The Smooth Selection Embedding Method with Chebyshev Polynomials
This work provides a more efficient implementation of an existing numerical method for solving boundary value problems in complex domains, offering faster convergence for practitioners in computational science.
The paper extends the Smooth Selection Embedding Method (SSEM) to Chebyshev polynomials, achieving faster convergence on smaller grids compared to the Fourier-based version. Numerical experiments demonstrate the method's simplicity, robustness, efficiency, and high-order accuracy for complex geometries, non-constant coefficients, and general boundary conditions.
We propose an implementation of the Smooth Selection Embedding Method (SSEM) in the setting of Chebyshev polynomials. The SSEM is a hybrid fictitious domain / collocation method which solves boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, the SSEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non-periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve problems in complex geometries, with non-constant coefficients, and for general boundary conditions.