An overview of cut cell algorithms and their accuracy, stability, and performance on GPUs

I will present a class of embedded boundary (EB) "cut cell" discretizations that accurately represent material boundaries on a regular block-structured mesh. This introduces potentially very small cells but removes the need for global mesh generation and adjustments to improve matrix conditioning. EB solvers can use batch, block, and multigrid algorithms, and because they are regular (matrix-free) discretizations away from boundaries, they can be efficiently implemented on graphical processing units (GPUs). The challenge with this approach is not accuracy, but a lack of theory - when are they stable, what is their spectrum, how robust are they to round-off errors, etc.? I will discuss some progress in this area - local discrete maximum principles and pseudo-spectrum analysis. Questions of accuracy and stability become particularly difficult when looking at EB matrices in mixed-precision algorithms, which are common in AI and HPC linear algebra libraries. Although lower precision approaches greatly improve GPU memory and bandwidth, when applied blindly they destroy the PDE-based structure of the resulting matrix. I will present a "fix" for these problems and motivate the need for "co-design" in computational science and engineering, connecting all of numerical analysis, algorithm design, and software performance engineering.