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.
경계함유 유한요소법이 타원형 방정식에서 좋은 수렴성을 보여주었지만 이론적인 최적의 수렴성을 갖지 못하는 부분을 지적하고, 선적분을 포함하는 개선된 경계함유 유한요소법을 개발하여 최적의 수렴성을 갖도록 발전시켰다. 또한, 한 원소에 하나의 경계면만 지나야 하는 기존의 경계함유 유한요소법의 한계를 극복한 유한요소법을 개발하여 매우 가까운 두 개 이상의 경계면을 가지는 문제에 대해서도 경계함유 유한요소법의 장점을 적용할 수 있게 하였다. 경계함유 유한요소법을 정상상태 스톡스 문제에 적용하는 방법을 연구하였는데, 먼저는 경계에서 속도 미분에 관한 점프와 압력의 점프를 각각 알고 있을 때에 대한 문제를 해결할 수 있는 방법을 제안하였다. 이론적인 최적의 수렴성을 갖는 것을 수치적으로 확인하였다. 나아가 경계에서 속도 미분에 관한 점프와 압력의 점프를 각각 알지 못하고 연관된 식만 알고 있을 때 해결할 수 있는 방법을 제시하여 좋은 수렴성을 갖는 결과를 도출하였다.
