Solving Elliptic PDE's on Domains with Curved Boundaries with an Immersed Penalized Boundary Method

The purpose of this paper is to describe a method (which we call IPBM) for solving boundary value problems on domains with curved boundaries. The method combines two ideas from the PDE literature: (a) the idea of immersing the problem in a larger and simpler domain, and (b) the idea of enforcing boundary conditions by using a penalty term. The method has a number of advantages as compared to existing methods in the literature and can be considered as a viable alternative to the very popular isogeometric analysis methods. It can be used with a wide variety of spline spaces including tensor-product splines and splines on triangulations. It also works with splines on H-triangulations and splines on T-meshes, which opens the door to adaptive methods. The paper contains a series of examples both in 2D and 3D to illustrate the capability of the method to produce high order approximations.