A fictitious domain formulation embeds an irregular physical domain in a simple rectangular mesh and imposes the Dirichlet boundary condition with a Lagrange multiplier, producing a symmetric saddle-point problem whose solution is non-smooth across the embedded interface. This concept shows how to solve that saddle-point system with a nested, inexact, wavelet-preconditioned Uzawa iteration whose inner elliptic problems — which carry H^-1 forcing data supported on the interface curve — are approximated by an adaptive finite element method driven by bulk-chasing on a residual estimator. The payoff a student should be able to argue is that the combined outer/inner scheme still converges at the best rate the trial spaces allow, and how that best rate degrades in three or more dimensions because of the interface jump.
This paper presents a fictitious domain method for solving elliptic PDEs by embedding irregular domains into simpler rectangular domains and enforcing boundary conditions via a saddle-point formulati…