Special Seminar in Applied Mathematics

Boundary integral methods (BIMs) (or rather boundary integral equation methods (BIEMs)) originating from those problems of physics and mechanics have become one of the essential tools of solving boundary value problems for linear, elliptic partial differential equations. Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as boundary element methods (BEMs). In fact, today, any numerical realizations of BIMs are generally referred to as BEMs.  They have been proven to be as efficient computationally as finite element methods (FEMs) for constructing numerical solutions of partial differential equations.

This lecture gives an overview of BIMs from both theoretical and numerical points of views.  It summarizes some main results obtained by the speaker and his collaborators over the last 40 years.  Emphasis will be placed upon basic concepts and fundamental theory concerning solution spaces, variational formulations, mapping, properties of boundary integral operators, etc.  These mathematical ingredients play crucial roles and may serve as mathematical foundations of BEMs.  A glimpse into BIMs for time-dependent problems will also be reviewed. BIMs for time-dependent problems have received increasing attention in recent years; however, their theoretical developments are still in their infancy.  In closing, hybrid methods based on boundary-field equation methods (BFEMs) with their applicabilities will be remarked upon.  Some numerical experiments in elasticity as well as in fluid mechanics will be included for the illustration of BIMs.

This lecture is exploratory and particularly aimed for audiences in applied mathematics, physics and engineering.