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A new approach to the solution of boundary value problems within theso-called fictitious domain methods philosophy is proposed which avoids wellknown shortcomings of other fictitious domain methods, including the need togenerate extensions of the data. The salient feature of the novel method, whichwe refer to as SSEM (Smooth Selection Embedding Method), is that it reduces thewhole boundary value problem to a linear constraint for an appropriateoptimization problem formulated in a larger simpler set containing the domainon which the boundary value problem is posed and which allows for the use ofstraightforward discretizations. The proposed method in essence computes a(discrete) extension of the solution to the boundary value problem by selectingit as a smooth element of the complete affine family of solutions of theextended, yet unmodified, under-determined problem. The actual regularity ofthis extension is determined by that of the analytic solution and the choice ofobejctive functional. Numerical experiments will demonstrate that it can bestably used to efficiently deal with non-constant coefficients, generalgeometries, and different boundary conditions in dimensions d=1,2,3 and that itproduces solutions of tunable (and high) accuracy.

A Novel Optimization Approach to Fictitious Domain Methods