In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is $\mathcal{O}\left(h^2\right)$ in $L^2$ norm and $\mathcal{O}\left(h\right)$ in the $H^1$ semi-norm. Further the condition number increases with the mesh size similar to the FEM.

在本文中，我们提出了一种基于比例边界有限元方法的鲁棒框架，用于在非均质介质中对二维微分方程中的隐式界面进行建模。拟议工作的显着特征是：（a）界面可以隐式定义并且不需要符合背景网格；（b）可以直接沿界面施加狄利克雷边界条件；（c）不需要特殊的数值积分技术来计算双线性和线性形式，并且（d）可以使用分层背景网格进行有效的局部网格细化。解决了涉及直界面，圆形界面和移动界面问题的数值示例，以验证所提出的技术。进一步，在准确性和收敛性方面，将本文提出的技术与有限元方法进行了比较。从数值研究中可以看出，所提出的框架产生了误差为$\mathcal{Ø}（H^2）$ 在 ${\mathrm{大号}}^2$ 规范和 $\mathcal{Ø}（H）$ 在里面 $H^{\mathrm{1个}}$半规范。此外，条件数随着网格尺寸的增加而增加，类似于FEM。