Multiscale Methods for High-Contrast Heterogeneous Sign-Changing Problems

Explore a cutting-edge numerical method for solving sign-changing problems in partial differential equations. Delve into the mathematical formulation of these problems, which involve linear second-order PDEs in divergence form with coefficients that can be both positive and negative in different subdomains. Learn about the physical applications of these problems in negative-index metamaterials, either as inclusions in common materials or vice versa. Discover the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) adapted for sign-changing problems, including the construction of tailored auxiliary spaces. Examine numerical results demonstrating the method's effectiveness in handling complex coefficient profiles and its robustness across various coefficient contrast ratios. Gain insights into the theoretical foundations, including inf-sup stability and a priori error estimates, established using T-coercivity theory. This 44-minute lecture by Eric Chung at the Hausdorff Center for Mathematics offers a deep dive into advanced numerical techniques for solving challenging multiscale problems in applied mathematics.