A priori error estimates of a fictitious domain formulation of the Newtonian cooling problem

We present a theoretical analysis of a fictitious domain formulation of the Newtonian cooling problem, motivated by applications in topology optimization. The method reformulates the classical heat conduction model with Robin-type boundary conditions on a fixed computational domain using a so-called weak material approximation. In this setting, the conductivity equals one in the solid subdomain Ω ₛ Ω s and a small positive parameter ε ϵ in the surrounding fictitious region. We derive a priori error estimates that quantify the consistency error between the extended and original formulations and prove that the solution restricted to Ω ₛ Ω s converges to the true solution with an O(ε ) O ( ϵ ) error in the H¹(Ω ₛ) H 1 ( Ω s ) norm. We further provide ε ϵ -dependent finite element (FE) error estimates and show that for a mesh with characteristic size h , the condition number of the FE systems scales as O(ε ⁻¹h⁻²) O ( ϵ - 1 h - 2 ) . To address the ε ϵ -induced ill-conditioning, we introduce a simple yet effective diagonal preconditioning strategy that removes the ε ϵ -dependency from the condition number. Numerical experiments confirm the theoretical convergence rate and demonstrate the effectiveness of the method. Thus, this work provides a theoretical foundation for utilizing weak material approximations for boundary-effect-dominated problems, thereby extending existing analyses to cases with Robin-type boundary conditions.