(A) modified $P_1$ immersed finite element method for the stationary Stokes equation

Abstract

In recent years, the immersed finite element methods (IFEM) introduced, to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface to cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence.

In chapter 1, An improved version of the $P_1$ based IFEM is proposed by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, this method has much less degrees of freedom than that of the DG methods since the discrete system has the same number of unknowns as the conventional $P_1$ finite element method. We prove $H^1$ and $L^2$ error estimates which are optimal both in order and regularity.

In chapter 2, the improvement of IFEM are given, which can be applied to the multi-interface elliptic problem. Previous IFEM must have an assumption : each element is cut by at most one interface. When the interfaces are very close to each other, this assumption requires very small element. To overcome this threshold in multi-interface problems, the basis functions on elements which are cut by two interface are introduced.

In chapters 3 and 4, we develop a finite element method using $P_1$ nonconforming, piecewise constant pair for a two phase, stationary incompressible Stokes flow with singular forces along interfaces. Contrary to a conventional way of generating fitted grid, we use a uniform grid to discretize the computational domain. Chapter 3 has an assumption : the jumps of the pressure and the velocity along the interface are given, respectively. We modify the basis functions to satisfy certain compatibility conditions along the interface. We provide a scheme for the case of homogeneous jumps, and solve the problem with nonhomogeneous case, by constructing appropriate singular functions and subtract them from the variational form. In chapter 4, we consider the case when velocity gradient jumps and pressure jumps are coupled along the interface. This assumption is weaker than the assumption in previous chapter. By constructing the pair of the singular parts, we derive a new scheme to obtain higher convergence rate.

Numerical experiments in each chapter are carried out for several examples, which show optimal orders.