Two-Scale FEMs for Non-Variational Elliptic PDEs

Explore a 50-minute lecture on two-scale finite element methods (FEMs) for non-variational elliptic partial differential equations (PDEs). Delve into the innovative approach of adding a larger scale ε to the usual meshsize h in FEM approximations for non-variational elliptic PDEs. Learn how this technique enables the computation of centered second differences of continuous functions, replacing wide stencil notions and enforcing monotonicity without strict mesh restrictions. Examine applications to linear PDEs in non-divergence form, the Monge-Ampere equation, and a fully nonlinear PDE for the convex envelope. Understand the role of scale ε as analogous to a finite horizon for integro-differential operators. Discover pointwise error estimates derived for all three cases, exploiting the separation of scales. Study the novel discrete Alexandroff estimate and the discrete Alexandroff-Bakelman-Pucci estimate, fundamental tools in controlling function properties and discrete Laplacians. Gain insights from this joint work by Ricardo Nochetto, Wenbo Li, Dimitris Ntogkas, and Wujun Zhang, presented at the Hausdorff Center for Mathematics.