The first lecture is about how elliptic partial differential equations
(PDEs) may be reformulated as Fredholm second kind integral equations
for the purpose of obtaining fast and accurate numerical solutions. A
discretization scheme for integral equations, called the Nyström
scheme, is presented. Advantages with this integral equation approach
to solving elliptic PDEs are reviewed, as are current trends and
challenges. In particular, I will discuss the difficulties that arise
on domains whose boundaries contain singularities such as corners, and
how to the Recursively Compressed Inverse Preconditioning (RCIP)
method is used to combat these difficulties. Numerical examples cover
applications to electromagnetic scattering and to solid mechanics.y

The first lecture is about how elliptic partial differential equations
(PDEs) may be reformulated as Fredholm second kind integral equations
for the purpose of obtaining fast and accurate numerical solutions. A
discretization scheme for integral equations, called the Nyström
scheme, is presented. Advantages with this integral equation approach
to solving elliptic PDEs are reviewed, as are current trends and
challenges. In particular, I will discuss the difficulties that arise
on domains whose boundaries contain singularities such as corners, and
how to the Recursively Compressed Inverse Preconditioning (RCIP)
method is used to combat these difficulties. Numerical examples cover
applications to electromagnetic scattering and to solid mechanics.