We introduce a new method for the numerical integration over curved surfaces and volumes
defined by a level set function. The method is based on the solution of a small linear system …

Singular terms in differential equations pose severe challenges for numerical
approximations on regular grids. Regularization of the singularities is a very useful …

Discretization of singular functions is an important component in many problems to which
level set methods have been applied. We present two methods for constructing consistent …

New Gaussian integration schemes are presented for the efficient and accurate evaluation
of weak form integrals in the extended finite element method. For discontinuous functions …

This article presents an immersed finite element (IFE) method for solving the typical three-
dimensional second order elliptic interface problem with an interface-independent Cartesian …

The rate of convergence for numerical methods approximating differential equations are
often drastically reduced from lack of regularity in the solution. Typical examples are …

IN-FLIGHT icing poses serious threats to aircraft airworthiness. Icing clouds are composed of
supercooled water droplets of various sizes. Even though the temperature may be well …

In [A.-K. Tornberg, B. Engquist, Numerical approximations of singular source terms in
differential equations, J. Comput. Phys. 200 (2004) 462–488], it was shown for simple …

In this article we consider regularizations of the Dirac delta distribution with applications to
prototypical elliptic and hyperbolic partial differential equations (PDEs). We study the …

This work presents a new approach to implementing a recently proposed optimal order Cut
Finite Element Method (CutFEM) for problems with moving embedded solid structures in …