SEEM: A Kernel-Based Fictitious Domain Method

This thesis presents SEEM (Smooth Extension Embedding Method), a novel approach to thesolution of boundary value problems within the framework of the fictitious domain methodphilosophy. The salient feature of the novel method is that it reduces the whole boundaryvalue problem to a linear constraint for an appropriate optimization problem formulated ina larger, simpler set which contains the domain on which the boundary value problem isposed and which allows for the use of straightforward discretizations. It can also be viewedas a fully discrete meshfree method which uses a novel class of basis functions, thus buildinga bridge between fictitious domain and meshfree methods.SEEM in essence computes a (discrete) extension of the solution to the boundary valueproblem by selecting it as a smooth element of the complete affine family of solutions of theoriginal equations, which now yield an underdetermined problem for an unknown defined inthe whole fictitious domain. The actual regularity of this extension is determined by thatof the analytic solution and by the choice of objective functional. Numerical experimentsare presented which demonstrate that the method can be stably used to efficiently solveboundary value problems on general geometries, and that it produces solutions of tunable(and high) accuracy. Divergence-free and time-dependent problems are considered as well