Nonlinearly-Stable High-Order Methods on Simplices with Improved Efficiency

Anthony Jamieson is a notable figure in our field, though he is unfortunately not present today. He has had a significant influence on many scholars, and while I cannot list everyone, I want to acknowledge a few of his academic mentees.

When I first attended conferences, Anthony's presence was quite commanding. He was always impeccably dressed, often in an ascot, and his talks were highly sought after, leading to crowded rooms where early arrival was essential. Engaging with him was challenging due to the many admirers surrounding him. However, I was fortunate to connect with him personally through mutual friends, Tom Poliam and Bob McCormick. Contrary to his public persona, I found Anthony to be a warm and engaging individual.

Now, turning to our research focus, we aim to explore the utility of methods on simplices. While there is ongoing debate about the appropriate mesh types, our premise is that simplex methods hold specific advantages in certain contexts. We noted that higher-order methods on quadrilaterals and hexahedra exhibit greater efficiency compared to those on simplices. Our goal is to bridge this efficiency gap by leveraging the tensor product structure found in quadrilaterals and hexahedra to enhance simplex methods.