  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.
