High-Order Numerical Methods on Complex Domains

Any physical theory relies on a mathematical model, typically in the form of a Partial Differential Equation (PDE), the solutions of which yield the predictions of the theory. Barring some simple cases, the solutions are found numerically by devising efficient algorithms. However, this task becomes harder as the physical domain becomes more complicated. Conventional techniques typically yield low order accuracy and end up requiring a balance between accuracy, and computational effort. In this talk, I will discuss some of my
efforts towards developing numerical methods for complex geometries. One such common yet largely unexplored domain is the three-dimensional cylinder. Employing carefully chosen basis functions, leads to powerful solvers and allows us to address challenges such as the nonlinear Faraday wave problem. Next, we shall move beyond domain-specific approaches and identify the subtle issues underlying the shortcomings of the generic low accuracy methods. After providing an overview of some of the proposed methods to overcome these issues, I shall present the Projection Extension method, a novel approach that yields highly  accurate solutions on a large array of complex domains. The technique leverages simple ideas
to great effect, with the result that it is straightforward to apply, implement, and generalize, and allows us to compute demonstrably accurate solutions to several challenging models, including heat flow and Newtonian and viscoelastic fluid problems.