A Novel Optimization Approach to Fictitious Domain Methods
For researchers in numerical PDEs, this method offers a stable and accurate alternative to existing fictitious domain methods, though it is an incremental improvement.
The paper proposes a new fictitious domain method (SSEM) that avoids data extension and reduces boundary value problems to a linear constraint in an optimization problem, achieving tunable high accuracy for non-constant coefficients, general geometries, and various boundary conditions in 1D, 2D, and 3D.
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.