In engineering applications, the description of the geometry and the mesh generation process are often

bottlenecks in nite element approximations of elliptic boundary value problems. Some eorts have been made

to develop meshless methods. However a central problem of such methods is to incorporate boundary conditions

of Dirichlet type.

This motivates the interest in immersed boundary and interface methods, also known as ctitious domain or

embedded domain methods. The traditional philosophy of immersed boundary methods is to embed the com-

putational domain in a structured grid and employ simple, mesh-aligned numerical schemes. Clearly, immersed

methods require a proper treatment of the cells that are cut by boundaries and/or interfaces with some special,

and often ad hoc, technique to achieve acceptably accurate results.

Moreover, one of the current challenges both in CAD and IgA is dealing with trimmed geometries. Indeed,

the most common description of CAD models is the B-rep, where an object is represented by its boundary

surfaces, described by suitable geometry maps on the parametric domain. Often only certain regions of a

surfaces are supposed to be part of the actual object and the unused areas are trimmed away. Trimming

results in identifying complex geometries in the parametric domain which can be treated following an immersed

approach.

In this talk we aim to present our ongoing results on immersed boundary Isogeometric analysis based on B-

splines/NURBS dened in both rectangular and triangular regular meshes, for general, non-constant coecients,

elliptic problems.