A priori error estimates of regularized elliptic problems

Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $$H^1$$ H 1 and $$L^2$$ L 2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

Numerical simulation of the formation of vortices around rigid cylinders inside a channel using Immersed Interface method

The numerical simulation of the flow of fluid through one or a set of objects that causes the flow to separate from the surface of them has been the subject of interest by researchers over the past few decades. One of the most important types of these objects is those with a square cross section which have important and diverse applications in different industries. One of the practical applications of these types of streams is flow around chimneys, high-rise buildings, naval structures, suspended bridges, airplane wings, ship propellers and ducts. In this research, the immersed interface method is used which is a non-conforming method to the boundary. Eulerian mesh for fluid field, and Lagrangian mesh for solid field is used. The connection of these two networks is established by the Dirac Delta function. Considering the cylinder as a rigid immersion boundary within the flow. First, the flow around a square cylinder was simulated and we surveyed different flow patterns. The changes in the number of Strouhal and the Drag coefficient were investigated in different Reynolds. The flow around the two cylinders was simulated. It was observed that with the increase of Reynolds number and the gap between cylinders, the vortex shedding (Strouhal number) would increase.