In the high Reynolds number regime, under what conditions do there exist steadily rising bubbles? This question has been studied extensively both experimentally and numerically, but current mathematical models and numerical discretizations suffer from large numerical errors that make the results less convincing. In the first part of this talk, we build an inviscid model for the steady rising problem and find different solution branches of bubble shapes characterized by the number of humps. These only exist when there is no gravity. When there is gravity, viscous potential flow is used to find different steady shapes. The corresponding dynamic problem is also studied. Techniques such as axisymmetric potential theory, Hou-Lowengrub-Shelley framework, and weak/hyper-singularity removal are applied to guarantee spectral accuracy.
Due to the importance of accurate evaluation of orthogonal polynomials in the boundary integral method used in the first part, in the second part of the talk I will introduce a new way to evaluate orthogonal polynomials more accurately near the endpoints of the integration interval. An associated family of orthogonal polynomials is evaluated at interior points to determine the values of the original polynomials near endpoints. The new method can achieve round-off error accuracy even for end-point evaluation of generic high-degree Jacobi polynomials and generalized Laguerre polynomials. More accurate quadrature abscissas and weights can be achieved accordingly.