Multiscale and adaptive methods for partial differential equations

In this talk, a number of recent results will be reported on multiscale and adaptive finite element methods for numerical solutions of partial differential equations, including sharp convergence theory on the general subspace correction iterative methods, two-grid methods for symmetrization, linearization and parallelization, partition of unity methods for nonmatching grids, a posteriori error estimators and grid adaptation involving refinement, coarsening and moving. Several numerical results will also be given, including applications for black-hole and multi-phase flows simulations.