. These include a synergistic combination of Newton and corrector iterations to reduce the algorithmic complexity, the use of corrector degrees of freedom in the Galerkin projection to eliminate the repeated recomputation of multiscale basis functions during Newton iterations, and a natural residual-based strategy for fully automatic fine-mesh adaptivity. We illustrate through numerical examples from phase-field fracture and plasticity that the MsFEM with residual-driven adaptive correction achieves full fine-scale fidelity while also being computationally more efficient than the pristine MsFEM. We also show that for localization problems, it significantly increases accuracy and robustness over standard oversampling.