Daniel Agress

Daniel Agress, Patrick Guidotti, Dong Yan

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.

Daniel Agress, Patrick Guidotti

A new approach to the solution of boundary value problems within the so-called fictitious domain methods philosophy is proposed which avoids well known shortcomings of other fictitious domain methods, including the need to generate extensions of the data. The salient feature of the novel method, which we refer to as SSEM (Smooth Selection Embedding Method), is that it reduces the whole boundary value problem to a linear constraint for an appropriate optimization problem formulated in a larger simpler set containing the domain on which the boundary value problem is posed and which allows for the use of straightforward discretizations. The proposed method in essence computes a (discrete) extension of the solution to the boundary value problem by selecting it as a smooth element of the complete affine family of solutions of the extended, yet unmodified, under-determined problem. The actual regularity of this extension is determined by that of the analytic solution and the choice of obejctive functional. Numerical experiments will demonstrate that it can be stably used to efficiently deal with non-constant coefficients, general geometries, and different boundary conditions in dimensions d=1,2,3 and that it produces solutions of tunable (and high) accuracy.