A new space transformed finite element method for elliptic interface problems in Rn

Abstract

Interface problems, where distinct materials or physical domains meet, pose significant challenges in numerical simulations due to the discontinuities and sharp gradients across interfaces. Traditional finite element methods struggle to capture such behavior accurately. A new space transformed finite element method (ST-FEM) is developed for solving elliptic interface problems in $R^n$. A homeomorphic stretching transformation is introduced to obtain an equivalent problem in the transformed domain which can be solved easily, and the solution can be projected back to original domain by the inverse transformation. Compared with the existing methods, this new scheme has capability of handling discontinuities across the interface. The proposed approach has advantages in circumventing interface approximation properties and reducing the degree of freedom. We initially develop ST-FEM for elliptic problems and subsequently expand upon this concept to address elliptic interface problems. We prove optimal a priori error estimates in the $H^1$ and $L^2$ norms, and quasi-optimal error estimate for the maximum norm. Finally, numerical experiments demonstrate the superior accuracy and convergence properties of the ST-FEM when compared to the standard finite element method. The interface is assumed to be a $(n-1)$-sphere, nevertheless, our analysis can cover symmetric domains such as an ellipsoid or a cylinder.