UC Irvine

This thesis presents SEEM (Smooth Extension Embedding Method), a novel approach to the

solution of boundary value problems within the framework of the fictitious domain method

philosophy. The salient feature of the novel 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 which contains the domain on which the boundary value problem is

posed and which allows for the use of straightforward discretizations. It can also be viewed

as a fully discrete meshfree method which uses a novel class of basis functions, thus building

a bridge between fictitious domain and meshfree methods.

SEEM 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

original equations, which now yield an underdetermined problem for an unknown defined in

the whole fictitious domain. The actual regularity of this extension is determined by that

of the analytic solution and by the choice of objective functional. Numerical experiments

are presented which demonstrate that the method can be stably used to efficiently solve

boundary 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.