UC Santa Barbara

Interface problems are naturally a good fit to model discontinuous phenomena andcan be applied to a wide range of applications. In this work, we look to solve a PDE on an arbitrary and irregular domain geometry. To this end, we use a sophistication of FEM called unfitted FEM to dissociate the relationship between the domain and its discretization. This disconnect comes at a cost when producing numerical results and must therefore be stabilized to ensure validity in the approximations. We propose an unfitted spectral element method for solving, both boundary-value and eigenvalue, elliptic interface problems. The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element method and the flexibility of the unfitted Nitsche’s method. We also use tailored ghost penalty terms to enhance its robustness. We establish optimal hp convergence rates for both types of interface problems. Additionally, we demonstrate spectral accuracy for model problems in terms of polynomial degree. Lastly, we qualitatively study a time-dependent elliptic interface problem. Given the added degree of freedom, we develop a machine-learning algorithm able to track the domain evolution as well as the solution itself. Promising results are obtained for various test cases.