A higher-order poly-region boundary element method for steady thermoviscous fluid flows

In this presentation, we re-visit the poly-region boundary element methods (BEM) proposed earlier for the steady Navier-Stokes and Boussinesq flows, and develop a novel higher-order BEM formulation for the thermoviscous fluid flows that involves the definition of the domains of kernel influences due to steady Oseenlets. We introduce region-by-region implementation of the steady-state Oseenlets within the poly-region boundary element fequatramework, and perform integration only over the (parts of) higher-order boundary elements and volume cells that are influenced by the kernels. No integration outside the domains of the kernel influences are needed. Owing to the properties of the convective Oseenlets, the kernel influences are very local and propagate upstream. The localization becomes more prominent as the Reynolds number of the flow increases. This improves the conditioning of the global matrix, which in turn, facilitates an efficient use of the iterative solvers for the sparse matrices. Here, we consider quartic boundary elements and bi-quartic volume cells to ensure a high level resolution in space. Similar to the previous developments, coefficients of the discrete boundary integral equations are evaluated with the sufficient precision using semi-analytic approach to ensure exceptional accuracy of the boundary element formulation. To demonstrate the attractiveness of the poly-region BEM formulation, we consider a numerical example of the well-known Rayleigh-Benard problem governed by the Boussinesq equations.