Fast algorithms for PDEs in simple and complex geometries

We review some fast and accurate numerical techniques for solving elliptic equations in complicated domains developed by the author and his colleagues in recent past. These are based on a combination of fast algorithms for regular domains and various domain embedding techniques. The fast algorithms for regular domains are derived from analysis of integral equation approach for solving elliptic equations in twoand three-dimensions. These algorithms are very accurate and easy to implement on serial as well as parallel computers. For an irregular domain, first the domain is embedded in a regular domain and then the problem is solved in the regular domain using either boundary or distributed control techniques and the fast algorithm for the regular domain. There are more additional considerations in these methods that make the method more efficient and accurate. We do not go into these details in this short paper. One of the key contributing ideas of the author in this broad set-up is the way fast algorithms in regular domains for various elliptic equations are derived. This algorithm was originally conceived during the course of subsonic airfoil design using complex Beltrami equation formulation of subsonic compressible flow equations [1]. We show the basic idea of the algorithm through the following simple model problem in the real plane.